A Bazilevic-type function associated with the class of univalent functions with study Coefficient Bounds and Fekete-Szego inequalities

A Bazilevic-type function associated with the class... | F1000Research "use strict";function _typeof(t){return(_typeof="function"==typeof Symbol&&"symbol"==typeof Symbol.iterator?function(t){return typeof t}:function(t){return t&&"function"==typeof Symbol&&t.constructor===Symbol&&t!==Symbol.prototype?"symbol":typeof t})(t)}!function(){var t=function(){var t,e,o=[],n=window,r=n;for(;r;){try{if(r.frames.__tcfapiLocator){t=r;break}}catch(t){}if(r===n.top)break;r=r.parent}t||(!function t(){var e=n.document,o=!!n.frames.__tcfapiLocator;if(!o)if(e.body){var r=e.createElement("iframe");r.style.cssText="display:none",r.name="__tcfapiLocator",e.body.appendChild(r)}else setTimeout(t,5);return!o}(),n.__tcfapi=function(){for(var t=arguments.length,n=new Array(t),r=0;r 3&&2===parseInt(n[1],10)&&"boolean"==typeof n[3]&&(e=n[3],"function"==typeof n[2]&&n[2]("set",!0)):"ping"===n[0]?"function"==typeof n[2]&&n[2]({gdprApplies:e,cmpLoaded:!1,cmpStatus:"stub"}):o.push(n)},n.addEventListener("message",(function(t){var e="string"==typeof t.data,o={};if(e)try{o=JSON.parse(t.data)}catch(t){}else o=t.data;var n="object"===_typeof(o)&&null!==o?o.__tcfapiCall:null;n&&window.__tcfapi(n.command,n.version,(function(o,r){var a={__tcfapiReturn:{returnValue:o,success:r,callId:n.callId}};t&&t.source&&t.source.postMessage&&t.source.postMessage(e?JSON.stringify(a):a,"*")}),n.parameter)}),!1))};"undefined"!=typeof module?module.exports=t:t()}(); dataLayer = dataLayer || []; // Standard GTM initialization - Google Consent Mode handles consent automatically (function(w,d,s,l,i){w[l]=w[l]||[];w[l].push({'gtm.start': new Date().getTime(),event:'gtm.js'});var f=d.getElementsByTagName(s)[0], j=d.createElement(s),dl=l!='dataLayer'?'&l='+l:'';j.async=true;j.src= 'https://www.googletagmanager.com/gtm.js?id='+i+dl+ '>m_auth=hzk0Vc3qFsQYhCrIoHz68A>m_preview=env-1>m_cookies_win=x';f.parentNode.insertBefore(j,f); })(window,document,'script','dataLayer','GTM-MWFK8L5J'); ;window.NREUM||(NREUM={});NREUM.init={distributed_tracing:{enabled:true},privacy:{cookies_enabled:true},ajax:{deny_list:["bam.nr-data.net"]}}; ;NREUM.loader_config={accountID:"438030",trustKey:"438030",agentID:"772317073",licenseKey:"97f8f67f26",applicationID:"772317073"} ;NREUM.info={beacon:"bam.nr-data.net",errorBeacon:"bam.nr-data.net",licenseKey:"97f8f67f26",applicationID:"772317073",sa:1} ;/*! For license information please see nr-loader-spa-1.236.0.min.js.LICENSE.txt */ (()=>{"use strict";var e,t,r={5763:(e,t,r)=>{r.d(t,{P_:()=>l,Mt:()=>g,C5:()=>s,DL:()=>v,OP:()=>T,lF:()=>D,Yu:()=>y,Dg:()=>h,CX:()=>c,GE:()=>b,sU:()=>_});var n=r(8632),i=r(9567);const o={beacon:n.ce.beacon,errorBeacon:n.ce.errorBeacon,licenseKey:void 0,applicationID:void 0,sa:void 0,queueTime:void 0,applicationTime:void 0,ttGuid:void 0,user:void 0,account:void 0,product:void 0,extra:void 0,jsAttributes:{},userAttributes:void 0,atts:void 0,transactionName:void 0,tNamePlain:void 0},a={};function s(e){if(!e)throw new Error("All info objects require an agent identifier!");if(!a[e])throw new Error("Info for ".concat(e," was never set"));return a[e]}function c(e,t){if(!e)throw new Error("All info objects require an agent identifier!");a[e]=(0,i.D)(t,o),(0,n.Qy)(e,a[e],"info")}var u=r(7056);const d=()=>{const e={blockSelector:"[data-nr-block]",maskInputOptions:{password:!0}};return{allow_bfcache:!0,privacy:{cookies_enabled:!0},ajax:{deny_list:void 0,enabled:!0,harvestTimeSeconds:10},distributed_tracing:{enabled:void 0,exclude_newrelic_header:void 0,cors_use_newrelic_header:void 0,cors_use_tracecontext_headers:void 0,allowed_origins:void 0},session:{domain:void 0,expiresMs:u.oD,inactiveMs:u.Hb},ssl:void 0,obfuscate:void 0,jserrors:{enabled:!0,harvestTimeSeconds:10},metrics:{enabled:!0},page_action:{enabled:!0,harvestTimeSeconds:30},page_view_event:{enabled:!0},page_view_timing:{enabled:!0,harvestTimeSeconds:30,long_task:!1},session_trace:{enabled:!0,harvestTimeSeconds:10},harvest:{tooManyRequestsDelay:60},session_replay:{enabled:!1,harvestTimeSeconds:60,sampleRate:.1,errorSampleRate:.1,maskTextSelector:"*",maskAllInputs:!0,get blockClass(){return"nr-block"},get ignoreClass(){return"nr-ignore"},get maskTextClass(){return"nr-mask"},get blockSelector(){return e.blockSelector},set blockSelector(t){e.blockSelector+=",".concat(t)},get maskInputOptions(){return e.maskInputOptions},set maskInputOptions(t){e.maskInputOptions={...t,password:!0}}},spa:{enabled:!0,harvestTimeSeconds:10}}},f={};function l(e){if(!e)throw new Error("All configuration objects require an agent identifier!");if(!f[e])throw new Error("Configuration for ".concat(e," was never set"));return f[e]}function h(e,t){if(!e)throw new Error("All configuration objects require an agent identifier!");f[e]=(0,i.D)(t,d()),(0,n.Qy)(e,f[e],"config")}function g(e,t){if(!e)throw new Error("All configuration objects require an agent identifier!");var r=l(e);if(r){for(var n=t.split("."),i=0;i {r.d(t,{D:()=>i});var n=r(50);function i(e,t){try{if(!e||"object"!=typeof e)return(0,n.Z)("Setting a Configurable requires an object as input");if(!t||"object"!=typeof t)return(0,n.Z)("Setting a Configurable requires a model to set its initial properties");const r=Object.create(Object.getPrototypeOf(t),Object.getOwnPropertyDescriptors(t)),o=0===Object.keys(r).length?e:r;for(let a in o)if(void 0!==e[a])try{"object"==typeof e[a]&&"object"==typeof t[a]?r[a]=i(e[a],t[a]):r[a]=e[a]}catch(e){(0,n.Z)("An error occurred while setting a property of a Configurable",e)}return r}catch(e){(0,n.Z)("An error occured while setting a Configurable",e)}}},6818:(e,t,r)=>{r.d(t,{Re:()=>i,gF:()=>o,q4:()=>n});const n="1.236.0",i="PROD",o="CDN"},385:(e,t,r)=>{r.d(t,{FN:()=>a,IF:()=>u,Nk:()=>f,Tt:()=>s,_A:()=>o,il:()=>n,pL:()=>c,v6:()=>i,w1:()=>d});const n="undefined"!=typeof window&&!!window.document,i="undefined"!=typeof WorkerGlobalScope&&("undefined"!=typeof self&&self instanceof WorkerGlobalScope&&self.navigator instanceof WorkerNavigator||"undefined"!=typeof globalThis&&globalThis instanceof WorkerGlobalScope&&globalThis.navigator instanceof WorkerNavigator),o=n?window:"undefined"!=typeof WorkerGlobalScope&&("undefined"!=typeof self&&self instanceof WorkerGlobalScope&&self||"undefined"!=typeof globalThis&&globalThis instanceof WorkerGlobalScope&&globalThis),a=""+o?.location,s=/iPad|iPhone|iPod/.test(navigator.userAgent),c=s&&"undefined"==typeof SharedWorker,u=(()=>{const e=navigator.userAgent.match(/Firefox[/\s](\d+\.\d+)/);return Array.isArray(e)&&e.length>=2?+e[1]:0})(),d=Boolean(n&&window.document.documentMode),f=!!navigator.sendBeacon},1117:(e,t,r)=>{r.d(t,{w:()=>o});var n=r(50);const i={agentIdentifier:"",ee:void 0};class o{constructor(e){try{if("object"!=typeof e)return(0,n.Z)("shared context requires an object as input");this.sharedContext={},Object.assign(this.sharedContext,i),Object.entries(e).forEach((e=>{let[t,r]=e;Object.keys(i).includes(t)&&(this.sharedContext[t]=r)}))}catch(e){(0,n.Z)("An error occured while setting SharedContext",e)}}}},8e3:(e,t,r)=>{r.d(t,{L:()=>d,R:()=>c});var n=r(2177),i=r(1284),o=r(4322),a=r(3325);const s={};function c(e,t){const r={staged:!1,priority:a.p[t]||0};u(e),s[e].get(t)||s[e].set(t,r)}function u(e){e&&(s[e]||(s[e]=new Map))}function d(){let e=arguments.length>0&&void 0!==arguments[0]?arguments[0]:"",t=arguments.length>1&&void 0!==arguments[1]?arguments[1]:"feature";if(u(e),!e||!s[e].get(t))return a(t);s[e].get(t).staged=!0;const r=[...s[e]];function a(t){const r=e?n.ee.get(e):n.ee,a=o.X.handlers;if(r.backlog&&a){var s=r.backlog[t],c=a[t];if(c){for(var u=0;s&&u {let[t,r]=e;return r.staged}))&&(r.sort(((e,t)=>e[1].priority-t[1].priority)),r.forEach((e=>{let[t]=e;a(t)})))}function f(e,t){var r=e[1];(0,i.D)(t[r],(function(t,r){var n=e[0];if(r[0]===n){var i=r[1],o=e[3],a=e[2];i.apply(o,a)}}))}},2177:(e,t,r)=>{r.d(t,{c:()=>f,ee:()=>u});var n=r(8632),i=r(2210),o=r(1284),a=r(5763),s="nr@context";let c=(0,n.fP)();var u;function d(){}function f(e){return(0,i.X)(e,s,l)}function l(){return new d}function h(){u.aborted=!0,u.backlog={}}c.ee?u=c.ee:(u=function e(t,r){var n={},c={},f={},g=!1;try{g=16===r.length&&(0,a.OP)(r).isolatedBacklog}catch(e){}var p={on:b,addEventListener:b,removeEventListener:y,emit:v,get:x,listeners:w,context:m,buffer:A,abort:h,aborted:!1,isBuffering:E,debugId:r,backlog:g?{}:t&&"object"==typeof t.backlog?t.backlog:{}};return p;function m(e){return e&&e instanceof d?e:e?(0,i.X)(e,s,l):l()}function v(e,r,n,i,o){if(!1!==o&&(o=!0),!u.aborted||i){t&&o&&t.emit(e,r,n);for(var a=m(n),s=w(e),d=s.length,f=0;fn,p:()=>i});var n=r(2177).ee.get("handle");function i(e,t,r,i,o){o?(o.buffer([e],i),o.emit(e,t,r)):(n.buffer([e],i),n.emit(e,t,r))}},4322:(e,t,r)=>{r.d(t,{X:()=>o});var n=r(5546);o.on=a;var i=o.handlers={};function o(e,t,r,o){a(o||n.E,i,e,t,r)}function a(e,t,r,i,o){o||(o="feature"),e||(e=n.E);var a=t[o]=t[o]||{};(a[r]=a[r]||[]).push([e,i])}},3239:(e,t,r)=>{r.d(t,{bP:()=>s,iz:()=>c,m$:()=>a});var n=r(385);let i=!1,o=!1;try{const e={get passive(){return i=!0,!1},get signal(){return o=!0,!1}};n._A.addEventListener("test",null,e),n._A.removeEventListener("test",null,e)}catch(e){}function a(e,t){return i||o?{capture:!!e,passive:i,signal:t}:!!e}function s(e,t){let r=arguments.length>2&&void 0!==arguments[2]&&arguments[2],n=arguments.length>3?arguments[3]:void 0;window.addEventListener(e,t,a(r,n))}function c(e,t){let r=arguments.length>2&&void 0!==arguments[2]&&arguments[2],n=arguments.length>3?arguments[3]:void 0;document.addEventListener(e,t,a(r,n))}},4402:(e,t,r)=>{r.d(t,{Ht:()=>u,M:()=>c,Rl:()=>a,ky:()=>s});var n=r(385);const i="xxxxxxxx-xxxx-4xxx-yxxx-xxxxxxxxxxxx";function o(e,t){return e?15&e[t]:16*Math.random()|0}function a(){const e=n._A?.crypto||n._A?.msCrypto;let t,r=0;return e&&e.getRandomValues&&(t=e.getRandomValues(new Uint8Array(31))),i.split("").map((e=>"x"===e?o(t,++r).toString(16):"y"===e?(3&o()|8).toString(16):e)).join("")}function s(e){const t=n._A?.crypto||n._A?.msCrypto;let r,i=0;t&&t.getRandomValues&&(r=t.getRandomValues(new Uint8Array(31)));const a=[];for(var s=0;s {r.d(t,{Bq:()=>n,Hb:()=>o,oD:()=>i});const n="NRBA",i=144e5,o=18e5},7894:(e,t,r)=>{function n(){return Math.round(performance.now())}r.d(t,{z:()=>n})},7243:(e,t,r)=>{r.d(t,{e:()=>o});var n=r(385),i={};function o(e){if(e in i)return i[e];if(0===(e||"").indexOf("data:"))return{protocol:"data"};let t;var r=n._A?.location,o={};if(n.il)t=document.createElement("a"),t.href=e;else try{t=new URL(e,r.href)}catch(e){return o}o.port=t.port;var a=t.href.split("://");!o.port&&a[1]&&(o.port=a[1].split("/")[0].split("@").pop().split(":")[1]),o.port&&"0"!==o.port||(o.port="https"===a[0]?"443":"80"),o.hostname=t.hostname||r.hostname,o.pathname=t.pathname,o.protocol=a[0],"/"!==o.pathname.charAt(0)&&(o.pathname="/"+o.pathname);var s=!t.protocol||":"===t.protocol||t.protocol===r.protocol,c=t.hostname===r.hostname&&t.port===r.port;return o.sameOrigin=s&&(!t.hostname||c),"/"===o.pathname&&(i[e]=o),o}},50:(e,t,r)=>{function n(e,t){"function"==typeof console.warn&&(console.warn("New Relic: ".concat(e)),t&&console.warn(t))}r.d(t,{Z:()=>n})},2587:(e,t,r)=>{r.d(t,{N:()=>c,T:()=>u});var n=r(2177),i=r(5546),o=r(8e3),a=r(3325);const s={stn:[a.D.sessionTrace],err:[a.D.jserrors,a.D.metrics],ins:[a.D.pageAction],spa:[a.D.spa],sr:[a.D.sessionReplay,a.D.sessionTrace]};function c(e,t){const r=n.ee.get(t);e&&"object"==typeof e&&(Object.entries(e).forEach((e=>{let[t,n]=e;void 0===u[t]&&(s[t]?s[t].forEach((e=>{n?(0,i.p)("feat-"+t,[],void 0,e,r):(0,i.p)("block-"+t,[],void 0,e,r),(0,i.p)("rumresp-"+t,[Boolean(n)],void 0,e,r)})):n&&(0,i.p)("feat-"+t,[],void 0,void 0,r),u[t]=Boolean(n))})),Object.keys(s).forEach((e=>{void 0===u[e]&&(s[e]?.forEach((t=>(0,i.p)("rumresp-"+e,[!1],void 0,t,r))),u[e]=!1)})),(0,o.L)(t,a.D.pageViewEvent))}const u={}},2210:(e,t,r)=>{r.d(t,{X:()=>i});var n=Object.prototype.hasOwnProperty;function i(e,t,r){if(n.call(e,t))return e[t];var i=r();if(Object.defineProperty&&Object.keys)try{return Object.defineProperty(e,t,{value:i,writable:!0,enumerable:!1}),i}catch(e){}return e[t]=i,i}},1284:(e,t,r)=>{r.d(t,{D:()=>n});const n=(e,t)=>Object.entries(e||{}).map((e=>{let[r,n]=e;return t(r,n)}))},4351:(e,t,r)=>{r.d(t,{P:()=>o});var n=r(2177);const i=()=>{const e=new WeakSet;return(t,r)=>{if("object"==typeof r&&null!==r){if(e.has(r))return;e.add(r)}return r}};function o(e){try{return JSON.stringify(e,i())}catch(e){try{n.ee.emit("internal-error",[e])}catch(e){}}}},3960:(e,t,r)=>{r.d(t,{K:()=>a,b:()=>o});var n=r(3239);function i(){return"undefined"==typeof document||"complete"===document.readyState}function o(e,t){if(i())return e();(0,n.bP)("load",e,t)}function a(e){if(i())return e();(0,n.iz)("DOMContentLoaded",e)}},8632:(e,t,r)=>{r.d(t,{EZ:()=>u,Qy:()=>c,ce:()=>o,fP:()=>a,gG:()=>d,mF:()=>s});var n=r(7894),i=r(385);const o={beacon:"bam.nr-data.net",errorBeacon:"bam.nr-data.net"};function a(){return i._A.NREUM||(i._A.NREUM={}),void 0===i._A.newrelic&&(i._A.newrelic=i._A.NREUM),i._A.NREUM}function s(){let e=a();return e.o||(e.o={ST:i._A.setTimeout,SI:i._A.setImmediate,CT:i._A.clearTimeout,XHR:i._A.XMLHttpRequest,REQ:i._A.Request,EV:i._A.Event,PR:i._A.Promise,MO:i._A.MutationObserver,FETCH:i._A.fetch}),e}function c(e,t,r){let i=a();const o=i.initializedAgents||{},s=o[e]||{};return Object.keys(s).length||(s.initializedAt={ms:(0,n.z)(),date:new Date}),i.initializedAgents={...o,[e]:{...s,[r]:t}},i}function u(e,t){a()[e]=t}function d(){return function(){let e=a();const t=e.info||{};e.info={beacon:o.beacon,errorBeacon:o.errorBeacon,...t}}(),function(){let e=a();const t=e.init||{};e.init={...t}}(),s(),function(){let e=a();const t=e.loader_config||{};e.loader_config={...t}}(),a()}},7956:(e,t,r)=>{r.d(t,{N:()=>i});var n=r(3239);function i(e){let t=arguments.length>1&&void 0!==arguments[1]&&arguments[1],r=arguments.length>2?arguments[2]:void 0,i=arguments.length>3?arguments[3]:void 0;return void(0,n.iz)("visibilitychange",(function(){if(t)return void("hidden"==document.visibilityState&&e());e(document.visibilityState)}),r,i)}},1214:(e,t,r)=>{r.d(t,{em:()=>v,u5:()=>N,QU:()=>S,_L:()=>I,Gm:()=>L,Lg:()=>M,gy:()=>U,BV:()=>Q,Kf:()=>ee});var n=r(2177);const i="nr@original";var o=Object.prototype.hasOwnProperty,a=!1;function s(e,t){return e||(e=n.ee),r.inPlace=function(e,t,n,i,o){n||(n="");var a,s,c,u="-"===n.charAt(0);for(c=0;c 2?n-2:0),o=2;o {r(A[T],e,w),r(E[T],e,w)})),r(l._A,"fetch",y),t.on(y+"end",(function(e,r){var n=this;if(r){var i=r.headers.get("content-length");null!==i&&(n.rxSize=i),t.emit(y+"done",[null,r],n)}else t.emit(y+"done",[e],n)})),t}const O={},j=["pushState","replaceState"];function S(e){const t=function(e){return(e||n.ee).get("history")}(e);return!l.il||O[t.debugId]++||(O[t.debugId]=1,s(t).inPlace(window.history,j,"-")),t}var P=r(3239);const C={},R=["appendChild","insertBefore","replaceChild"];function I(e){const t=function(e){return(e||n.ee).get("jsonp")}(e);if(!l.il||C[t.debugId])return t;C[t.debugId]=!0;var r=s(t),i=/[?