Study analytical function subordination properties by applying a novel linear operator

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Majel" }, { "@type": "Person", "name": "Mustafa I. Hameed" } ], "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 study of theory for analytic univalent and multivalent functions is an old subject in mathematics, particularly in complex analysis, that has captivated a great deal of scholars owing to the sheer sophistication of its geometrical features as well as its many research possibilities. The study of univalent functions is one of many significant elements of complex analysis for both single and multiple variables. Investigators have become keen on the conventional investigation of this topic since at least 1907. Numerous scholars in the area of complex analysis have emerged since then, including Euler, Gauss, Riemann, Cauchy, and other people. Geometric function theory combines geometry and analysis. Methods This study employs the differential subordination technique to derive multiple characteristics from the new linear operator M σ , μ n , ς Υ ( s ) . The concept of the differential subordination subclass of analytical univalent functions is analyzed. Results In this section, We studied some results on differential subordination and superordination using a specific class of univalent functions stated on a specific space of univalent functions stated on the open unit disc. Using properties of the operator, we discovered a number of properties of superordinations and subordinations related to the idea of the Hadamard product. We investigated several aspects of superordinations and subordinations using a new operator M σ , μ n , ς Υ ( s ) . Conclusions A new operator M σ , μ n , ς Υ ( s ) : Λ ⟶ Λ has been established in this paper connected to the Dziok-Srivastava operator T σ n and the Hadamard product corresponding to the Komatu integral operator Ω μ ς . The difference operator M σ , μ n , ς ϒ ( s ) can have specific properties derived by applying the differential subordination technique. And the objective of this paper is to make use of the connection ( β 1 μ + 1 ) M σ , μ n + 1 , ς ϒ ( s ) = w ( M σ , μ n , ς ϒ ( s ) ) ′ + β 1 μ ( M σ , μ n , ς ϒ ( s ) ) . 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F1000Research 2025, 14 :1479 ( https://doi.org/10.12688/f1000research.174492.1 ) 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 Study analytical function subordination properties by applying a novel linear operator [version 1; peer review: 2 approved] Maryam S. Majel https://orcid.org/0009-0006-4216-7847 1,2 , Mustafa I. Hameed https://orcid.org/0000-0003-0726-3180 1,2 Maryam S. Majel https://orcid.org/0009-0006-4216-7847 1,2 , Mustafa I. Hameed https://orcid.org/0000-0003-0726-3180 1,2 PUBLISHED 31 Dec 2025 Author details Author details 1 University of Anbar, Ramadi, Al Anbar Governorate, Iraq 2 University of Anbar, Ramadi, Al Anbar Governorate, Iraq Maryam S. Majel Roles: Data Curation, Formal Analysis, Investigation, Methodology, Writing – Original Draft Preparation Mustafa I. Hameed Roles: Project Administration, Supervision, Writing – Original Draft Preparation, Writing – Review & Editing OPEN PEER REVIEW DETAILS REVIEWER STATUS This article is included in the Fallujah Multidisciplinary Science and Innovation gateway. Abstract Background The study of theory for analytic univalent and multivalent functions is an old subject in mathematics, particularly in complex analysis, that has captivated a great deal of scholars owing to the sheer sophistication of its geometrical features as well as its many research possibilities. The study of univalent functions is one of many significant elements of complex analysis for both single and multiple variables. Investigators have become keen on the conventional investigation of this topic since at least 1907. Numerous scholars in the area of complex analysis have emerged since then, including Euler, Gauss, Riemann, Cauchy, and other people. Geometric function theory combines geometry and analysis. Methods This study employs the differential subordination technique to derive multiple characteristics from the new linear operator M σ , μ n , ς Υ ( s ) . The concept of the differential subordination subclass of analytical univalent functions is analyzed. Results In this section, We studied some results on differential subordination and superordination using a specific class of univalent functions stated on a specific space of univalent functions stated on the open unit disc. Using properties of the operator, we discovered a number of properties of superordinations and subordinations related to the idea of the Hadamard product. We investigated several aspects of superordinations and subordinations using a new operator M σ , μ n , ς Υ ( s ) . Conclusions A new