&](?:callback|cb)=([^&#]+)/,o=/(.*)\.([^.]+)/,a=/^(\w+)(\.|$)(.*)$/;function c(e,t){var r=e.match(a),n=r[1],i=r[3];return i?c(i,t[n]):t[n]}return r.inPlace(Node.prototype,R,"dom-"),t.on("dom-start",(function(e){!function(e){if(!e||"string"!=typeof e.nodeName||"script"!==e.nodeName.toLowerCase())return;if("function"!=typeof e.addEventListener)return;var n=(a=e.src,s=a.match(i),s?s[1]:null);var a,s;if(!n)return;var u=function(e){var t=e.match(o);if(t&&t.length>=3)return{key:t[2],parent:c(t[1],window)};return{key:e,parent:window}}(n);if("function"!=typeof u.parent[u.key])return;var d={};function f(){t.emit("jsonp-end",[],d),e.removeEventListener("load",f,(0,P.m$)(!1)),e.removeEventListener("error",l,(0,P.m$)(!1))}function l(){t.emit("jsonp-error",[],d),t.emit("jsonp-end",[],d),e.removeEventListener("load",f,(0,P.m$)(!1)),e.removeEventListener("error",l,(0,P.m$)(!1))}r.inPlace(u.parent,[u.key],"cb-",d),e.addEventListener("load",f,(0,P.m$)(!1)),e.addEventListener("error",l,(0,P.m$)(!1)),t.emit("new-jsonp",[e.src],d)}(e[0])})),t}var k=r(5763);const H={};function L(e){const t=function(e){return(e||n.ee).get("mutation")}(e);if(!l.il||H[t.debugId])return t;H[t.debugId]=!0;var r=s(t),i=k.Yu.MO;return i&&(window.MutationObserver=function(e){return this instanceof i?new i(r(e,"fn-")):i.apply(this,arguments)},MutationObserver.prototype=i.prototype),t}const z={};function M(e){const t=function(e){return(e||n.ee).get("promise")}(e);if(z[t.debugId])return t;z[t.debugId]=!0;var r=n.c,o=s(t),a=k.Yu.PR;return a&&function(){function e(r){var n=t.context(),i=o(r,"executor-",n,null,!1);const s=Reflect.construct(a,[i],e);return t.context(s).getCtx=function(){return n},s}l._A.Promise=e,Object.defineProperty(e,"name",{value:"Promise"}),e.toString=function(){return a.toString()},Object.setPrototypeOf(e,a),["all","race"].forEach((function(r){const n=a[r];e[r]=function(e){let i=!1;[...e||[]].forEach((e=>{this.resolve(e).then(a("all"===r),a(!1))}));const o=n.apply(this,arguments);return o;function a(e){return function(){t.emit("propagate",[null,!i],o,!1,!1),i=i||!e}}}})),["resolve","reject"].forEach((function(r){const n=a[r];e[r]=function(e){const r=n.apply(this,arguments);return e!==r&&t.emit("propagate",[e,!0],r,!1,!1),r}})),e.prototype=a.prototype;const n=a.prototype.then;a.prototype.then=function(){var e=this,i=r(e);i.promise=e;for(var a=arguments.length,s=new Array(a),c=0;c e())),t};function m(e,t){i.inPlace(t,["onreadystatechange"],"fn-",E)}function b(){var e=this,t=r.context(e);e.readyState>3&&!t.resolved&&(t.resolved=!0,r.emit("xhr-resolved",[],e)),i.inPlace(e,f,"fn-",E)}if(function(e,t){for(var r in e)t[r]=e[r]}(o,p),p.prototype=o.prototype,i.inPlace(p.prototype,J,"-xhr-",E),r.on("send-xhr-start",(function(e,t){m(e,t),function(e){h.push(e),a&&(y?y.then(A):u?u(A):(w=-w,x.data=w))}(t)})),r.on("open-xhr-start",m),a){var y=c&&c.resolve();if(!u&&!c){var w=1,x=document.createTextNode(w);new a(A).observe(x,{characterData:!0})}}else t.on("fn-end",(function(e){e[0]&&e[0].type===d||A()}));function A(){for(var e=0;e {r.d(t,{t:()=>n});const n=r(3325).D.ajax},6660:(e,t,r)=>{r.d(t,{A:()=>i,t:()=>n});const n=r(3325).D.jserrors,i="nr@seenError"},3081:(e,t,r)=>{r.d(t,{gF:()=>o,mY:()=>i,t9:()=>n,vz:()=>s,xS:()=>a});const n=r(3325).D.metrics,i="sm",o="cm",a="storeSupportabilityMetrics",s="storeEventMetrics"},4649:(e,t,r)=>{r.d(t,{t:()=>n});const n=r(3325).D.pageAction},7633:(e,t,r)=>{r.d(t,{Dz:()=>i,OJ:()=>a,qw:()=>o,t9:()=>n});const n=r(3325).D.pageViewEvent,i="firstbyte",o="domcontent",a="windowload"},9251:(e,t,r)=>{r.d(t,{t:()=>n});const n=r(3325).D.pageViewTiming},3614:(e,t,r)=>{r.d(t,{BST_RESOURCE:()=>i,END:()=>s,FEATURE_NAME:()=>n,FN_END:()=>u,FN_START:()=>c,PUSH_STATE:()=>d,RESOURCE:()=>o,START:()=>a});const n=r(3325).D.sessionTrace,i="bstResource",o="resource",a="-start",s="-end",c="fn"+a,u="fn"+s,d="pushState"},7836:(e,t,r)=>{r.d(t,{BODY:()=>A,CB_END:()=>E,CB_START:()=>u,END:()=>x,FEATURE_NAME:()=>i,FETCH:()=>_,FETCH_BODY:()=>v,FETCH_DONE:()=>m,FETCH_START:()=>p,FN_END:()=>c,FN_START:()=>s,INTERACTION:()=>l,INTERACTION_API:()=>d,INTERACTION_EVENTS:()=>o,JSONP_END:()=>b,JSONP_NODE:()=>g,JS_TIME:()=>T,MAX_TIMER_BUDGET:()=>a,REMAINING:()=>f,SPA_NODE:()=>h,START:()=>w,originalSetTimeout:()=>y});var n=r(5763);const i=r(3325).D.spa,o=["click","submit","keypress","keydown","keyup","change"],a=999,s="fn-start",c="fn-end",u="cb-start",d="api-ixn-",f="remaining",l="interaction",h="spaNode",g="jsonpNode",p="fetch-start",m="fetch-done",v="fetch-body-",b="jsonp-end",y=n.Yu.ST,w="-start",x="-end",A="-body",E="cb"+x,T="jsTime",_="fetch"},5938:(e,t,r)=>{r.d(t,{W:()=>o});var n=r(5763),i=r(2177);class o{constructor(e,t,r){this.agentIdentifier=e,this.aggregator=t,this.ee=i.ee.get(e,(0,n.OP)(this.agentIdentifier).isolatedBacklog),this.featureName=r,this.blocked=!1}}},9144:(e,t,r)=>{r.d(t,{j:()=>m});var n=r(3325),i=r(5763),o=r(5546),a=r(2177),s=r(7894),c=r(8e3),u=r(3960),d=r(385),f=r(50),l=r(3081),h=r(8632);function g(){const e=(0,h.gG)();["setErrorHandler","finished","addToTrace","inlineHit","addRelease","addPageAction","setCurrentRouteName","setPageViewName","setCustomAttribute","interaction","noticeError","setUserId"].forEach((t=>{e[t]=function(){for(var r=arguments.length,n=new Array(r),i=0;i 1?r-1:0),i=1;i {e.exposed&&e.api[t]&&o.push(e.api[t](...n))})),o.length>1?o:o[0]}(t,...n)}}))}var p=r(2587);function m(e){let t=arguments.length>1&&void 0!==arguments[1]?arguments[1]:{},m=arguments.length>2?arguments[2]:void 0,v=arguments.length>3?arguments[3]:void 0,{init:b,info:y,loader_config:w,runtime:x={loaderType:m},exposed:A=!0}=t;const E=(0,h.gG)();y||(b=E.init,y=E.info,w=E.loader_config),(0,i.Dg)(e,b||{}),(0,i.GE)(e,w||{}),(0,i.sU)(e,x),y.jsAttributes??={},d.v6&&(y.jsAttributes.isWorker=!0),(0,i.CX)(e,y),g();const T=function(e,t){t||(0,c.R)(e,"api");const h={};var g=a.ee.get(e),p=g.get("tracer"),m="api-",v=m+"ixn-";function b(t,r,n,o){const a=(0,i.C5)(e);return null===r?delete a.jsAttributes[t]:(0,i.CX)(e,{...a,jsAttributes:{...a.jsAttributes,[t]:r}}),x(m,n,!0,o||null===r?"session":void 0)(t,r)}function y(){}["setErrorHandler","finished","addToTrace","inlineHit","addRelease"].forEach((e=>h[e]=x(m,e,!0,"api"))),h.addPageAction=x(m,"addPageAction",!0,n.D.pageAction),h.setCurrentRouteName=x(m,"routeName",!0,n.D.spa),h.setPageViewName=function(t,r){if("string"==typeof t)return"/"!==t.charAt(0)&&(t="/"+t),(0,i.OP)(e).customTransaction=(r||"http://custom.transaction")+t,x(m,"setPageViewName",!0)()},h.setCustomAttribute=function(e,t){let r=arguments.length>2&&void 0!==arguments[2]&&arguments[2];if("string"==typeof e){if(["string","number"].includes(typeof t)||null===t)return b(e,t,"setCustomAttribute",r);(0,f.Z)("Failed to execute setCustomAttribute.\nNon-null value must be a string or number type, but a type of was provided."))}else(0,f.Z)("Failed to execute setCustomAttribute.\nName must be a string type, but a type of was provided."))},h.setUserId=function(e){if("string"==typeof e||null===e)return b("enduser.id",e,"setUserId",!0);(0,f.Z)("Failed to execute setUserId.\nNon-null value must be a string type, but a type of was provided."))},h.interaction=function(){return(new y).get()};var w=y.prototype={createTracer:function(e,t){var r={},i=this,a="function"==typeof t;return(0,o.p)(v+"tracer",[(0,s.z)(),e,r],i,n.D.spa,g),function(){if(p.emit((a?"":"no-")+"fn-start",[(0,s.z)(),i,a],r),a)try{return t.apply(this,arguments)}catch(e){throw p.emit("fn-err",[arguments,this,"string"==typeof e?new Error(e):e],r),e}finally{p.emit("fn-end",[(0,s.z)()],r)}}}};function x(e,t,r,i){return function(){return(0,o.p)(l.xS,["API/"+t+"/called"],void 0,n.D.metrics,g),i&&(0,o.p)(e+t,[(0,s.z)(),...arguments],r?null:this,i,g),r?void 0:this}}function A(){r.e(439).then(r.bind(r,7438)).then((t=>{let{setAPI:r}=t;r(e),(0,c.L)(e,"api")})).catch((()=>(0,f.Z)("Downloading runtime APIs failed...")))}return["actionText","setName","setAttribute","save","ignore","onEnd","getContext","end","get"].forEach((e=>{w[e]=x(v,e,void 0,n.D.spa)})),h.noticeError=function(e,t){"string"==typeof e&&(e=new Error(e)),(0,o.p)(l.xS,["API/noticeError/called"],void 0,n.D.metrics,g),(0,o.p)("err",[e,(0,s.z)(),!1,t],void 0,n.D.jserrors,g)},d.il?(0,u.b)((()=>A()),!0):A(),h}(e,v);return(0,h.Qy)(e,T,"api"),(0,h.Qy)(e,A,"exposed"),(0,h.EZ)("activatedFeatures",p.T),T}},3325:(e,t,r)=>{r.d(t,{D:()=>n,p:()=>i});const n={ajax:"ajax",jserrors:"jserrors",metrics:"metrics",pageAction:"page_action",pageViewEvent:"page_view_event",pageViewTiming:"page_view_timing",sessionReplay:"session_replay",sessionTrace:"session_trace",spa:"spa"},i={[n.pageViewEvent]:1,[n.pageViewTiming]:2,[n.metrics]:3,[n.jserrors]:4,[n.ajax]:5,[n.sessionTrace]:6,[n.pageAction]:7,[n.spa]:8,[n.sessionReplay]:9}}},n={};function i(e){var t=n[e];if(void 0!==t)return t.exports;var o=n[e]={exports:{}};return r[e](o,o.exports,i),o.exports}i.m=r,i.d=(e,t)=>{for(var r in t)i.o(t,r)&&!i.o(e,r)&&Object.defineProperty(e,r,{enumerable:!0,get:t[r]})},i.f={},i.e=e=>Promise.all(Object.keys(i.f).reduce(((t,r)=>(i.f[r](e,t),t)),[])),i.u=e=>(({78:"page_action-aggregate",147:"metrics-aggregate",242:"session-manager",317:"jserrors-aggregate",348:"page_view_timing-aggregate",412:"lazy-feature-loader",439:"async-api",538:"recorder",590:"session_replay-aggregate",675:"compressor",733:"session_trace-aggregate",786:"page_view_event-aggregate",873:"spa-aggregate",898:"ajax-aggregate"}[e]||e)+"."+{78:"ac76d497",147:"3dc53903",148:"1a20d5fe",242:"2a64278a",317:"49e41428",348:"bd6de33a",412:"2f55ce66",439:"30bd804e",538:"1b18459f",590:"cf0efb30",675:"ae9f91a8",733:"83105561",786:"06482edd",860:"03a8b7a5",873:"e6b09d52",898:"998ef92b"}[e]+"-1.236.0.min.js"),i.o=(e,t)=>Object.prototype.hasOwnProperty.call(e,t),e={},t="NRBA:",i.l=(r,n,o,a)=>{if(e[r])e[r].push(n);else{var s,c;if(void 0!==o)for(var u=document.getElementsByTagName("script"),d=0;d {s.onerror=s.onload=null,clearTimeout(h);var i=e[r];if(delete e[r],s.parentNode&&s.parentNode.removeChild(s),i&&i.forEach((e=>e(n))),t)return t(n)},h=setTimeout(l.bind(null,void 0,{type:"timeout",target:s}),12e4);s.onerror=l.bind(null,s.onerror),s.onload=l.bind(null,s.onload),c&&document.head.appendChild(s)}},i.r=e=>{"undefined"!=typeof Symbol&&Symbol.toStringTag&&Object.defineProperty(e,Symbol.toStringTag,{value:"Module"}),Object.defineProperty(e,"__esModule",{value:!0})},i.j=364,i.p="https://js-agent.newrelic.com/",(()=>{var e={364:0,953:0};i.f.j=(t,r)=>{var n=i.o(e,t)?e[t]:void 0;if(0!==n)if(n)r.push(n[2]);else{var o=new Promise(((r,i)=>n=e[t]=[r,i]));r.push(n[2]=o);var a=i.p+i.u(t),s=new Error;i.l(a,(r=>{if(i.o(e,t)&&(0!==(n=e[t])&&(e[t]=void 0),n)){var o=r&&("load"===r.type?"missing":r.type),a=r&&r.target&&r.target.src;s.message="Loading chunk "+t+" failed.\n("+o+": "+a+")",s.name="ChunkLoadError",s.type=o,s.request=a,n[1](s)}}),"chunk-"+t,t)}};var t=(t,r)=>{var n,o,[a,s,c]=r,u=0;if(a.some((t=>0!==e[t]))){for(n in s)i.o(s,n)&&(i.m[n]=s[n]);if(c)c(i)}for(t&&t(r);u {i.r(o);var e=i(3325),t=i(5763);const r=Object.values(e.D);function n(e){const n={};return r.forEach((r=>{n[r]=function(e,r){return!1!==(0,t.Mt)(r,"".concat(e,".enabled"))}(r,e)})),n}var a=i(9144);var s=i(5546),c=i(385),u=i(8e3),d=i(5938),f=i(3960),l=i(50);class h extends d.W{constructor(e,t,r){let n=!(arguments.length>3&&void 0!==arguments[3])||arguments[3];super(e,t,r),this.auto=n,this.abortHandler,this.featAggregate,this.onAggregateImported,n&&(0,u.R)(e,r)}importAggregator(){let e=arguments.length>0&&void 0!==arguments[0]?arguments[0]:{};if(this.featAggregate||!this.auto)return;const r=c.il&&!0===(0,t.Mt)(this.agentIdentifier,"privacy.cookies_enabled");let n;this.onAggregateImported=new Promise((e=>{n=e}));const o=async()=>{let t;try{if(r){const{setupAgentSession:e}=await Promise.all([i.e(860),i.e(242)]).then(i.bind(i,3228));t=e(this.agentIdentifier)}}catch(e){(0,l.Z)("A problem occurred when starting up session manager. This page will not start or extend any session.",e)}try{if(!this.shouldImportAgg(this.featureName,t))return void(0,u.L)(this.agentIdentifier,this.featureName);const{lazyFeatureLoader:r}=await i.e(412).then(i.bind(i,8582)),{Aggregate:o}=await r(this.featureName,"aggregate");this.featAggregate=new o(this.agentIdentifier,this.aggregator,e),n(!0)}catch(e){(0,l.Z)("Downloading and initializing ".concat(this.featureName," failed..."),e),this.abortHandler?.(),n(!1)}};c.il?(0,f.b)((()=>o()),!0):o()}shouldImportAgg(r,n){return r!==e.D.sessionReplay||!1!==(0,t.Mt)(this.agentIdentifier,"session_trace.enabled")&&(!!n?.isNew||!!n?.state.sessionReplay)}}var g=i(7633),p=i(7894);class m extends h{static featureName=g.t9;constructor(r,n){let i=!(arguments.length>2&&void 0!==arguments[2])||arguments[2];if(super(r,n,g.t9,i),("undefined"==typeof PerformanceNavigationTiming||c.Tt)&&"undefined"!=typeof PerformanceTiming){const n=(0,t.OP)(r);n[g.Dz]=Math.max(Date.now()-n.offset,0),(0,f.K)((()=>n[g.qw]=Math.max((0,p.z)()-n[g.Dz],0))),(0,f.b)((()=>{const t=(0,p.z)();n[g.OJ]=Math.max(t-n[g.Dz],0),(0,s.p)("timing",["load",t],void 0,e.D.pageViewTiming,this.ee)}))}this.importAggregator()}}var v=i(1117),b=i(1284);class y extends v.w{constructor(e){super(e),this.aggregatedData={}}store(e,t,r,n,i){var o=this.getBucket(e,t,r,i);return o.metrics=function(e,t){t||(t={count:0});return t.count+=1,(0,b.D)(e,(function(e,r){t[e]=w(r,t[e])})),t}(n,o.metrics),o}merge(e,t,r,n,i){var o=this.getBucket(e,t,n,i);if(o.metrics){var a=o.metrics;a.count+=r.count,(0,b.D)(r,(function(e,t){if("count"!==e){var n=a[e],i=r[e];i&&!i.c?a[e]=w(i.t,n):a[e]=function(e,t){if(!t)return e;t.c||(t=x(t.t));return t.min=Math.min(e.min,t.min),t.max=Math.max(e.max,t.max),t.t+=e.t,t.sos+=e.sos,t.c+=e.c,t}(i,a[e])}}))}else o.metrics=r}storeMetric(e,t,r,n){var i=this.getBucket(e,t,r);return i.stats=w(n,i.stats),i}getBucket(e,t,r,n){this.aggregatedData[e]||(this.aggregatedData[e]={});var i=this.aggregatedData[e][t];return i||(i=this.aggregatedData[e][t]={params:r||{}},n&&(i.custom=n)),i}get(e,t){return t?this.aggregatedData[e]&&this.aggregatedData[e][t]:this.aggregatedData[e]}take(e){for(var t={},r="",n=!1,i=0;i t.max&&(t.max=e),e 2&&void 0!==arguments[2])||arguments[2];super(e,r,j.t,n),c.il&&((0,t.OP)(e).initHidden=Boolean("hidden"===document.visibilityState),(0,N.N)((()=>(0,s.p)("docHidden",[(0,p.z)()],void 0,j.t,this.ee)),!0),(0,O.bP)("pagehide",(()=>(0,s.p)("winPagehide",[(0,p.z)()],void 0,j.t,this.ee))),this.importAggregator())}}var P=i(3081);class C extends h{static featureName=P.t9;constructor(e,t){let r=!(arguments.length>2&&void 0!==arguments[2])||arguments[2];super(e,t,P.t9,r),this.importAggregator()}}var R,I=i(2210),k=i(1214),H=i(2177),L={};try{R=localStorage.getItem("__nr_flags").split(","),console&&"function"==typeof console.log&&(L.console=!0,-1!==R.indexOf("dev")&&(L.dev=!0),-1!==R.indexOf("nr_dev")&&(L.nrDev=!0))}catch(e){}function z(e){try{L.console&&z(e)}catch(e){}}L.nrDev&&H.ee.on("internal-error",(function(e){z(e.stack)})),L.dev&&H.ee.on("fn-err",(function(e,t,r){z(r.stack)})),L.dev&&(z("NR AGENT IN DEVELOPMENT MODE"),z("flags: "+(0,b.D)(L,(function(e,t){return e})).join(", ")));var M=i(6660);class B extends h{static featureName=M.t;constructor(r,n){let i=!(arguments.length>2&&void 0!==arguments[2])||arguments[2];super(r,n,M.t,i),this.skipNext=0;try{this.removeOnAbort=new AbortController}catch(e){}const o=this;o.ee.on("fn-start",(function(e,t,r){o.abortHandler&&(o.skipNext+=1)})),o.ee.on("fn-err",(function(t,r,n){o.abortHandler&&!n[M.A]&&((0,I.X)(n,M.A,(function(){return!0})),this.thrown=!0,(0,s.p)("err",[n,(0,p.z)()],void 0,e.D.jserrors,o.ee))})),o.ee.on("fn-end",(function(){o.abortHandler&&!this.thrown&&o.skipNext>0&&(o.skipNext-=1)})),o.ee.on("internal-error",(function(t){(0,s.p)("ierr",[t,(0,p.z)(),!0],void 0,e.D.jserrors,o.ee)})),this.origOnerror=c._A.onerror,c._A.onerror=this.onerrorHandler.bind(this),c._A.addEventListener("unhandledrejection",(t=>{const r=function(e){let t="Unhandled Promise Rejection: ";if(e instanceof Error)try{return e.message=t+e.message,e}catch(t){return e}if(void 0===e)return new Error(t);try{return new Error(t+(0,D.P)(e))}catch(e){return new Error(t)}}(t.reason);(0,s.p)("err",[r,(0,p.z)(),!1,{unhandledPromiseRejection:1}],void 0,e.D.jserrors,this.ee)}),(0,O.m$)(!1,this.removeOnAbort?.signal)),(0,k.gy)(this.ee),(0,k.BV)(this.ee),(0,k.em)(this.ee),(0,t.OP)(r).xhrWrappable&&(0,k.Kf)(this.ee),this.abortHandler=this.#e,this.importAggregator()}#e(){this.removeOnAbort?.abort(),this.abortHandler=void 0}onerrorHandler(t,r,n,i,o){"function"==typeof this.origOnerror&&this.origOnerror(...arguments);try{this.skipNext?this.skipNext-=1:(0,s.p)("err",[o||new F(t,r,n),(0,p.z)()],void 0,e.D.jserrors,this.ee)}catch(t){try{(0,s.p)("ierr",[t,(0,p.z)(),!0],void 0,e.D.jserrors,this.ee)}catch(e){}}return!1}}function F(e,t,r){this.message=e||"Uncaught error with no additional information",this.sourceURL=t,this.line=r}let U=1;const q="nr@id";function G(e){const t=typeof e;return!e||"object"!==t&&"function"!==t?-1:e===c._A?0:(0,I.X)(e,q,(function(){return U++}))}function V(e){if("string"==typeof e&&e.length)return e.length;if("object"==typeof e){if("undefined"!=typeof ArrayBuffer&&e instanceof ArrayBuffer&&e.byteLength)return e.byteLength;if("undefined"!=typeof Blob&&e instanceof Blob&&e.size)return e.size;if(!("undefined"!=typeof FormData&&e instanceof FormData))try{return(0,D.P)(e).length}catch(e){return}}}var X=i(7243);class W{constructor(e){this.agentIdentifier=e,this.generateTracePayload=this.generateTracePayload.bind(this),this.shouldGenerateTrace=this.shouldGenerateTrace.bind(this)}generateTracePayload(e){if(!this.shouldGenerateTrace(e))return null;var r=(0,t.DL)(this.agentIdentifier);if(!r)return null;var n=(r.accountID||"").toString()||null,i=(r.agentID||"").toString()||null,o=(r.trustKey||"").toString()||null;if(!n||!i)return null;var a=(0,_.M)(),s=(0,_.Ht)(),c=Date.now(),u={spanId:a,traceId:s,timestamp:c};return(e.sameOrigin||this.isAllowedOrigin(e)&&this.useTraceContextHeadersForCors())&&(u.traceContextParentHeader=this.generateTraceContextParentHeader(a,s),u.traceContextStateHeader=this.generateTraceContextStateHeader(a,c,n,i,o)),(e.sameOrigin&&!this.excludeNewrelicHeader()||!e.sameOrigin&&this.isAllowedOrigin(e)&&this.useNewrelicHeaderForCors())&&(u.newrelicHeader=this.generateTraceHeader(a,s,c,n,i,o)),u}generateTraceContextParentHeader(e,t){return"00-"+t+"-"+e+"-01"}generateTraceContextStateHeader(e,t,r,n,i){return i+"@nr=0-1-"+r+"-"+n+"-"+e+"----"+t}generateTraceHeader(e,t,r,n,i,o){if(!("function"==typeof c._A?.btoa))return null;var a={v:[0,1],d:{ty:"Browser",ac:n,ap:i,id:e,tr:t,ti:r}};return o&&n!==o&&(a.d.tk=o),btoa((0,D.P)(a))}shouldGenerateTrace(e){return this.isDtEnabled()&&this.isAllowedOrigin(e)}isAllowedOrigin(e){var r=!1,n={};if((0,t.Mt)(this.agentIdentifier,"distributed_tracing")&&(n=(0,t.P_)(this.agentIdentifier).distributed_tracing),e.sameOrigin)r=!0;else if(n.allowed_origins instanceof Array)for(var i=0;i 2&&void 0!==arguments[2])||arguments[2];super(r,n,Z.t,i),(0,t.OP)(r).xhrWrappable&&(this.dt=new W(r),this.handler=(e,t,r,n)=>(0,s.p)(e,t,r,n,this.ee),(0,k.u5)(this.ee),(0,k.Kf)(this.ee),function(r,n,i,o){function a(e){var t=this;t.totalCbs=0,t.called=0,t.cbTime=0,t.end=E,t.ended=!1,t.xhrGuids={},t.lastSize=null,t.loadCaptureCalled=!1,t.params=this.params||{},t.metrics=this.metrics||{},e.addEventListener("load",(function(r){_(t,e)}),(0,O.m$)(!1)),c.IF||e.addEventListener("progress",(function(e){t.lastSize=e.loaded}),(0,O.m$)(!1))}function s(e){this.params={method:e[0]},T(this,e[1]),this.metrics={}}function u(e,n){var i=(0,t.DL)(r);i.xpid&&this.sameOrigin&&n.setRequestHeader("X-NewRelic-ID",i.xpid);var a=o.generateTracePayload(this.parsedOrigin);if(a){var s=!1;a.newrelicHeader&&(n.setRequestHeader("newrelic",a.newrelicHeader),s=!0),a.traceContextParentHeader&&(n.setRequestHeader("traceparent",a.traceContextParentHeader),a.traceContextStateHeader&&n.setRequestHeader("tracestate",a.traceContextStateHeader),s=!0),s&&(this.dt=a)}}function d(e,t){var r=this.metrics,i=e[0],o=this;if(r&&i){var a=V(i);a&&(r.txSize=a)}this.startTime=(0,p.z)(),this.listener=function(e){try{"abort"!==e.type||o.loadCaptureCalled||(o.params.aborted=!0),("load"!==e.type||o.called===o.totalCbs&&(o.onloadCalled||"function"!=typeof t.onload)&&"function"==typeof o.end)&&o.end(t)}catch(e){try{n.emit("internal-error",[e])}catch(e){}}};for(var s=0;s 1?e[1]=i:e.push(i)}else e[0]&&e[0].headers&&s(e[0].headers,n)&&(this.dt=n);function s(e,t){var r=!1;return t.newrelicHeader&&(e.set("newrelic",t.newrelicHeader),r=!0),t.traceContextParentHeader&&(e.set("traceparent",t.traceContextParentHeader),t.traceContextStateHeader&&e.set("tracestate",t.traceContextStateHeader),r=!0),r}}function x(e,t){this.params={},this.metrics={},this.startTime=(0,p.z)(),this.dt=t,e.length>=1&&(this.target=e[0]),e.length>=2&&(this.opts=e[1]);var r,n=this.opts||{},i=this.target;"string"==typeof i?r=i:"object"==typeof i&&i instanceof Y?r=i.url:c._A?.URL&&"object"==typeof i&&i instanceof URL&&(r=i.href),T(this,r);var o=(""+(i&&i instanceof Y&&i.method||n.method||"GET")).toUpperCase();this.params.method=o,this.txSize=V(n.body)||0}function A(t,r){var n;this.endTime=(0,p.z)(),this.params||(this.params={}),this.params.status=r?r.status:0,"string"==typeof this.rxSize&&this.rxSize.length>0&&(n=+this.rxSize);var o={txSize:this.txSize,rxSize:n,duration:(0,p.z)()-this.startTime};i("xhr",[this.params,o,this.startTime,this.endTime,"fetch"],this,e.D.ajax)}function E(t){var r=this.params,n=this.metrics;if(!this.ended){this.ended=!0;for(var o=0;o 2&&void 0!==arguments[2])||arguments[2];super(e,t,we.t,r),this.importAggregator()}}new class{constructor(e){let t=arguments.length>1&&void 0!==arguments[1]?arguments[1]:(0,_.ky)(16);c._A?(this.agentIdentifier=t,this.sharedAggregator=new y({agentIdentifier:this.agentIdentifier}),this.features={},this.desiredFeatures=new Set(e.features||[]),this.desiredFeatures.add(m),Object.assign(this,(0,a.j)(this.agentIdentifier,e,e.loaderType||"agent")),this.start()):(0,l.Z)("Failed to initial the agent. Could not determine the runtime environment.")}get config(){return{info:(0,t.C5)(this.agentIdentifier),init:(0,t.P_)(this.agentIdentifier),loader_config:(0,t.DL)(this.agentIdentifier),runtime:(0,t.OP)(this.agentIdentifier)}}start(){const t="features";try{const r=n(this.agentIdentifier),i=[...this.desiredFeatures];i.sort(((t,r)=>e.p[t.featureName]-e.p[r.featureName])),i.forEach((t=>{if(r[t.featureName]||t.featureName===e.D.pageViewEvent){const n=function(t){switch(t){case e.D.ajax:return[e.D.jserrors];case e.D.sessionTrace:return[e.D.ajax,e.D.pageViewEvent];case e.D.sessionReplay:return[e.D.sessionTrace];case e.D.pageViewTiming:return[e.D.pageViewEvent];default:return[]}}(t.featureName);n.every((e=>r[e]))||(0,l.Z)("".concat(t.featureName," is enabled but one or more dependent features has been disabled (").concat((0,D.P)(n),"). This may cause unintended consequences or missing data...")),this.features[t.featureName]=new t(this.agentIdentifier,this.sharedAggregator)}})),(0,T.Qy)(this.agentIdentifier,this.features,t)}catch(e){(0,l.Z)("Failed to initialize all enabled instrument classes (agent aborted) -",e);for(const e in this.features)this.features[e].abortHandler?.();const r=(0,T.fP)();return delete r.initializedAgents[this.agentIdentifier]?.api,delete r.initializedAgents[this.agentIdentifier]?.[t],delete this.sharedAggregator,r.ee?.abort(),delete r.ee?.get(this.agentIdentifier),!1}}}({features:[J,m,S,class extends h{static featureName=oe;constructor(t,r){if(super(t,r,oe,!(arguments.length>2&&void 0!==arguments[2])||arguments[2]),!c.il)return;const n=this.ee;let i;(0,k.QU)(n),this.eventsEE=(0,k.em)(n),this.eventsEE.on(se,(function(e,t){this.bstStart=(0,p.z)()})),this.eventsEE.on(ae,(function(t,r){(0,s.p)("bst",[t[0],r,this.bstStart,(0,p.z)()],void 0,e.D.sessionTrace,n)})),n.on(ce+ne,(function(e){this.time=(0,p.z)(),this.startPath=location.pathname+location.hash})),n.on(ce+ie,(function(t){(0,s.p)("bstHist",[location.pathname+location.hash,this.startPath,this.time],void 0,e.D.sessionTrace,n)}));try{i=new PerformanceObserver((t=>{const r=t.getEntries();(0,s.p)(te,[r],void 0,e.D.sessionTrace,n)})),i.observe({type:re,buffered:!0})}catch(e){}this.importAggregator({resourceObserver:i})}},C,xe,B,class extends h{static featureName=de;constructor(e,r){if(super(e,r,de,!(arguments.length>2&&void 0!==arguments[2])||arguments[2]),!c.il)return;if(!(0,t.OP)(e).xhrWrappable)return;try{this.removeOnAbort=new AbortController}catch(e){}let n,i=0;const o=this.ee.get("tracer"),a=(0,k._L)(this.ee),s=(0,k.Lg)(this.ee),u=(0,k.BV)(this.ee),d=(0,k.Kf)(this.ee),f=this.ee.get("events"),l=(0,k.u5)(this.ee),h=(0,k.QU)(this.ee),g=(0,k.Gm)(this.ee);function m(e,t){h.emit("newURL",[""+window.location,t])}function v(){i++,n=window.location.hash,this[ve]=(0,p.z)()}function b(){i--,window.location.hash!==n&&m(0,!0);var e=(0,p.z)();this[pe]=~~this[pe]+e-this[ve],this[ye]=e}function y(e,t){e.on(t,(function(){this[t]=(0,p.z)()}))}this.ee.on(ve,v),s.on(be,v),a.on(be,v),this.ee.on(ye,b),s.on(ge,b),a.on(ge,b),this.ee.buffer([ve,ye,"xhr-resolved"],this.featureName),f.buffer([ve],this.featureName),u.buffer(["setTimeout"+le,"clearTimeout"+fe,ve],this.featureName),d.buffer([ve,"new-xhr","send-xhr"+fe],this.featureName),l.buffer([me+fe,me+"-done",me+he+fe,me+he+le],this.featureName),h.buffer(["newURL"],this.featureName),g.buffer([ve],this.featureName),s.buffer(["propagate",be,ge,"executor-err","resolve"+fe],this.featureName),o.buffer([ve,"no-"+ve],this.featureName),a.buffer(["new-jsonp","cb-start","jsonp-error","jsonp-end"],this.featureName),y(l,me+fe),y(l,me+"-done"),y(a,"new-jsonp"),y(a,"jsonp-end"),y(a,"cb-start"),h.on("pushState-end",m),h.on("replaceState-end",m),window.addEventListener("hashchange",m,(0,O.m$)(!0,this.removeOnAbort?.signal)),window.addEventListener("load",m,(0,O.m$)(!0,this.removeOnAbort?.signal)),window.addEventListener("popstate",(function(){m(0,i>1)}),(0,O.m$)(!0,this.removeOnAbort?.signal)),this.abortHandler=this.#e,this.importAggregator()}#e(){this.removeOnAbort?.abort(),this.abortHandler=void 0}}],loaderType:"spa"})})(),window.NRBA=o})(); window.jQuery || document.write(' ') CKEDITOR_BASEPATH='https://f1000research.com/js/vendor/ckeditor/' window.reactTheme = 'research'; window.MathJax = { CommonHTML: { linebreaks: { automatic: true } }, 'HTML-CSS': { linebreaks: { automatic: true } }, SVG: { linebreaks: { automatic: true } }, AuthorInit: function() { MathJax.Hub.Register.MessageHook('End Process', function () { let timeout = false; // holder for timeout id const delay = 250; // delay after event is "complete" to run callback const reflowMath = function() { const dispFormulas = document.querySelectorAll('.disp-formula.panel'); if (!dispFormulas) { return; } for (const dispFormula of dispFormulas) { const child = dispFormula.querySelector('.MathJax_Preview').nextSibling.firstChild; const isMultiline = MathJax.Hub.getAllJax(dispFormula)[0].root.isMultiline; if (dispFormula.offsetWidth < child.offsetWidth || isMultiline) { MathJax.Hub.Queue(['Rerender', MathJax.Hub, dispFormula]); } } }; window.addEventListener('resize', function() { clearTimeout(timeout); // clear the timeout timeout = setTimeout(reflowMath, delay); // start timing for event "completion" }); }); }, }; if (window.location.hash == '#_=_'){ window.location = window.location.href.split('#')[0] } !function(f,b,e,v,n,t,s){if(f.fbq)return;n=f.fbq=function() {n.callMethod? n.callMethod.apply(n,arguments):n.queue.push(arguments)} ;if(!f._fbq)f._fbq=n; n.push=n;n.loaded=!0;n.version='2.0';n.queue=[];t=b.createElement(e);t.async=!0; t.src=v;s=b.getElementsByTagName(e)[0];s.parentNode.insertBefore(t,s)}(window, document,'script','https://connect.facebook.net/en_US/fbevents.js'); fbq('init', '1641728616063202'); fbq('track', "PixelInitialized", {}); (function(h,o,t,j,a,r){ h.hj=h.hj||function(){(h.hj.q=h.hj.q||[]).push(arguments)}; h._hjSettings={hjid:2318163,hjsv:6}; a=o.getElementsByTagName('head')[0]; r=o.createElement('script');r.async=1; r.src=t+h._hjSettings.hjid+j+h._hjSettings.hjsv; a.appendChild(r); })(window,document,'https://static.hotjar.com/c/hotjar-','.js?sv='); search file_upload Submit your research search menu close search Browse Gateways & Collections How to Publish Submit your Research My Submissions Article Guidelines Article Guidelines (New Versions) Open Data, Software and Code Guidelines Open Data and Accessible Source Materials Guidelines (HSS) Open Data, Software and Code Guidelines (PSE) Prepublication Checks Production Process Posters and Slides Guidelines Document Guidelines Article Processing Charges Peer Review Finding Article Reviewers About How it Works For Reviewers Our Advisors Policies Glossary FAQs For Developers Newsroom Contact My Research Submissions Content and Tracking Alerts My Details Sign In file_upload Submit your research { "@context": "https://schema.org", "@type": "ScholarlyArticle", "mainEntityOfPage": { "@type": "WebPage", "@id": "https://f1000research.com/articles/15-10" }, "headline": "A Bazilevic-type function associated with the class of univalent functions with study Coefficient Bounds and...", "datePublished": "2026-01-06T16:25:17", "dateModified": "2026-01-06T16:25:17", "author": [ { "@type": "Person", "name": "Mays S.Abdul Ameer" }, { "@type": "Person", "name": "Abdul Rahman S.Juma" }, { "@type": "Person", "name": "Hassan Hussien Ebrahim" } ], "publisher": { "@type": "Organization", "name": "F1000Research", "logo": { "@type": "ImageObject", "url": "https://f1000research.com/img/AMP/F1000Research_image.png", "height": 480, "width": 60 } }, "image": { "@type": "ImageObject", "url": "https://f1000research.com/img/AMP/F1000Research_image.png", "height": 1200, "width": 150 }, "description": " Background The Fekete–Szegö inequality and coefficient bounds play a fundamental role in the study of Bazilevič-type functions, a subclass of univalent functions with significant applications in complex analysis and related mathematical fields. These functions have been extensively studied for their geometric properties and coefficient behavior, yet there remains a need to explore subclasses defined via differential operators and subordination techniques to obtain sharper and more general bounds. Understanding the coefficient structure of these functions provides insight into their analytic behavior and extends classical results in the theory of univalent functions. Methods This study focuses on a specific subclass of Bazilevič-type functions and investigates its properties using differential subordination and differential operators. These mathematical tools are employed to derive explicit coefficient estimates and establish the Fekete–Szegö inequalities. The analysis involves careful application of operator techniques to obtain bounds that reflect the influence of the subclass parameters on the analytic functions under consideration. Results The derived inequalities demonstrate the effectiveness of differential operators in obtaining precise coefficient constraints. The results highlight how variations in the defining parameters of the subclass influence the function behavior, providing clear and explicit bounds for the coefficients. These findings extend existing results in the literature and offer new insights into the structure of Bazilevič-type functions. Conclusions Overall, this study provides a systematic approach to understanding the coefficient structure of Bazilevič-type functions. The findings establish a foundation for further theoretical research on univalent functions and offer tools that can be applied in both theoretical investigations and practical problems in complex analysis. The results contribute to the broader understanding of analytic functions and their applications in mathematical modeling. " } { "@context": "http://schema.org", "@type": "BreadcrumbList", "itemListElement": [ { "@type": "ListItem", "position": "1", "item": { "@id": "https://f1000research.com/", "name": "Home" } }, { "@type": "ListItem", "position": "2", "item": { "@id": "https://f1000research.com/browse/articles", "name": "Browse" } }, { "@type": "ListItem", "position": "3", "item": { "@id": "https://f1000research.com/articles/15-10", "name": "A Bazilevic-type function associated with the class of univalent functions..." } } ] } Home Browse A Bazilevic-type function associated with the class of univalent functions... ALL Metrics - Views Downloads Get PDF Get XML Cite How to cite this article S.Abdul Ameer M, S.Juma AR and Hussien Ebrahim H. A Bazilevic-type function associated with the class of univalent functions with study Coefficient Bounds and Fekete-Szego inequalities [version 1; peer review: 1 approved, 1 approved with reservations] . F1000Research 2026, 15 :10 ( https://doi.org/10.12688/f1000research.172490.1 ) NOTE: If applicable, it is important to ensure the information in square brackets after the title is included in all citations of this article. Close Copy Citation Details Export Export Citation Sciwheel EndNote Ref. Manager Bibtex ProCite Sente EXPORT Select a format first Track Share ▬ ✚ Research Article A Bazilevic-type function associated with the class of univalent functions with study Coefficient Bounds and Fekete-Szego inequalities [version 1; peer review: 1 approved, 1 approved with reservations] Mays S.Abdul Ameer https://orcid.org/0009-0002-0471-0483 1 , Abdul Rahman S.Juma 2 , Hassan Hussien Ebrahim 3 Mays S.Abdul Ameer https://orcid.org/0009-0002-0471-0483 1 , Abdul Rahman S.Juma 2 , Hassan Hussien Ebrahim 3 PUBLISHED 06 Jan 2026 Author details Author details 1 Mathematics, Tikrit University, Tikrit, Saladin Governorate, Iraq 2 Mathematics, University of Anbar, Ramadi, Al Anbar Governorate, Iraq 3 Mathematics, Tikrit University, Tikrit, Saladin Governorate, Iraq Mays S.Abdul Ameer Roles: Writing – Original Draft Preparation Abdul Rahman S.Juma Roles: Writing – Review & Editing Hassan Hussien Ebrahim Roles: Writing – Review & Editing OPEN PEER REVIEW DETAILS REVIEWER STATUS This article is included in the Fallujah Multidisciplinary Science and Innovation gateway. Abstract Background The Fekete–Szegö inequality and coefficient bounds play a fundamental role in the study of Bazilevič-type functions, a subclass of univalent functions with significant applications in complex analysis and related mathematical fields. These functions have been extensively studied for their geometric properties and coefficient behavior, yet there remains a need to explore subclasses defined via differential operators and subordination techniques to obtain sharper and more general bounds. Understanding the coefficient structure of these functions provides insight into their analytic behavior and extends classical results in the theory of univalent functions. Methods This study focuses on a specific subclass of Bazilevič-type functions and investigates its properties using differential subordination and differential operators. These mathematical tools are employed to derive explicit coefficient estimates and establish the Fekete–Szegö inequalities. The analysis involves careful application of operator techniques to obtain bounds that reflect the influence of the subclass parameters on the analytic functions under consideration. Results The derived inequalities demonstrate the effectiveness of differential operators in obtaining precise coefficient constraints. The results highlight how variations in the defining parameters of the subclass influence the function behavior, providing clear and explicit bounds for the coefficients. These findings extend existing results in the literature and offer new insights into the structure of Bazilevič-type functions. Conclusions Overall, this study provides a systematic approach to understanding the coefficient structure of Bazilevič-type functions. The findings establish a foundation for further theoretical research on univalent functions and offer tools that can be applied in both theoretical investigations and practical problems in complex analysis. The results contribute to the broader understanding of analytic functions and their applications in mathematical modeling. READ ALL READ LESS Keywords Analytic function, Univalent functions, Differential operator, Subordination, Bazilevic type, Fekete-Szego inequalities. Corresponding Author(s) Mays S.Abdul Ameer ( [email protected] ) Close Corresponding author: Mays S.Abdul Ameer Competing interests: No competing interests were disclosed. Grant information: The author(s) declared that no grants were involved in supporting this work. Copyright: © 2026 S.Abdul Ameer M et al . This is an open access article distributed under the terms of the Creative Commons Attribution License , which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. How to cite: S.Abdul Ameer M, S.Juma AR and Hussien Ebrahim H. A Bazilevic-type function associated with the class of univalent functions with study Coefficient Bounds and Fekete-Szego inequalities [version 1; peer review: 1 approved, 1 approved with reservations] . F1000Research 2026, 15 :10 ( https://doi.org/10.12688/f1000research.172490.1 ) First published: 06 Jan 2026, 15 :10 ( https://doi.org/10.12688/f1000research.172490.1 ) Latest published: 06 Jan 2026, 15 :10 ( https://doi.org/10.12688/f1000research.172490.1 ) Introduction Geometric function theory is a subfield of complex analysis that studies the geometric characteristics of analytic functions. The foundation of complex analysis is academic research on univalent and multivalent function theory. find it fascinating because of its complex geometry and variety of research options. Understanding univalent functions is crucial for the complicated analysis of single and multiple variables. Univalent functions, which are analytic and one-to-one in the unit disk, form a fundamental class in complex analysis and geometric function theory, with numerous applications in mathematical modeling and theoretical investigations. 1 – 3 Among the various subclasses, Bazilevič-type functions, introduced by Bazilevich, 4 – 7 have drawn significant attention due to their rich geometric properties and connections with differential subordination. 8 These functions have been widely studied for their coefficient bounds and associated inequalities, including the classical Fekete–Szegö problem, which provides important constraints on the coefficients of analytic and multivalent functions. 9 – 12 Recent developments have focused on extending these studies to subclasses defined via differential operators and linear transformations, allowing for the derivation of sharper coefficient estimates and Fekete–Szegö inequalities. 13 , 14 Investigations on bi-Bazilevič functions, Faber polynomial coefficients, and operator-based approaches have further highlighted the effectiveness of these methods in obtaining explicit bounds and revealing structural properties of analytic functions. 15 Despite these advances, there remains a need to explore new subclasses of Bazilevič-type functions associated with specific differential or linear operators, particularly in relation to their coefficient estimates and Fekete–Szegö inequalities. 16 – 18 This study addresses this gap by examining the subclass, establishing explicit coefficient bounds, and formulating relevant Fekete–Szegö inequalities. The findings provide a systematic framework for understanding the analytic and geometric properties of Bazilevič-type functions and offer potential directions for future research in the theory of univalent functions. Methods Consider the open unit disc to be Ω = { z : | z | < 1 } and let D stand for the class of analytic functions of the type (1) f ( z ) = z + ∑ n = 2 ∞ a n z n . The functions are normalized with f ( 0 ) = 0 and f ′ ( 0 ) = 1 , and they are analytic in the open unit disk. In the open unit disk Ω , let E represent the subclass of all univalent functions. A refers to the class of functions h ( z ) with a positive real fraction in Ω . As follows. h ( z ) = 1 + ∑ n = 1 ∞ c n z n . Since w ( z ) = z ¯ is an analytic function on ∂ D , we define w ( z ) as the Schwarz function defined on the domain's border ∂ D in ℂ . 1 , 2 The class A has the following relationship with the class of Schwarz functions w see Ref. 3 . (2) h ∈ A ↔ h = 1 + w 1 − w Consider two analytic functions in Ω , f and h , then we say f ( z ) is subordinate to g ( z ) . Write as follows. (3) f ≺ g or f ( z ) ≺ g ( z ) , ( z ∈ Ω ) , If there is an analytic Schwarz function in Ω with w ( 0 ) = 0 and | w ( z ) | < 1 , then f ( z ) = g ( w ( z ) ) . If a function g ( z ) be univalent in Ω , then f ( z ) ≺ g ( z ) i s equivalent to f ( 0 ) = g ( 0 ) and f ( Ω ) ⊂ g ( Ω ) (see Ref. 4 ). If a function f ( z ) meets the following criteria, it is referred to as univalent starlike in Ω : Re { z f ( z ) ´ f ( z ) } > 0 , ( z ∈ Ω ) . S ∗ is the name given to the old class of functions (see Ref. 3 ). We can write from Equation (1) (4) ( f ( z ) ) β = ( z + ∑ n = 2 ∞ a n z n ) β , If β is a real number greater than zero, we can obtain (5) ( f ( z ) ) β = ( z + a 2 z 2 + a 3 z 3 + a 4 z 4 + … ) β . ( f ( z ) ) β = z β ( 1 + a 2 z + a 3 z 2 + a 4 z 3 + ⋯ ) β . By applying Equation (5) binomial expansion, we get ( f ( z ) ) β = z β [ 1 + β ( a 2 z + a 3 z 2 + a 4 z 3 + ⋯ ) + β ( β − 1 ) 2 ! ( a 2 z + a 3 z 2 + a 4 z 3 + ⋯ ) 2 + … ] β ( f ( z ) ) β = z β ( 1 + β ( a 2 z + a 3 z 2 + a 4 z 3 + ⋯ ) ) β . Therefore ( f ( z ) ) β = z β + β a 2 z β + 1 + β a 3 z β + 2 + β a 4 z β + 3 + ⋯ . The following represents the class of analytic functions D β : (6) ( f ( z ) ) β = z β + ∑ n = 2 ∞ a n ( β ) z β + n − 1 . G ( D β ) is the differential operator determined for a function ( f ( z ) ) β given in Equation (6) on the space of analytic functions. H ξ , μ m , λ : G ( D β ) ⟶ G ( D β ) as follows: (7) H ξ , μ m , λ ( ( f ( z ) ) β ) = ( f ( z ) β ) . H 1 = ( λ − ξ ) z μ + λ H + ( 1 − ( λ − ξ ) μ + λ ) H ξ , μ 1 , λ ( ( f ( z ) ) β ) = H 1 ( H ξ , μ 0 , λ ( ( f ( z ) ) β ) ) = H 1 [ z β + ∑ n = 2 ∞ a n ( β ) z β + n − 1 ] H ξ , μ 1 , λ ( ( f ( z ) ) β ) = ( 1 + ( 1 − β ) ( λ − ξ ) μ + λ ) 1 z β + ∑ n = 2 ∞ ( 1 + ( n − β ) ( λ − ξ ) μ + λ ) 1 a n ( β ) z β + n − 1 H ξ , μ 2 , λ ( ( f ( z ) ) β ) = H 1 ( H ξ , μ 1 , λ ( ( f ( z ) ) β ) ) = ( 1 + ( 1 − β ) ( λ − ξ ) μ + λ ) 2 z β + ∑ n = 2 ∞ ( 1 + ( λ − ξ ) ( β + n − 2 ) μ + λ ) 2 a n ( β ) z β + n − 1 . In general, we have (8) H ξ , μ m , λ ( ( f ( z ) ) β ) = H 1 ( H ξ , μ m − 1 , λ ( ( f ( z ) ) β ) ) = ( 1 + ( 1 − β ) ( λ − ξ ) μ + λ ) m z β + ∑ n = 2 ∞ ( 1 + ( λ − ξ ) ( β + n − 2 ) μ + λ ) m a n ( β ) z β + n − 1 . where ( μ > 0 , ξ , λ ≥ 0 , m ∈ N 0 = N ∪ { 0 } ; β > 0 , z ∈ Ω ) . The goal of this study is to find “coefficient bounds and Fekete-Szego inequalities for the subclass L ξ , μ m , λ ( δ , α , γ ) of Bazilevic type functions”. Now, using the differential operator, define a class of Bazilevic functions (see Ref. 5 ) as follows: Definition 1. The function f ( z ) β which has the form Equation (5) belongs to the class L ξ , μ m , λ ( δ , α , γ ) satisfies the following conditions (9) Re { H ξ , μ m , λ f ( z ) β ( 1 + ( 1 − β ) ( λ − ξ ) μ + λ ) m z β } > γ | H ξ , μ m , λ f ( z ) β ( 1 + ( 1 − β ) ( λ − ξ ) μ + λ ) m z β − 1 | + α . Where ( μ > 0 , ξ , λ , γ ≥ 0 , m ∈ N 0 = N ∪ { 0 } ; β > 0 , z ∈ Ω ) (see Ref. 6 ). The above condition is equivalent to (10) H ξ , μ m , λ f ( z ) β ( 1 + ( 1 − β ) ( λ − ξ ) μ + λ ) m z β ≺ γ ( z ) , where γ ( z ) = 1 + αz 1 − δαz be univalent function with γ ( 0 ) = 1 and γ ´ ( z ) > 0 . If ξ = δ = 0 and m = λ = μ = α = 1 , we derive the following definition of the subclass of Bazilevic univalent functions. 4 (11) | f ́ ( z ) ( f ( z ) z ) β − 1 | < 1 . If ξ = 0 and m = λ = μ = 1 , L 0 , 1 1 , 1 ( δ , α , γ ) ( f ( z ) β ) (12) f ´ ( z ) ( f ( z ) z ) β − 1 ≺ γ ( z ) . If ξ = 0 , m = λ = β = 1 and placing γ ( z ) = 1 + z 1 − z , f ´ ( z ) ≺ 1 + z 1 − z . Bazilevic was generalized to provide the class described in (9) (see Ref. 7 ). Where he presented this job and studied it as follows: (13) f ( z ) = { ξ 1 + ε 2 ∫ 0 z ( h ( v ) − iε ) v − ( 1 + iξε 1 + ε 2 ) g ( v ) ξ 1 + ε 2 dv } 1 + iε ξ , where the function h ( v ) belongs to A and g ( v ) ∈ S ∗ with β > 0 . The following lemmas are necessary for you to discuss the main results. Lemma 2. Let h ( z ) = 1 + c 1 z + c 2 z 2 + c 3 z 3 + … , z ∈ Ω belongs to the class A and μ ∈ C , then | c 2 − μ c 1 2 | ≤ 2 max { 1 , | 2 μ − 1 | } . The following functions can produce crisp results. 8 h ( z ) = 1 + z 2 1 − z 2 and h ( z ) = 1 + z 1 − z , z ∈ Ω . Lemma 3. Let the function h ( z ) is analytic in Ω , with | h ( z ) | 0 . Lemma 4. If h ( z ) = 1 + c 1 z + c 2 z 2 + c 3 z 3 + ⋯ , z ∈ Ω belongs to the class A then | c 2 − γ c 1 2 | ≤ { − 4 γ + 2 if γ ≤ 0 2 if 0 ≤ γ ≤ 1 4 γ − 2 if γ ≥ 0 . For γ 1 , equality holds if and only if h ( z ) equals 1 + z 1 − z or one of them is a rotation. If 0 < γ < 1 , then equality holds if and only if h ( z ) is equal to 1 + z 2 1 − z 2 or one of them is rotations. Inequality becomes equality when γ = 0 if and only if h ( z ) = ( 1 + λ 2 ) 1 + z 1 − z + ( 1 − λ 2 ) 1 + z 1 − z , One of its rotations, or 0 ≤ γ ≤ 1 . If and only if the function h is the reciprocal of one of the functions such that equality holds in the case of γ = 0 , equality holds for γ = 1 . If the highest limit stated above is sharp, it can be made better when 0 < γ < 1 : | c 2 − γ c 1 2 | + γ | c 1 | 2 ≤ 2 0 < γ ≤ 1 2 , and | c 2 − γ c 1 2 | + ( 1 − γ ) | c 1 | 2 ≤ 2 1 2 ≤ γ < 1 . Many writers have examined Bazilevic functions, including Refs. 5 , 10 – 13 and 14 . Studied Singh in 1973, “two subclasses of the class B-the class of Bazilevic functions”. 15 In 2020, Particle Umar et al. investigated a subclass related to Bazilevic functions. 16 Juma and AL-khafaj 2019 The characteristics and properties of a family of Bazilevic (type) functions that are provided by particular linear operators. 17 Faber polynomial coefficient estimates for a subclass of analytic Bi-Bazilevic functions defined by a differential operator were introduced in Refs. 6 , 18 . Results and discussion Theorem 1. Let ( f ( z ) ) β ∈ D β which is given in Equation (6) . If ( f ( z ) ) β belongs to the class L ξ , μ m , λ ( δ , α , γ ) . Then | a 2 ( β ) | ≤ α ( 1 + δ ) 2 α ( 1 + ( λ − ξ ) ( − 1 − β ) 1 + ( λ − ξ ) ( 1 − β ) ) m , | a 3 ( β ) | ≤ α ( 1 + δ ) 2 α ( 1 + ( λ − ξ ) ( − 2 − β ) 1 + ( λ − ξ ) ( 1 − β ) ) m [ 1 − ( 1 − δα ) A 1 + α ( 1 + δ ) A 3 2 A 1 ] and | a 4 ( β ) | ≤ α ( 1 + δ ) 2 β ( 1 + ( λ − ξ ) ( − 3 − β ) 1 + ( λ − ξ ) ( 1 − β ) ) m [ 1 − ( β − 1 ) A 3 + 2 ( 1 − δα ) A 2 2 A 2 + 3 ( β − 1 ) ( 1 − δα ) A 1 A 4 + 3 α ( β − 1 ) ( 1 + δ ) A 3 A 4 12 A 1 A 2 − α ( β − 2 ) ( 1 + δ ) A 3 A 4 12 A 1 A 2 − 3 12 ( 1 − 2 δα ) ] where A 1 = 2 β ( 1 + ( λ − ξ ) ( − 1 − β ) 1 + ( λ − ξ ) ( 1 − β ) ) 2 m , A 2 = β ( 1 + ( λ − ξ ) ( − 1 − β ) 1 + ( λ − ξ ) ( 1 − β ) ) m ( 1 + ( λ − ξ ) ( − 2 − β ) 1 + ( λ − ξ ) ( 1 − β ) ) m , A 3 = ( β − 1 ) ( 1 + ( λ − ξ ) ( − 2 − β ) 1 + ( λ − ξ ) ( 1 − β ) ) m , A 4 = α ( 1 + δ ) ( 1 + ( λ − ξ ) ( − 3 − β ) 1 + ( λ − ξ ) ( 1 − β ) ) m . Proof: If ( f ( z ) ) β ∈ L ξ , μ m , λ ( δ , α , γ ) , then from Equation (10) there is the Schwarz function w ( z ) . This is analytic in Ω, with w ( 0 ) = 0 and | w ( z ) | < 1 such that (14) H ξ , μ m , λ f ( z ) β ( 1 + ( 1 − β ) ( λ − ξ ) μ + λ ) m z β ≺ γ ( w ( z ) ) , where (15) γ ( z ) = 1 + αz 1 − δαz = 1 + α ( 1 + δ ) z + δ α 2 ( 1 + δ ) z 2 + δ 2 α 3 ( 1 + δ ) z 3 + ⋯ . The definition of the function h ( z ) is: h ( z ) = 1 + w ( z ) 1 − w ( z ) = 1 + c 1 h + c 2 h 2 + c 3 h 3 + ⋯ . As may be seen from Equation (14) , h ∈ A and (16) w ( z ) = 1 − w ( z ) 1 + w ( z ) = 1 2 c 1 z + 1 2 ( c 2 − 1 2 c 1 2 ) z 2 + 1 2 ( c 3 − c 1 c 2 + 1 4 c 1 3 ) z 3 + ⋯ . Considering Equations (14) , (15) , and (16) , we get at (17) H ξ , μ m , λ f ( z ) β ( 1 + ( 1 − β ) ( λ − ξ ) μ + λ ) m z β ≺ γ ( w ( z ) ) = γ ( z ) ( h ( z ) − 1 h ( z ) + 1 ) . = γ ( 1 2 c 1 z + 1 2 ( c 2 − 1 2 c 1 2 ) z 2 + 1 2 ( c 3 − c 1 c 2 + 1 4 c 1 3 ) z 3 + ⋯ ) , = 1 + 1 2 α ( 1 + δ ) c 1 z + [ 1 2 α ( 1 + δ ) ( c 2 − 1 2 c 1 2 ) + 1 4 δ α 2 ( 1 + δ ) c 1 2 ] z 2 + + [ 1 2 α ( 1 + δ ) ( ( c 3 − c 1 c 2 + 1 4 c 1 3 ) + 1 2 δ α 2 ( 1 + δ ) ( c 2 − 1 2 c 1 2 ) c 1 + 1 8 δ 2 α 2 ( 1 + δ ) c 1 2 ] … By use of the series expansion of 1 + ∑ n = 2 ∞ ( 1 + ( λ − ξ ) ( β + n − 1 ) 1 + ( ( λ − ξ ) ( 1 − β ) ) m a n ( β ) z n − 1 , we have (18) H ξ , μ m , λ f ( z ) β ( 1 + ( 1 − β ) ( λ − ξ ) μ + λ ) m z β = 1 + β ( 1 + ( λ − ξ ) ( − 1 − β ) 1 + ( λ − ξ ) ( 1 − β ) ) m a 2 ( β ) z + [ β a 3 ( β ) β ( β − 1 ) 2 ! a 2 2 ( β ) ] ( 1 + ( λ − ξ ) ( − 2 − β ) 1 + ( λ − ξ ) ( 1 − β ) ) m z 2 + [ β a 4 ( β ) + β ( β − 1 ) a 2 ( β ) a 3 ( β ) + β ( β − 1 ) ( β − 2 ) 3 ! a 2 3 ( β ) ] ( 1 + ( λ − ξ ) ( − 3 − β ) 1 + ( λ − ξ ) ( 1 − β ) ) m z 3 + … . Based on Equations (17) and (18) , we compare the coefficients of z , z 2 , and z 3 in Equation (14) to obtain (19) a 2 ( β ) = α ( 1 + δ ) c 1 2 α ( 1 + ( λ − ξ ) ( − 1 − β ) 1 + ( λ − ξ ) ( 1 − β ) ) m , (20) a 3 ( β ) ≤ α ( 1 + δ ) 2 α ( 1 + ( λ − ξ ) ( − 2 − β ) 1 + ( λ − ξ ) ( 1 − β ) ) m [ 1 − ] [ c 2 − ( 1 − δα ) A 1 + α ( 1 + δ ) A 3 2 A 1 c 1 2 ] , and (21) a 4 ( β ) = α ( 1 + δ ) 2 β ( 1 + ( λ − ξ ) ( − 3 − β ) 1 + ( λ − ξ ) ( 1 − β ) ) m [ c 3 − ( β − 1 ) A 3 + 2 ( 1 − δα ) A 2 2 A 2 c 1 c 2 + ( 3 ( β − 1 ) ( 1 − δα ) A 1 A 4 + 3 α ( β − 1 ) ( 1 + δ ) A 3 A 4 12 A 1 A 2 − α ( β − 2 ) ( 1 + δ ) A 3 A 4 12 A 1 A 2 − 3 12 ( 1 − 2 δα ) ) c 1 3 ] . Given that h ( z ) has an analytic function and is confined to Ω , (22) | c 0 | ≤ 1 and | c k | ≤ 1 − | c 0 | 2 for k > 0 . By using Equation (22) , we get | a 2 ( β ) | ≤ α ( 1 + δ ) 2 α ( 1 + ( λ − ξ ) ( − 1 − β ) 1 + ( λ − ξ ) ( 1 − β ) ) m , | a 3 ( β ) | ≤ α ( 1 + δ ) 2 α ( 1 + ( λ − ξ ) ( − 2 − β ) 1 + ( λ − ξ ) ( 1 − β ) ) m [ 1 − ( 1 − δα ) A 1 + α ( 1 + δ ) A 3 2 A 1 ] | a 4 ( β ) | ≤ α ( 1 + δ ) 2 β ( 1 + ( λ − ξ ) ( − 3 − β ) 1 + ( λ − ξ ) ( 1 − β ) ) m [ 1 − ( β − 1 ) A 3 + 2 ( 1 − δα ) A 2 2 A 2 + 3 ( β − 1 ) ( 1 − δα ) A 1 A 4 + 3 α ( β − 1 ) ( 1 + δ ) A 3 A 4 12 A 1 A 2 − α ( β − 2 ) ( 1 + δ ) A 3 A 4 12 A 1 A 2 − 3 12 ( 1 − 2 δα ) ] . The proof is complete. Using Theorem 1 and γ ( z ) = 1 + z 1 − z , we obtain the following. Corollary 2. Let ( f ( z ) ) β ∈ D β which is given in Equation (6) . If ( f ( z ) ) β belonging to the class L ξ , μ m , λ ( 1 , 1 , γ ) . Then | a 2 ( β ) | ≤ 1 α ( 1 + ( λ − ξ ) ( − 1 − β ) 1 + ( λ − ξ ) ( 1 − β ) ) m , Proof: Based on Equations (19) and (20) , we have | a 3 ( β ) | ≤ 1 α ( 1 + ( λ − ξ ) ( − 2 − β ) 1 + ( λ − ξ ) ( 1 − β ) ) m − A 1 α ( 1 + ( λ − ξ ) ( − 2 − β ) 1 + ( λ − ξ ) ( 1 − β ) ) m A 3 and | a 4 ( β ) | ≤ 1 β ( 1 + ( λ − ξ ) ( − 3 − β ) 1 + ( λ − ξ ) ( 1 − β ) ) m . [ 1 − 2 ( β − 1 ) ( 1 + ( λ − ξ ) ( − 2 − β ) 1 + ( λ − ξ ) ( 1 − β ) ) m 2 A 2 + 4 ( 2 β − 1 ) ( 1 + ( λ − ξ ) ( − 2 − β ) 1 + ( λ − ξ ) ( 1 − β ) ) m A 3 12 A 1 A 2 ] We obtain the result by entering α = 1 and m = 0 in Corollary 2 , Corollary 3. Let f ( z ) ∈ D which is given in Equation (6) . If f ( z ) belongs to the class L ξ , μ 0 , λ ( 1 , 1 , γ ) . Then | a 2 ( β ) | ≤ 1 | a 3 ( β ) | ≤ 1 , and | a 4 ( β ) | ≤ 1 . Theorem 4. Let γ ( z ) = 1 + α ( 1 + δ ) z + δ α 2 ( 1 + δ ) z 2 + … , where γ ( z ) ∈ D and γ ´ ( z ) > 0 . If ( f ( z ) ) β which is given in Equation (6) belongs to the class L ξ , μ m , λ ( δ , α , γ ) and φ ∈ ℕ , then | a 3 ( β ) − φ a 2 2 ( β ) | ≤ α ( 1 + δ ) β ( 1 + ( λ − ξ ) ( − 2 − β ) 1 + ( λ − ξ ) ( 1 − β ) ) m max { 1 , | ( δα + δ ) ( [ 2 λ + β − 1 ] ( 1 + ( λ − ξ ) ( − 2 − β ) 1 + ( λ − ξ ) ( 1 − β ) ) m 2 β ( 1 + ( λ − ξ ) ( − 1 − β ) 1 + ( λ − ξ ) ( 1 − β ) ) m ) | } . a 3 ( β ) − φ a 2 2 ( β ) = α ( 1 + δ ) c 2 2 β ( 1 + ( λ − ξ ) ( − 2 − β ) 1 + ( λ − ξ ) ( 1 − β ) ) m − [ 2 βα ( 1 + δ ) ( 1 − δα ) ( 1 + ( λ − ξ ) ( − 1 − β ) 1 + ( λ − ξ ) ( 1 − β ) ) 2 m + ( β − 1 ) β 2 ( 1 + δ ) 2 ( 1 + ( λ − ξ ) ( − 2 − β ) 1 + ( λ − ξ ) ( 1 − β ) ) m ] 8 β 2 ( 1 + ( λ − ξ ) ( − 1 − β ) 1 + ( λ − ξ ) ( 1 − β ) ) 2 m ( 1 + ( λ − ξ ) ( − 2 − β ) 1 + ( λ − ξ ) ( 1 − β ) ) m c 1 2 − φ β 2 ( 1 + δ ) 2 c 1 2 4 α 2 ( 1 + ( λ − ξ ) ( − 1 − β ) 1 + ( λ − ξ ) ( 1 − β ) ) 2 m . Therefore | a 3 ( β ) − φ a 2 2 ( β ) | = α ( 1 + δ ) c 2 2 β ( 1 + ( λ − ξ ) ( − 2 − β ) 1 + ( λ − ξ ) ( 1 − β ) ) m [ c 2 − v c 1 2 ] , where v = 1 2 [ 1 − δβ + β ( 1 + δ ) ( ( 2 φ + β − 1 ) ( 1 + ( λ − ξ ) ( − 2 − β ) 1 + ( λ − ξ ) ( 1 − β ) ) m 2 β ( 1 + ( λ − ξ ) ( − 1 − β ) 1 + ( λ − ξ ) ( 1 − β ) ) 2 m ) ] . Using Lemma 2 , we obtain | a 3 ( β ) − φ a 2 2 ( β ) | ≤ α ( 1 + δ ) β ( 1 + ( λ − ξ ) ( − 2 − β ) 1 + ( λ − ξ ) ( 1 − β ) ) m max { 1 , | δα + α ( 1 + δ ) ( [ 2 λ + β − 1 ] ( 1 + ( λ − ξ ) ( − 2 − β ) 1 + ( λ − ξ ) ( 1 − β ) ) m 2 β ( 1 + ( λ − ξ ) ( − 1 − β ) 1 + ( λ − ξ ) ( 1 − β ) ) m ) | } . The outcome for ( f ( z ) ) β , which is provided as follows: H ξ , μ m , λ f ( z ) β ( 1 + ( 1 − β ) ( λ − ξ ) μ + λ ) m z β ≺ γ ( z 2 ) , or H ξ , μ m , λ f ( z ) β ( 1 + ( 1 − β ) ( λ − ξ ) μ + λ ) m z β ≺ γ ( z ) . Theroofsomplete. The results of Lemma 4 are as follows: Theorem 5. Let γ ( z ) = 1 + α ( 1 + δ ) z + δ α 2 ( 1 + δ ) z 2 + ⋯ , 0 ≤ δ ≤ 1 and 0 < β ≤ 1 . If ( f ( z ) ) β given by Equation (6) belongs to the class L ξ , μ m , λ ( δ , α , γ ) , then τ ∈ ℂ | a 3 ( β ) − φ a 2 2 ( β ) | ≤ { α ( 1 + δ ) β ( 1 + ( λ − ξ ) ( − 2 − β ) 1 + ( λ − ξ ) ( 1 − β ) ) m [ δα − α ( 1 + δ ) ( [ 2 τ + β − 1 ] ( 1 + ( λ − ξ ) ( − 2 − β ) 1 + ( λ − ξ ) ( 1 − β ) ) m 2 β ( 1 + ( λ − ξ ) ( − 1 − β ) 1 + ( λ − ξ ) ( 1 − β ) ) m ) ] if τ ≤ τ 1 α ( 1 + δ ) β ( 1 + ( λ − ξ ) ( − 2 − β ) 1 + ( λ − ξ ) ( 1 − β ) ) m , if τ 1 ≤ τ ≤ τ 2 α ( 1 + δ ) β ( 1 + ( λ − ξ ) ( − 2 − β ) 1 + ( λ − ξ ) ( 1 − β ) ) m [ α ( 1 + δ ) ( [ 2 τ + β − 1 ] ( 1 + ( λ − ξ ) ( − 2 − β ) 1 + ( λ − ξ ) ( 1 − β ) ) m 2 β ( 1 + ( λ − ξ ) ( − 1 − β ) 1 + ( λ − ξ ) ( 1 − β ) ) m ) − δα ] , if τ ≥ τ 2 , Where τ 1 = 2 α ( δα − 1 ) ( 1 + ( λ − ξ ) ( − 1 − β ) 1 + ( λ − ξ ) ( 1 − β ) ) 2 m − α ( 1 + δ ) ( β − 1 ) ( 1 + ( λ − ξ ) ( − 2 − β ) 1 + ( λ − ξ ) ( 1 − β ) ) m 2 α ( 1 + δ ) ( 1 + ( λ − ξ ) ( − 2 − β ) 1 + ( λ − ξ ) ( 1 − β ) ) m , τ 2 = 2 α ( δα + 1 ) ( 1 + ( λ − ξ ) ( − 1 − β ) 1 + ( λ − ξ ) ( 1 − β ) ) 2 m − α ( 1 + δ ) ( β − 1 ) ( 1 + ( λ − ξ ) ( − 2 − β ) 1 + ( λ − ξ ) ( 1 − β ) ) m 2 α ( 1 + δ ) ( 1 + ( λ − ξ ) ( − 2 − β ) 1 + ( λ − ξ ) ( 1 − β ) ) m . Proof. Since ( f ( z ) ) β ∈ L ξ , μ m , λ ( δ , α , γ ) and γ ( z ) given by Equation (17) , then a 2 ( β ) and a 3 ( β ) are given in Theorem 1 . Furthermore, | a 3 ( β ) − φ a 2 2 ( β ) | = α ( 1 + δ ) 2 β ( 1 + ( λ − ξ ) ( − 2 − β ) 1 + ( λ − ξ ) ( 1 − β ) ) m [ c 2 − t c 1 2 ] , where t = 1 2 [ 1 − δ + α ( 1 + δ ) ( [ 2 τ + β − 1 ] ( 1 + ( λ − ξ ) ( − 2 − β ) 1 + ( λ − ξ ) ( 1 − β ) ) m 2 β ( 1 + ( λ − ξ ) ( − 1 − β ) 1 + ( λ − ξ ) ( 1 − β ) ) 2 m ) ] . Using Lemma 4 , we can establish the inequality Equation (10) as follows. | a 3 ( β ) − φ a 2 2 ( β ) | ≤ α ( 1 + δ ) β ( 1 + ( λ − ξ ) ( − 2 − β ) 1 + ( λ − ξ ) ( 1 − β ) ) m [ δα − α ( 1 + δ ) ( [ 2 τ + β − 1 ] ( 1 + ( λ − ξ ) ( − 2 − β ) 1 + ( λ − ξ ) ( 1 − β ) ) m 2 β ( 1 + ( λ − ξ ) ( − 1 − β ) 1 + ( λ − ξ ) ( 1 − β ) ) m ) ] . When 0 ≤ t ≤ 1 , then τ 1 ≤ τ ≤ τ 2 , and using Lemma 4 , yields | a 3 ( β ) − φ a 2 2 ( β ) | ≤ α ( 1 + δ ) β ( 1 + ( λ − ξ ) ( − 2 − β ) 1 + ( λ − ξ ) ( 1 − β ) ) m . Assuming t ≥ 1 , then τ 2 ≤ τ . Additionally, using Lemma 4 , we get | a 3 ( β ) − φ a 2 2 ( β ) | ≤ α ( 1 + δ ) β ( 1 + ( λ − ξ ) ( − 2 − β ) 1 + ( λ − ξ ) ( 1 − β ) ) m [ α ( 1 + δ ) ( [ 2 τ + β − 1 ] ( 1 + ( λ − ξ ) ( − 2 − β ) 1 + ( λ − ξ ) ( 1 − β ) ) m 2 β ( 1 + ( λ − ξ ) ( − 1 − β ) 1 + ( λ − ξ ) ( 1 − β ) ) m ) − δα ] Using γ ( z ) = 1 + z 1 − z and Theorem 5 , we arrive at the following conclusion: Corollary 6. Let γ ( z ) = 1 + 2 z + 2 z 2 + ⋯ . if ( f ( z ) ) β which is given in Equation (6) belongs to the class L ξ , μ m , λ ( δ , α , γ ) , then for any real complex number σ | a 3 ( β ) − φ a 2 2 ( β ) | ≤ { 2 β ( 1 + ( λ − ξ ) ( − 2 − β ) 1 + ( λ − ξ ) ( 1 − β ) ) m [ 1 − ( [ 2 τ + β − 1 ] ( 1 + ( λ − ξ ) ( − 2 − β ) 1 + ( λ − ξ ) ( 1 − β ) ) m 2 β ( 1 + ( λ − ξ ) ( − 1 − β ) 1 + ( λ − ξ ) ( 1 − β ) ) m ) ] if τ ≤ τ 1 2 β ( 1 + ( λ − ξ ) ( − 2 − β ) 1 + ( λ − ξ ) ( 1 − β ) ) m , if τ 1 ≤ τ ≤ τ 2 2 β ( 1 + ( λ − ξ ) ( − 2 − β ) 1 + ( λ − ξ ) ( 1 − β ) ) m [ ( [ 2 τ + β − 1 ] ( 1 + ( λ − ξ ) ( − 2 − β ) 1 + ( λ − ξ ) ( 1 − β ) ) m 2 β ( 1 + ( λ − ξ ) ( − 1 − β ) 1 + ( λ − ξ ) ( 1 − β ) ) m ) − 1 ] , if τ ≥ τ 2 , where τ 1 = − ( 1 − β ) 2 ( 1 + δ ) , τ 2 = 4 β ( 1 + ( λ − ξ ) ( − 1 − β ) 1 + ( λ − ξ ) ( 1 − β ) ) 2 m − 2 ( 1 − β ) ( 1 + ( λ − ξ ) ( − 2 − β ) 1 + ( λ − ξ ) ( 1 − β ) ) m 4 ( 1 + ( λ − ξ ) ( − 2 − β ) 1 + ( λ − ξ ) ( 1 − β ) ) m . Observe that we obtain the following Corollary for and in Corollary 6 . For β = 1 and m = 0 , Corollary 6 yields the following. Corollary 7. If γ ( z ) = 1 + 2 z + 2 z 2 + ⋯ . if ( f ( z ) ) β given by Equation (6) belongs to the class L ξ , μ 0 , λ ( 1 , 1 , γ ) . Then for any real complex number σ | a 2 ( β ) − τ a 1 2 ( β ) | ≤ { 2 − 4 τ if τ ≤ τ 1 , 2 if τ 1 ≤ τ ≤ 4 τ − 2 if τ ≥ τ 2 , τ 2 , The result of W. Ma. D. Minda [Ref. 8 , Lemma 1] is obtained when τ 1 = 0 and τ 2 = 1 . Corollary 8. Let γ ( z ) = 1 + α ( 1 + δ ) z + δ α 2 ( 1 + δ ) z 2 + ⋯ , 0 ≤ δ ≤ 1 and 0 < β ≤ 1 . If ( f ( z ) ) β given by Equation (6) belongs to the class L ξ , μ m , λ ( δ , α , γ ) , and τ 1 ≤ τ 3 ≤ τ 2 . Let τ 3 = 1 2 α ( 1 + δ ) ( 1 + ( λ − ξ ) ( − 2 − β ) 1 + ( λ − ξ ) ( 1 − β ) ) m ( 2 βδαβ ( 1 + ( λ − ξ ) ( − 1 − β ) 1 + ( λ − ξ ) ( 1 − β ) ) 2 m − α ( 1 + δ ) ( 1 − β ) ( 1 + ( λ − ξ ) ( − 2 − β ) 1 + ( λ − ξ ) ( 1 − β ) ) m ) . If τ 1 ≤ τ ≤ τ 3 , then (23) ≤ β ( 1 + δ ) 2 β ( 1 + ( λ − ξ ) ( − 2 − β ) 1 + ( λ − ξ ) ( 1 − β ) ) m . If τ 3 ≤ τ ≤ τ 2 , then (24) | a 3 ( β ) − τ a 2 2 ( β ) | + 2 β ( 1 + ( λ − ξ ) ( − 1 − β ) 1 + ( λ − ξ ) ( 1 − β ) ) 2 m α ( 1 + δ ) ( 1 + ( λ − ξ ) ( − 2 − β ) 1 + ( λ − ξ ) ( 1 − β ) ) m × [ 1 − δα + α ( 1 + δ ) ( [ 2 τ + β − 1 ] ( 1 + ( λ − ξ ) ( − 2 − β ) 1 + ( λ − ξ ) ( 1 − β ) ) m 2 β ( 1 + ( λ − ξ ) ( − 1 − β ) 1 + ( λ − ξ ) ( 1 − β ) ) m ) ] | a 2 ( β ) | 2 | a 3 ( β ) − τ a 2 2 ( β ) | + 2 β ( 1 + ( λ − ξ ) ( − 1 − β ) 1 + ( λ − ξ ) ( 1 − β ) ) 2 m α ( 1 + δ ) ( 1 + ( λ − ξ ) ( − 2 − β ) 1 + ( λ − ξ ) ( 1 − β ) ) m × [ 1 − δα + α ( 1 + δ ) ( [ 2 τ + β − 1 ] ( 1 + ( λ − ξ ) ( − 2 − β ) 1 + ( λ − ξ ) ( 1 − β ) ) m 2 β ( 1 + ( λ − ξ ) ( − 1 − β ) 1 + ( λ − ξ ) ( 1 − β ) ) m ) ] | a 2 ( β ) | 2 ≤ β ( 1 + δ ) 2 β ( 1 + ( λ − ξ ) ( − 2 − β ) 1 + ( λ − ξ ) ( 1 − β ) ) m Proof. For | a 3 ( β ) − τ a 2 2 ( β ) | + ( τ − τ 1 ) | a 2 ( β ) | 2 = ( τ − 1 2 α ( 1 + δ ) ( 1 + ( λ − ξ ) ( − 2 − β ) 1 + ( λ − ξ ) ( 1 − β ) ) m [ 2 β ( δα − 1 ) ( 1 + ( λ − ξ ) ( − 1 − β ) 1 + ( λ − ξ ) ( 1 − β ) ) 2 m + α ( 1 + δ ) ( β − 1 ) ( 1 + ( λ − ξ ) ( − 2 − β ) 1 + ( λ − ξ ) ( 1 − β ) ) m ] ) × β 2 ( 1 + δ ) 2 ( 1 + ( λ − ξ ) ( − 1 − β ) 1 + ( λ − ξ ) ( 1 − β ) ) 2 m | c 1 | 2 + α ( 1 + δ ) 2 α ( 1 + ( λ − ξ ) ( − 2 − β ) 1 + ( λ − ξ ) ( 1 − β ) ) m [ c 2 − t c 1 2 ] = α ( 1 + δ ) 2 α ( 1 + ( λ − ξ ) ( − 2 − β ) 1 + ( λ − ξ ) ( 1 − β ) ) m [ c 2 − t c 1 2 ( β ) + t | c 1 ( β ) | 2 ] . It follows that using Lemma 4 yields | a 3 ( β ) − τ a 2 2 ( β ) | + ( τ − τ 1 ) | a 2 ( β ) | 2 ≤ α ( 1 + δ ) 2 α ( 1 + ( λ − ξ ) ( − 2 − β ) 1 + ( λ − ξ ) ( 1 − β ) ) m , This is the inequality of Equation (24) . For | a 3 ( β ) − τ a 2 2 ( β ) | + ( τ − τ 1 ) | a 2 ( β ) | 2 = ( 2 β ( αδ + 1 ) ( 1 + ( λ − ξ ) ( − 1 − β ) 1 + ( λ − ξ ) ( 1 − β ) ) 2 m 2 β ( 1 + δ ) ( 1 + ( λ − ξ ) ( − 2 − β ) 1 + ( λ − ξ ) ( 1 − β ) ) m − α ( 1 + δ ) ( β − 1 ) ( 1 + ( λ − ξ ) ( − 2 − β ) 1 + ( λ − ξ ) ( 1 − β ) ) m β ( 1 + δ ) ( 1 + ( λ − ξ ) ( − 2 − β ) 1 + ( λ − ξ ) ( 1 − β ) ) m − τ ) × α 2 ( 1 + δ ) 2 4 β ( 1 + ( λ − ξ ) ( − 1 − β ) 1 + ( λ − ξ ) ( 1 − β ) ) 2 m | c 1 ( β ) | 2 + α ( 1 + δ ) 2 β ( 1 + ( λ − ξ ) ( − 2 − β ) 1 + ( λ − ξ ) ( 1 − β ) ) m [ c 2 − t c 1 2 ] = α ( 1 + δ ) 2 β ( 1 + ( λ − ξ ) ( − 2 − β ) 1 + ( λ − ξ ) ( 1 − β ) ) m [ | c 2 − t c 1 2 | + t | c 1 | 2 ] . Using Lemma 4 , the following results | a 3 ( β ) − τ a 2 2 ( β ) | + ( τ − τ 1 ) | a 2 ( β ) | 2 ≤ α ( 1 + δ ) 2 α ( 1 + ( λ − ξ ) ( − 2 − β ) 1 + ( λ − ξ ) ( 1 − β ) ) m , which is the inequality of Equation (24) . Conclusions In this study, we have investigated a subclass of Bazilevič-type functions associated with the class of univalent functions and examined its analytic properties using differential subordination and differential operators. By applying these techniques, new coefficient estimates and Fekete–Szegö inequalities were derived, providing sharper and more general results than several existing findings in the literature. The obtained results demonstrate that the parameters defining the subclass play a significant role in determining the coefficient behavior and geometric characteristics of the associated analytic functions. The study contributes to the deeper understanding of the coefficient structure of Bazilevič-type functions and extends the theoretical framework of univalent function theory. Furthermore, the approach used in this work may be applied to other subclasses of analytic or multivalent functions to establish similar inequalities and functional relationships. Future research may focus on exploring additional subclasses generated by modified differential operators or by incorporating complex parameters that enhance the geometric interpretation of these functions in applied fields such as fluid dynamics and conformal mapping. Ethical considerations This study did not involve human participants or animals, and therefore, ethical approval was not required. Data availability No data are associated with this article. Reporting guidelines This manuscript is paper in the field of Complex Analysis. Therefore, no specific reporting guidelines (such as CONSORT, PRISMA, or STROBE, which apply to empirical or experimental studies) are applicable. The paper follows standard mathematical reporting practices: all definitions, theorems, and proofs are presented with full logical rigor, and all symbols and notations are clearly defined upon first use. The structure and exposition conform to conventional mathematical standards for reproducibility and clarity. References 1. Abdul Ameer MS, Juma ARS, Al-Saphory RA: On differential subordination of higher-order derivatives of multivalent functions. J. Phys. Conf. Ser. 2021. IOP Publishing. 2. Duren P: Grundlehren der mathematischen wissenchaffen. Vol. 259 . . New York, NY, USA; Berlin/Heidelberg, Germany: Univalent Functions; Springer; 1983. 3. Graham I, Kohr G: Geometric function theory in one and higher dimensions. CRC Press; 2003. 4. Hamzat J, Fagbemiro O: Some properties of a new subclass of Bazilevic functions defined by Catas et al differential operator. FUW Trends Food Sci. Technol. 2018; 3 (28): 909–917. 5. Orhan H, Raducanu D: The Fekete–Szegö functional for generalized starlike and convex functions of complex order. Asian-Eur. J. Math. 2021; 14 (03): 2150036. Publisher Full Text 6. Juma ARS, Abdulhussain MS, Al-khafaji SN: Faber Polynomial Coefficient Estimates for Subclass of Analytic Bi-Bazilevic Functions Defined by Differential Operator. Baghdad Sci. J. 2019; 16 (1): 33. Publisher Full Text 7. Bazilevich IE: On a case of integrability in quadratures of the Loewner-Kufarev equation. Matematicheskii Sbornik. 1955; 79 (3): 471–476. 8. Ma W: A unified treatment of some special classes of univalent functions. in Proceedings of the Conference on Complex Analysis. Vol. 1992 . . International Press Inc.; 1992. 9. Keogh F, Merkes E: A coefficient inequality for certain classes of analytic functions. Proc. Am. Math. Soc. 1969; 20 (1): 8–12. Publisher Full Text 10. Juma A, Ularu N: Fekete-Szego inequality for certain subclasses of multivalent functions associated with quasi-sub ordination. Lib. Math. 2016; 36 (1): 45–52. 11. Silverman H: Univalent functions with negative coefficients. Proc. Am. Math. Soc. 1975; 51 (1): 109–116. Publisher Full Text 12. Srivastava H, Mishra A, Das M: The fekete-szegö-problem for a subclass of close-to-convex functions. Complex Var. Elliptic Equat. 2001; 44 (2): 145–163. 13. Bucur R, Andrei L, Breaz D: Coefficient bounds and Fekete-Szegö problem for a class of analytic functions defined by using a new differential operator. Appl. Math. Sci. 2015; 9 (25-28): 1355–1368. Publisher Full Text 14. Ravichandran V, et al. : Fekete-Szegö inequality for certain class of analytic functions. Aust. J. Math. Anal. Appl. 2004; 1 (2): 4–7. 15. Singh R: On Bazilevič functions. Proc. Am. Math. Soc. 1973; 38 (2): 261–271. 16. Umar S, et al. : On a subclass related to Bazilevic functions. AIMS MATHEMATICS. 2020; 5 (3): 2040–2056. Publisher Full Text 17. Juma A, Al-khafaji S, Irmak H: Properties and characteristics of a family consisting of Bazilevic (type) functions specified by certain linear operators. Electron. J. Math. Anal. Appl. 2019; 7 : 39–47. 18. Srivastava H, et al. : Faber polynomial coefficient inequalities for bi-Bazilevič functions associated with the Fibonacci-number series and the square-root functions. J. Inequal. Appl. 2024; 2024 (1): 16. Publisher Full Text Comments on this article Comments (0) Version 1 VERSION 1 PUBLISHED 06 Jan 2026 ADD YOUR COMMENT Comment Author details Author details 1 Mathematics, Tikrit University, Tikrit, Saladin Governorate, Iraq 2 Mathematics, University of Anbar, Ramadi, Al Anbar Governorate, Iraq 3 Mathematics, Tikrit University, Tikrit, Saladin Governorate, Iraq Mays S.Abdul Ameer Roles: Writing – Original Draft Preparation Abdul Rahman S.Juma Roles: Writing – Review & Editing Hassan Hussien Ebrahim Roles: Writing – Review & Editing Competing interests No competing interests were disclosed. Grant information The author(s) declared that no grants were involved in supporting this work. Article Versions (1) version 1 Published: 06 Jan 2026, 15:10 https://doi.org/10.12688/f1000research.172490.1 Copyright © 2026 S.Abdul Ameer M et al . This is an open access article distributed under the terms of the Creative Commons Attribution License , which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Download Export To Sciwheel Bibtex EndNote ProCite Ref. Manager (RIS) Sente metrics Views Downloads F1000Research - - PubMed Central info_outline Data from PMC are received and updated monthly. - - Citations open_in_new 0 open_in_new 0 open_in_new SEE MORE DETAILS CITE how to cite this article S.Abdul Ameer M, S.Juma AR and Hussien Ebrahim H. A Bazilevic-type function associated with the class of univalent functions with study Coefficient Bounds and Fekete-Szego inequalities [version 1; peer review: 1 approved, 1 approved with reservations] . F1000Research 2026, 15 :10 ( https://doi.org/10.12688/f1000research.172490.1 ) NOTE: If applicable, it is important to ensure the information in square brackets after the title is included in all citations of this article. COPY CITATION DETAILS track receive updates on this article Track an article to receive email alerts on any updates to this article. TRACK THIS ARTICLE Share Open Peer Review Current Reviewer Status: ? Key to Reviewer Statuses VIEW HIDE Approved The paper is scientifically sound in its current form and only minor, if any, improvements are suggested Approved with reservations A number of small changes, sometimes more significant revisions are required to address specific details and improve the papers academic merit. Not approved Fundamental flaws in the paper seriously undermine the findings and conclusions Version 1 VERSION 1 PUBLISHED 06 Jan 2026 Views 0 Cite How to cite this report: Oros GI. Reviewer Report For: A Bazilevic-type function associated with the class of univalent functions with study Coefficient Bounds and Fekete-Szego inequalities [version 1; peer review: 1 approved, 1 approved with reservations] . F1000Research 2026, 15 :10 ( https://doi.org/10.5256/f1000research.190223.r459388 ) The direct URL for this report is: https://f1000research.com/articles/15-10/v1#referee-response-459388 NOTE: it is important to ensure the information in square brackets after the title is included in this citation. Close Copy Citation Details Reviewer Report 10 Mar 2026 Georgia Irina Oros , University of Oradea, Oradea, Romania Approved VIEWS 0 https://doi.org/10.5256/f1000research.190223.r459388 As the Abstract adequately describes, the paper investigates a specific subclass of Bazilevič-type functions, utilizing differential operators and subordination techniques to establish Fekete–Szegö inequalities and coefficient bounds. The methods used for developing the new results are classical in Geometric Function Theory, ... Continue reading READ ALL As the Abstract adequately describes, the paper investigates a specific subclass of Bazilevič-type functions, utilizing differential operators and subordination techniques to establish Fekete–Szegö inequalities and coefficient bounds. The methods used for developing the new results are classical in Geometric Function Theory, particularly differential subordination, but well used for giving complete and correct proofs for the new results. The motivation of the study is well argued in Introduction. The history of the study is given supported by well chosen and properly cited references. However, it is suggested to add a few references of recent studies from 2025, 2025 on Bazilevič-type functions. The paper is adequately structured and all the necessary information is offered in the Methods part. It is advised to highlight the known results mentioned by adding the reference in brackets, next to the definitions or lemmas. There is no reference given for Lemma 2, 3 and 4. Please indicate where they were taken from. The new outcome is established in the Results and Discussion part. The proofs are complete and correct. This is reinforced by the mention of previously obtained results that are generalized by the original outcome of this study that can be obtained by taking specific values of the parameters involved in the newly proved theorems. The corollaries that follow the new results are given for functions with particular geometric properties which makes those corollaries interesting and inspiring for applications. Overall, the paper is well written and presents interesting new results. Is the work clearly and accurately presented and does it cite the current literature? Yes Is the study design appropriate and is the work technically sound? Yes Are sufficient details of methods and analysis provided to allow replication by others? Yes If applicable, is the statistical analysis and its interpretation appropriate? Not applicable Are all the source data underlying the results available to ensure full reproducibility? Yes Are the conclusions drawn adequately supported by the results? Yes Competing Interests: No competing interests were disclosed. Reviewer Expertise: Geometric Function Theory I confirm that I have read this submission and believe that I have an appropriate level of expertise to confirm that it is of an acceptable scientific standard. Close READ LESS CITE CITE HOW TO CITE THIS REPORT Oros GI. Reviewer Report For: A Bazilevic-type function associated with the class of univalent functions with study Coefficient Bounds and Fekete-Szego inequalities [version 1; peer review: 1 approved, 1 approved with reservations] . F1000Research 2026, 15 :10 ( https://doi.org/10.5256/f1000research.190223.r459388 ) The direct URL for this report is: https://f1000research.com/articles/15-10/v1#referee-response-459388 NOTE: it is important to ensure the information in square brackets after the title is included in all citations of this article. COPY CITATION DETAILS Report a concern Author Response 27 Mar 2026 Mays *Saleh , Mathematics, Tikrit University, Tikrit, Iraq 27 Mar 2026 Author Response We sincerely thank the reviewer for their positive and encouraging evaluation of our manuscript. We are grateful for your confirmation that the work is clearly presented, technically sound, and ... Continue reading We sincerely thank the reviewer for their positive and encouraging evaluation of our manuscript. We are grateful for your confirmation that the work is clearly presented, technically sound, and supported by appropriate methodology and analysis. We also appreciate your acknowledgment of the adequacy of the data, reproducibility, and the validity of the conclusions. Your expertise in Geometric Function Theory and your supportive assessment are highly valued. Thank you again for your time and thoughtful review. We sincerely thank the reviewer for their positive and encouraging evaluation of our manuscript. We are grateful for your confirmation that the work is clearly presented, technically sound, and supported by appropriate methodology and analysis. We also appreciate your acknowledgment of the adequacy of the data, reproducibility, and the validity of the conclusions. Your expertise in Geometric Function Theory and your supportive assessment are highly valued. Thank you again for your time and thoughtful review. Competing Interests: No competing interests were disclosed. Close Report a concern Respond or Comment COMMENTS ON THIS REPORT Author Response 27 Mar 2026 Mays *Saleh , Mathematics, Tikrit University, Tikrit, Iraq 27 Mar 2026 Author Response We sincerely thank the reviewer for their positive and encouraging evaluation of our manuscript. We are grateful for your confirmation that the work is clearly presented, technically sound, and ... Continue reading We sincerely thank the reviewer for their positive and encouraging evaluation of our manuscript. We are grateful for your confirmation that the work is clearly presented, technically sound, and supported by appropriate methodology and analysis. We also appreciate your acknowledgment of the adequacy of the data, reproducibility, and the validity of the conclusions. Your expertise in Geometric Function Theory and your supportive assessment are highly valued. Thank you again for your time and thoughtful review. We sincerely thank the reviewer for their positive and encouraging evaluation of our manuscript. We are grateful for your confirmation that the work is clearly presented, technically sound, and supported by appropriate methodology and analysis. We also appreciate your acknowledgment of the adequacy of the data, reproducibility, and the validity of the conclusions. Your expertise in Geometric Function Theory and your supportive assessment are highly valued. Thank you again for your time and thoughtful review. Competing Interests: No competing interests were disclosed. Close Report a concern COMMENT ON THIS REPORT Views 0 Cite How to cite this report: Murugusundaramoorthy G. Reviewer Report For: A Bazilevic-type function associated with the class of univalent functions with study Coefficient Bounds and Fekete-Szego inequalities [version 1; peer review: 1 approved, 1 approved with reservations] . F1000Research 2026, 15 :10 ( https://doi.org/10.5256/f1000research.190223.r459389 ) The direct URL for this report is: https://f1000research.com/articles/15-10/v1#referee-response-459389 NOTE: it is important to ensure the information in square brackets after the title is included in this citation. Close Copy Citation Details Reviewer Report 09 Mar 2026 Gangadharan Murugusundaramoorthy , Vellore Institute of Technology, Vellore, India Approved with Reservations VIEWS 0 https://doi.org/10.5256/f1000research.190223.r459389 In page 4, after line 5, by applying Equation (5) binomial expansion, we get -------- to be checked and ^\beta to be removed May english corrections Notational mistakes. Certain things are not defined like H and ... Continue reading READ ALL In page 4, after line 5, by applying Equation (5) binomial expansion, we get -------- to be checked and ^\beta to be removed May english corrections Notational mistakes. Certain things are not defined like H and H_1 The linear transformation in (7) is not defined properly. References are given properly in standard style format Results are correct Methodology techniques are okay. The definition of the operator is to be given clearly. After incorporating these points I will recommend this article for indexing. Is the work clearly and accurately presented and does it cite the current literature? Yes Is the study design appropriate and is the work technically sound? Yes Are sufficient details of methods and analysis provided to allow replication by others? Yes If applicable, is the statistical analysis and its interpretation appropriate? Not applicable Are all the source data underlying the results available to ensure full reproducibility? Yes Are the conclusions drawn adequately supported by the results? Yes Competing Interests: No competing interests were disclosed. Reviewer Expertise: COMPLEX ANALYSIS GEOMETRIC FUNCTION THEORY-30C45 I confirm that I have read this submission and believe that I have an appropriate level of expertise to confirm that it is of an acceptable scientific standard, however I have significant reservations, as outlined above. Close READ LESS CITE CITE HOW TO CITE THIS REPORT Murugusundaramoorthy G. Reviewer Report For: A Bazilevic-type function associated with the class of univalent functions with study Coefficient Bounds and Fekete-Szego inequalities [version 1; peer review: 1 approved, 1 approved with reservations] . F1000Research 2026, 15 :10 ( https://doi.org/10.5256/f1000research.190223.r459389 ) The direct URL for this report is: https://f1000research.com/articles/15-10/v1#referee-response-459389 NOTE: it is important to ensure the information in square brackets after the title is included in all citations of this article. COPY CITATION DETAILS Report a concern Respond or Comment COMMENT ON THIS REPORT Comments on this article Comments (0) Version 1 VERSION 1 PUBLISHED 06 Jan 2026 ADD YOUR COMMENT Comment keyboard_arrow_left keyboard_arrow_right Open Peer Review Reviewer Status info_outline Alongside their report, reviewers assign a status to the article: Approved The paper is scientifically sound in its current form and only minor, if any, improvements are suggested Approved with reservations A number of small changes, sometimes more significant revisions are required to address specific details and improve the papers academic merit. Not approved Fundamental flaws in the paper seriously undermine the findings and conclusions Reviewer Reports Invited Reviewers 1 2 Version 1 06 Jan 26 read read Gangadharan Murugusundaramoorthy , Vellore Institute of Technology, Vellore, India Georgia Irina Oros , University of Oradea, Oradea, Romania Comments on this article All Comments (0) Add a comment Sign up for content alerts Sign Up You are now signed up to receive this alert Browse by related subjects keyboard_arrow_left Back to all reports Reviewer Report 0 Views copyright © 2026 Oros G. This is an open access peer review report distributed under the terms of the Creative Commons Attribution License , which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. 10 Mar 2026 | for Version 1 Georgia Irina Oros , University of Oradea, Oradea, Romania 0 Views copyright © 2026 Oros G. This is an open access peer review report distributed under the terms of the Creative Commons Attribution License , which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. format_quote Cite this report speaker_notes Responses (1) Approved info_outline Alongside their report, reviewers assign a status to the article: Approved The paper is scientifically sound in its current form and only minor, if any, improvements are suggested Approved with reservations A number of small changes, sometimes more significant revisions are required to address specific details and improve the papers academic merit. Not approved Fundamental flaws in the paper seriously undermine the findings and conclusions As the Abstract adequately describes, the paper investigates a specific subclass of Bazilevič-type functions, utilizing differential operators and subordination techniques to establish Fekete–Szegö inequalities and coefficient bounds. The methods used for developing the new results are classical in Geometric Function Theory, particularly differential subordination, but well used for giving complete and correct proofs for the new results. The motivation of the study is well argued in Introduction. The history of the study is given supported by well chosen and properly cited references. However, it is suggested to add a few references of recent studies from 2025, 2025 on Bazilevič-type functions. The paper is adequately structured and all the necessary information is offered in the Methods part. It is advised to highlight the known results mentioned by adding the reference in brackets, next to the definitions or lemmas. There is no reference given for Lemma 2, 3 and 4. Please indicate where they were taken from. The new outcome is established in the Results and Discussion part. The proofs are complete and correct. This is reinforced by the mention of previously obtained results that are generalized by the original outcome of this study that can be obtained by taking specific values of the parameters involved in the newly proved theorems. The corollaries that follow the new results are given for functions with particular geometric properties which makes those corollaries interesting and inspiring for applications. Overall, the paper is well written and presents interesting new results. Is the work clearly and accurately presented and does it cite the current literature? Yes Is the study design appropriate and is the work technically sound? Yes Are sufficient details of methods and analysis provided to allow replication by others? Yes If applicable, is the statistical analysis and its interpretation appropriate? Not applicable Are all the source data underlying the results available to ensure full reproducibility? Yes Are the conclusions drawn adequately supported by the results? Yes Competing Interests No competing interests were disclosed. Reviewer Expertise Geometric Function Theory I confirm that I have read this submission and believe that I have an appropriate level of expertise to confirm that it is of an acceptable scientific standard. reply Respond to this report Responses (1) Author Response 27 Mar 2026 Mays *Saleh, Mathematics, Tikrit University, Tikrit, Iraq We sincerely thank the reviewer for their positive and encouraging evaluation of our manuscript. We are grateful for your confirmation that the work is clearly presented, technically sound, and supported by appropriate methodology and analysis. We also appreciate your acknowledgment of the adequacy of the data, reproducibility, and the validity of the conclusions. Your expertise in Geometric Function Theory and your supportive assessment are highly valued. Thank you again for your time and thoughtful review. View more View less Competing Interests No competing interests were disclosed. reply Respond Report a concern Oros GI. Peer Review Report For: A Bazilevic-type function associated with the class of univalent functions with study Coefficient Bounds and Fekete-Szego inequalities [version 1; peer review: 1 approved, 1 approved with reservations] . F1000Research 2026, 15 :10 ( https://doi.org/10.5256/f1000research.190223.r459388) NOTE: it is important to ensure the information in square brackets after the title is included in this citation. The direct URL for this report is: https://f1000research.com/articles/15-10/v1#referee-response-459388 keyboard_arrow_left Back to all reports Reviewer Report 0 Views copyright © 2026 Murugusundaramoorthy G. This is an open access peer review report distributed under the terms of the Creative Commons Attribution License , which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. 09 Mar 2026 | for Version 1 Gangadharan Murugusundaramoorthy , Vellore Institute of Technology, Vellore, India 0 Views copyright © 2026 Murugusundaramoorthy G. This is an open access peer review report distributed under the terms of the Creative Commons Attribution License , which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. format_quote Cite this report speaker_notes Responses (0) Approved With Reservations info_outline Alongside their report, reviewers assign a status to the article: Approved The paper is scientifically sound in its current form and only minor, if any, improvements are suggested Approved with reservations A number of small changes, sometimes more significant revisions are required to address specific details and improve the papers academic merit. Not approved Fundamental flaws in the paper seriously undermine the findings and conclusions In page 4, after line 5, by applying Equation (5) binomial expansion, we get -------- to be checked and ^\beta to be removed May english corrections Notational mistakes. Certain things are not defined like H and H_1 The linear transformation in (7) is not defined properly. References are given properly in standard style format Results are correct Methodology techniques are okay. The definition of the operator is to be given clearly. After incorporating these points I will recommend this article for indexing. Is the work clearly and accurately presented and does it cite the current literature? Yes Is the study design appropriate and is the work technically sound? Yes Are sufficient details of methods and analysis provided to allow replication by others? Yes If applicable, is the statistical analysis and its interpretation appropriate? Not applicable Are all the source data underlying the results available to ensure full reproducibility? Yes Are the conclusions drawn adequately supported by the results? Yes Competing Interests No competing interests were disclosed. Reviewer Expertise COMPLEX ANALYSIS GEOMETRIC FUNCTION THEORY-30C45 I confirm that I have read this submission and believe that I have an appropriate level of expertise to confirm that it is of an acceptable scientific standard, however I have significant reservations, as outlined above. reply Respond to this report Responses (0) Murugusundaramoorthy G. Peer Review Report For: A Bazilevic-type function associated with the class of univalent functions with study Coefficient Bounds and Fekete-Szego inequalities [version 1; peer review: 1 approved, 1 approved with reservations] . F1000Research 2026, 15 :10 ( https://doi.org/10.5256/f1000research.190223.r459389) NOTE: it is important to ensure the information in square brackets after the title is included in this citation. The direct URL for this report is: https://f1000research.com/articles/15-10/v1#referee-response-459389 Alongside their report, reviewers assign a status to the article: Approved - the paper is scientifically sound in its current form and only minor, if any, improvements are suggested Approved with reservations - A number of small changes, sometimes more significant revisions are required to address specific details and improve the papers academic merit. Not approved - fundamental flaws in the paper seriously undermine the findings and conclusions Adjust parameters to alter display View on desktop for interactive features Includes Interactive Elements View on desktop for interactive features Competing Interests Policy Provide sufficient details of any financial or non-financial competing interests to enable users to assess whether your comments might lead a reasonable person to question your impartiality. Consider the following examples, but note that this is not an exhaustive list: Examples of 'Non-Financial Competing Interests' Within the past 4 years, you have held joint grants, published or collaborated with any of the authors of the selected paper. You have a close personal relationship (e.g. parent, spouse, sibling, or domestic partner) with any of the authors. You are a close professional associate of any of the authors (e.g. scientific mentor, recent student). You work at the same institute as any of the authors. You hope/expect to benefit (e.g. favour or employment) as a result of your submission. You are an Editor for the journal in which the article is published. Examples of 'Financial Competing Interests' You expect to receive, or in the past 4 years have received, any of the following from any commercial organisation that may gain financially from your submission: a salary, fees, funding, reimbursements. You expect to receive, or in the past 4 years have received, shared grant support or other funding with any of the authors. You hold, or are currently applying for, any patents or significant stocks/shares relating to the subject matter of the paper you are commenting on. Stay Updated Sign up for content alerts and receive a weekly or monthly email with all newly published articles Register with F1000Research Already registered? Sign in Not now, thanks close PLEASE NOTE If you are an AUTHOR of this article, please check that you signed in with the account associated with this article otherwise we cannot automatically identify your role as an author and your comment will be labelled as a “User Comment”. If you are a REVIEWER of this article, please check that you have signed in with the account associated with this article and then go to your account to submit your report, please do not post your review here. If you do not have access to your original account, please contact us . All commenters must hold a formal affiliation as per our Policies . The information that you give us will be displayed next to your comment. User comments must be in English, comprehensible and relevant to the article under discussion. We reserve the right to remove any comments that we consider to be inappropriate, offensive or otherwise in breach of the User Comment Terms and Conditions . Commenters must not use a comment for personal attacks. When criticisms of the article are based on unpublished data, the data should be made available. I accept the User Comment Terms and Conditions Please confirm that you accept the User Comment Terms and Conditions. Affiliation ✕ refresh Please enter your institution. Note: To add your institution or organisation, start typing the name and then select the correct name from the list. Where applicable, the name will appear in both the original language and in English. Do not paste in the name. If the name does not appear in the drop-down list, we will display the information you have entered. ✕ refresh Country/Region * USA UK Canada China France Germany Afghanistan Aland Islands Albania Algeria American Samoa Andorra Angola Anguilla Antarctica Antigua and Barbuda Argentina Armenia Aruba Australia Austria Azerbaijan Bahamas Bahrain Bangladesh Barbados Belarus Belgium Belize Benin Bermuda Bhutan Bolivia Bosnia and Herzegovina Botswana Bouvet Island Brazil British Indian Ocean Territory British Virgin Islands Brunei Bulgaria Burkina Faso Burundi Cambodia Cameroon Canada Cape Verde Cayman Islands Central African Republic Chad Chile China Christmas Island Cocos (Keeling) Islands Colombia Comoros Congo Cook Islands Costa Rica Cote d'Ivoire Croatia Cuba Cyprus Czech Republic Democratic Republic of the Congo Denmark Djibouti Dominica Dominican Republic Ecuador Egypt El Salvador Equatorial Guinea Eritrea Estonia Ethiopia Falkland Islands Faroe Islands Federated States of Micronesia Fiji Finland France French Guiana French Polynesia French Southern Territories Gabon Georgia Germany Ghana Gibraltar Greece Greenland Grenada Guadeloupe Guam Guatemala Guernsey Guinea Guinea-Bissau Guyana Haiti Heard Island and Mcdonald Islands Holy See (Vatican City State) Honduras Hong Kong Hungary Iceland India Indonesia Iran Iraq Ireland Israel Italy Jamaica Japan Jersey Jordan Kazakhstan Kenya Kiribati Kosovo (Serbia and Montenegro) Kuwait Kyrgyzstan Lao People's Democratic Republic Latvia Lebanon Lesotho Liberia Libya Liechtenstein Lithuania Luxembourg Macao Madagascar Malawi Malaysia Maldives Mali Malta Marshall Islands Martinique Mauritania Mauritius Mayotte Mexico Minor Outlying Islands of the United States Moldova Monaco Mongolia Montenegro Montserrat Morocco Mozambique Myanmar Namibia Nauru Nepal Netherlands Antilles New Caledonia New Zealand Nicaragua Niger Nigeria Niue Norfolk Island North Korea North Macedonia Northern Mariana Islands Norway Oman Pakistan Palau Palestinian Territory Panama Papua New Guinea Paraguay Peru Philippines Pitcairn Poland Portugal Puerto Rico Qatar Reunion Romania Russian Federation Rwanda Saint Helena Saint Kitts and Nevis Saint Lucia Saint Pierre and Miquelon Saint Vincent and the Grenadines Samoa San Marino Sao Tome and Principe Saudi Arabia Senegal Serbia Seychelles Sierra Leone Singapore Slovakia Slovenia Solomon Islands Somalia South Africa South Georgia and the South Sandwich Is South Korea South Sudan Spain Sri Lanka Sudan Suriname Svalbard and Jan Mayen Swaziland Sweden Switzerland Syria Taiwan Tajikistan Tanzania Thailand The Gambia The Netherlands Timor-Leste Togo Tokelau Tonga Trinidad and Tobago Tunisia Turkey Turkmenistan Turks and Caicos Islands Tuvalu UK USA Uganda Ukraine United Arab Emirates United States Virgin Islands Uruguay Uzbekistan Vanuatu Venezuela Vietnam Wallis and Futuna West Bank and Gaza Strip Western Sahara Yemen Zambia Zimbabwe Please select your country/region. You must enter a comment. Competing Interests Please disclose any competing interests that might be construed to influence your judgment of the article's or peer review report's validity or importance. Competing Interests Policy Provide sufficient details of any financial or non-financial competing interests to enable users to assess whether your comments might lead a reasonable person to question your impartiality. Consider the following examples, but note that this is not an exhaustive list: Examples of 'Non-Financial Competing Interests' Within the past 4 years, you have held joint grants, published or collaborated with any of the authors of the selected paper. You have a close personal relationship (e.g. parent, spouse, sibling, or domestic partner) with any of the authors. You are a close professional associate of any of the authors (e.g. scientific mentor, recent student). You work at the same institute as any of the authors. You hope/expect to benefit (e.g. favour or employment) as a result of your submission. You are an Editor for the journal in which the article is published. Examples of 'Financial Competing Interests' You expect to receive, or in the past 4 years have received, any of the following from any commercial organisation that may gain financially from your submission: a salary, fees, funding, reimbursements. You expect to receive, or in the past 4 years have received, shared grant support or other funding with any of the authors. You hold, or are currently applying for, any patents or significant stocks/shares relating to the subject matter of the paper you are commenting on. Please state your competing interests The comment has been saved. An error has occurred. Please try again. Cancel Post var lTitle = "A Bazilevic-type function associated with...".replace("'", ''); var linkedInUrl = "http://www.linkedin.com/shareArticle?url=https://f1000research.com/articles/15-10/v1" + "&title=" + encodeURIComponent(lTitle) + "&summary=" + encodeURIComponent('Read the article by '); var deliciousUrl = "https://del.icio.us/post?url=https://f1000research.com/articles/15-10/v1&title=" + encodeURIComponent(lTitle); var redditUrl = "http://reddit.com/submit?url=https://f1000research.com/articles/15-10/v1" + "&title=" + encodeURIComponent(lTitle); linkedInUrl += encodeURIComponent('S.Abdul Ameer M et al.'); var offsetTop = /chrome/i.test( navigator.userAgent ) ? 4 : -10; var addthis_config = { ui_offset_top: offsetTop, services_compact : "facebook,twitter,www.linkedin.com,www.mendeley.com,reddit.com", services_expanded : "facebook,twitter,www.linkedin.com,www.mendeley.com,reddit.com", services_custom : [ { name: "LinkedIn", url: linkedInUrl, icon:"/img/icon/at_linkedin.svg" }, { name: "Mendeley", url: "http://www.mendeley.com/import/?url=https://f1000research.com/articles/15-10/v1/mendeley", icon:"/img/icon/at_mendeley.svg" }, { name: "Reddit", url: redditUrl, icon:"/img/icon/at_reddit.svg" }, ] }; var addthis_share = { url: "https://f1000research.com/articles/15-10", templates : { twitter : "A Bazilevic-type function associated with the class of univalent.... S.Abdul Ameer M et al., published by " + "@F1000Research" + ", https://f1000research.com/articles/15-10/v1" } }; if (typeof(addthis) != "undefined"){ addthis.addEventListener('addthis.ready', checkCount); addthis.addEventListener('addthis.menu.share', checkCount); } $(".f1r-shares-twitter").attr("href", "https://twitter.com/intent/tweet?text=" + addthis_share.templates.twitter); $(".f1r-shares-facebook").attr("href", "https://www.facebook.com/sharer/sharer.php?u=" + addthis_share.url); $(".f1r-shares-linkedin").attr("href", addthis_config.services_custom[0].url); $(".f1r-shares-reddit").attr("href", addthis_config.services_custom[2].url); $(".f1r-shares-mendelay").attr("href", addthis_config.services_custom[1].url); function checkCount(){ setTimeout(function(){ $(".addthis_button_expanded").each(function(){ var count = $(this).text(); if (count !== "" && count != "0") $(this).removeClass("is-hidden"); else $(this).addClass("is-hidden"); }); }, 1000); } close How to cite this report {{reportCitation}} Cancel Copy Citation Details $(function(){R.ui.buttonDropdowns('.dropdown-for-downloads');}); $(function(){R.ui.toolbarDropdowns('.toolbar-dropdown-for-downloads');}); $.get("/articles/acj/172490/190223") new F1000.Clipboard(); new F1000.ThesaurusTermsDisplay("articles", "article", "190223"); $(document).ready(function() { $( "#frame1" ).on('load', function() { var mydiv = $(this).contents().find("div"); var h = mydiv.height(); console.log(h) }); var tooltipLivingFigure = jQuery(".interactive-living-figure-label .icon-more-info"), titleLivingFigure = tooltipLivingFigure.attr("title"); tooltipLivingFigure.simpletip({ fixed: true, position: ["-115", "30"], baseClass: 'small-tooltip', content:titleLivingFigure + " " }); tooltipLivingFigure.removeAttr("title"); $("body").on("click", ".cite-living-figure", function(e) { e.preventDefault(); var ref = $(this).attr("data-ref"); $(this).closest(".living-figure-list-container").find("#" + ref).fadeIn(200); }); $("body").on("click", ".close-cite-living-figure", function(e) { e.preventDefault(); $(this).closest(".popup-window-wrapper").fadeOut(200); }); $(document).on("mouseup", function(e) { var metricsContainer = $(".article-metrics-popover-wrapper"); if (!metricsContainer.is(e.target) && metricsContainer.has(e.target).length === 0) { $(".article-metrics-close-button").click(); } }); var articleId = $('#articleId').val(); if($("#main-article-count-box").attachArticleMetrics) { $("#main-article-count-box").attachArticleMetrics(articleId, { articleMetricsView: true }); } }); var figshareWidget = $(".new_figshare_widget"); if (figshareWidget.length > 0) { window.figshare.load("f1000", function(Widget) { // Select a tag/tags defined in your page. In this tag we will place the widget. _.map(figshareWidget, function(el){ var widget = new Widget({ articleId: $(el).attr("figshare_articleId") //height:300 // this is the height of the viewer part. [Default: 550] }); widget.initialize(); // initialize the widget widget.mount(el); // mount it in a tag that's on your page // this will save the widget on the global scope for later use from // your JS scripts. This line is optional. //window.widget = widget; }); }); } close Error Close Add Reset F1000.MICROSERVICES.AFFILIATION = ''; $(document).ready(function () { $('.js-affiliations-form').each((index, form) => { new AffiliationForm({ formId: form.id, institutionErrorSelector: '.comment-enter-institution', departmentErrorSelector: '.comment-enter-department', placeSelector: '.js-add-comment-place', stateSelector: '.js-add-comment-state', zipCodeSelector: '.js-add-comment-zipcode', countrySelector: '.js-add-comment-country', countryErrorSelector: '.comment-enter-country', }); }); }); $(document).ready(function () { var reportIds = { "459383": 0, "448310": 0, "459382": 0, "448311": 0, "459381": 0, "459380": 0, "448309": 0, "448318": 0, "459389": 13, "448316": 0, "459388": 10, "448317": 0, "459387": 0, "448314": 0, "459386": 0, "448315": 0, "459385": 0, "448312": 0, "459384": 0, "448313": 0, }; $(".referee-response-container,.js-referee-report").each(function(index, el) { var reportId = $(el).attr("data-reportid"), reportCount = reportIds[reportId] || 0; $(el).find(".comments-count-container,.js-referee-report-views").html(reportCount); }); var uuidInput = $("#article_uuid"), oldUUId = uuidInput.val(), newUUId = "d1c88231-a48e-4601-8ecc-9d10e26051e4"; uuidInput.val(newUUId); $("a[href*='article_uuid=']").each(function(index, el) { var newHref = $(el).attr("href").replace(oldUUId, newUUId); $(el).attr("href", newHref); }); }); An innovative open access publishing platform offering rapid publication and open peer review, whilst supporting data deposition and sharing. Browse Gateways Collections How it Works Contact For Developers Cookie Notice Privacy Notice RSS Submit Your Research Follow us © 2012-2026 F1000 Research Ltd. ISSN 2046-1402 | Legal | Partner of Research4Life • CrossRef • ORCID • FAIRSharing R.templateTests.simpleTemplate = R.template(' $text $text $text $text $text '); R.templateTests.runTests(); var F1000platform = new F1000.Platform({ name: "f1000research", displayName: "F1000Research", hostName: "f1000research.com", id: "1", editorialEmail: "[email protected]", infoEmail: "[email protected]", usePmcStats: true }); $(function(){R.ui.dropdowns('.dropdown-for-authors, .dropdown-for-about, .dropdown-for-myresearch');}); // $(function(){R.ui.dropdowns('.dropdown-for-referees');}); $(document).ready(function () { if ($(".cookie-warning").is(":visible")) { $(".sticky").css("margin-bottom", "35px"); $(".devices").addClass("devices-and-cookie-warning"); } $(".cookie-warning .close-button").click(function (e) { $(".devices").removeClass("devices-and-cookie-warning"); $(".sticky").css("margin-bottom", "0"); }); $("#tweeter-feed .tweet-message").each(function (i, message) { var self = $(message); self.html(linkify(self.html())); }); $(".partner").on("mouseenter mouseleave", function() { $(this).find(".gray-scale, .colour").toggleClass("is-hidden"); }); }); Sign In Remember me Forgotten your password? Sign In Cancel Email or password not correct. Please try again Please wait... $(function(){ // Note: All the setup needs to run against a name attribute and *not* the id due the clonish // nature of facebox... $("a[id=googleSignInButton]").click(function(event){ event.preventDefault(); $("input[id=oAuthSystem]").val("GOOGLE"); $("form[id=oAuthForm]").submit(); }); $("a[id=facebookSignInButton]").click(function(event){ event.preventDefault(); $("input[id=oAuthSystem]").val("FACEBOOK"); $("form[id=oAuthForm]").submit(); }); $("a[id=orcidSignInButton]").click(function(event){ event.preventDefault(); $("input[id=oAuthSystem]").val("ORCID"); $("form[id=oAuthForm]").submit(); }); }); If you've forgotten your password, please enter your email address below and we'll send you instructions on how to reset your password. The email address should be the one you originally registered with F1000. Email address not valid, please try again You registered with F1000 via Google, so we cannot reset your password. To sign in, please click here . If you still need help with your Google account password, please click here . You registered with F1000 via Facebook, so we cannot reset your password. To sign in, please click here . If you still need help with your Facebook account password, please click here . Code not correct, please try again Reset password Cancel Email us for further assistance. Server error, please try again. If your email address is registered with us, we will email you instructions to reset your password. If you think you should have received this email but it has not arrived, please check your spam filters and/or contact for further assistance. Please wait... Register $(document).ready(function () { signIn.createSignInAsRow($("#sign-in-form-gfb-popup")); $(".target-field").each(function () { var uris = $(this).val().split("/"); if (uris.pop() === "login") { $(this).val(uris.toString().replace(",","/")); } }); });