operator M σ , μ n , ς Υ ( s ) : Λ ⟶ Λ has been established in this paper connected to the Dziok-Srivastava operator T σ n and the Hadamard product corresponding to the Komatu integral operator Ω μ ς . The difference operator M σ , μ n , ς ϒ ( s ) can have specific properties derived by applying the differential subordination technique. And the objective of this paper is to make use of the connection ( β 1 μ + 1 ) M σ , μ n + 1 , ς ϒ ( s ) = w ( M σ , μ n , ς ϒ ( s ) ) ′ + β 1 μ ( M σ , μ n , ς ϒ ( s ) ) . READ ALL READ LESS Keywords Univalent Function, Best Dominant, Derivative Operator, Differential Subordination, Convex Function, Hadamard Product. Corresponding Author(s) Mustafa I. Hameed ( [email protected] ) Close Corresponding author: Mustafa I. Hameed Competing interests: No competing interests were disclosed. Grant information: The author(s) declared that no grants were involved in supporting this work. Copyright: © 2025 Majel MS and Hameed MI. 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: Majel MS and Hameed MI. Study analytical function subordination properties by applying a novel linear operator [version 1; peer review: 2 approved] . F1000Research 2025, 14 :1479 ( https://doi.org/10.12688/f1000research.174492.1 ) First published: 31 Dec 2025, 14 :1479 ( https://doi.org/10.12688/f1000research.174492.1 ) Latest published: 31 Dec 2025, 14 :1479 ( https://doi.org/10.12688/f1000research.174492.1 ) 1. Introduction Gronwall’s Area Theorem, published in 1914, was a significant contribution to the theory of univalent functions. It is used for getting bounds on the coefficient values for meromorphic functions. Bieberbach resolved a similar issue regarding the class in 1916, as well as his known conjecture, which was mentioned in the exact same year but not applied until 1984, influenced the development of different methods in the geometric theory of complex variable functions. Estimates on the first two Taylor-Maclaurin coefficients are common in the study of bi-univalent functions since they are in the classes investigated by Gronwall and Bieberbach. Littlewood 1 is the source of the fundamental findings on subordination that Rogosinski 2 created and established. Subordination was recently utilized by Srivastava and Owa 3 to investigate the fascinating properties of the generalized hypergeometric function. A paper on differential subordinations, a generalization of differential disparities, was written by Miller and Mocanu. 4 The article is mainly concerned with the differential superordination of univalent functions in an open unit disk. To determine subordination properties, utilize the capabilities of the recently introduced operator M σ , μ n , ς ϒ ( s ) . Miller as well as Mocanu created the differential subordinations method in 1978, and the theory started to take shape in 1981. Refer to 5 – 9 for further information. Suppose Λ indicates the type of functions of the given form (1) ϒ ( s ) = s + ∑ η = 2 ∞ e η s η , e η ≥ 0 , is analytic function in unit disk Δ ∗ = { s : s ∈ ℂ , | s | < 1 } . Allow ϕ in Λ be provided by (2) ϕ ( s ) = s + ∑ η = 2 ∞ d η s η , d η ≥ 0 . The following is the Hadamard product of ϒ and ϕ : (3) ( ϒ ∗ ϕ ) ( s ) = z + ∑ η = 2 ∞ e η d η s η , e η d η ≥ 0 . When ϒ and ϕ are analytic in Δ ∗ , the function ϒ is considered subservient to ϕ , expressed as ϒ ( s ) ≺ ϕ ( s ) ( s ∈ Δ ∗ ) . The previously Schwarz function ω is present in Δ ∗ if and only if | w ( ω ) | < 1 as well as ω ( 0 ) = 0 , if and only if there exists the Schwarz function ω , in Δ ∗ , with ω ( 0 ) = 0 and | w ( ω ) | < 1 such that ϒ ( s ) = ϕ ( ω ( s ) ) , ( s ∈ Δ ∗ ) . Additionally, we’re left with the following equivalency if ϕ is univalent in Δ ∗ by, 10 – 13 : ϒ ≺ ϕ if and only if ϒ ( 0 ) = ϕ ( 0 ) and ϒ ( Δ ∗ ) ⊂ ϕ ( Δ ∗ ) . 2. Definitions and Lemmas Definition 2.1. 14 The definition of the Komatu Integral operator Ω μ ς : Λ → Λ for ϒ ∈ Λ is (4) Ω μ ς ( s ) = s + ∑ η = 2 ∞ ( μ + η − 1 μ ) − ς s n , ( μ > 0 ; ς ≥ 0 ) . Definition 2.2. 15 The definition of the Dziok-Srivastava operator Ω μ ς : Λ → Λ for ϒ ∈ Λ is (5) T σ n ( β 1 , β 2 , . . , β n ; π 1 , π 2 . . , π σ ) ϒ ( s ) = s + ∑ η = 2 ∞ ( β 1 ) η − 1 ( β 2 ) η − 1 … ( β n ) η − 1 ( π 1 ) η − 1 ( π 2 ) η − 1 … ( π σ ) η − 1 ( μ + η − 1 μ ) e η s η , where β n ∈ ℂ , n = 1 , 2 , 3 , … , n , π n ∈ ℂ ∖ { 0 , − 1 , − 2 , . . } , n = 1 , 2 , . . , σ . Definition 2.3. If a function ϒ ( s ) ∈ Λ and Hadamard products between operator T σ n as well as operator Ω μ ς given a new linear operator M σ , μ n , ς ϒ ( s ) : Λ → Λ , define (6) M σ , μ n , ς ϒ ( s ) = T σ n ϒ ( s ) ∗ Ω μ ς ( s ) , ( s ∈ Δ ∗ ) , and M σ , μ n , ς ϒ ( s ) = s + ∑ η = 2 ∞ ( β 1 μ ς ) η − 1 ( β 2 ) η − 1 … ( β n ) η − 1 ( π 1 ) η − 1 ( π 2 ) η − 1 … ( π σ ) η − 1 ( 1 μ + η − 1 ) − ς + 1 e η s η . Form Eq. 6 , we have (7) ( β 1 μ + 1 ) M σ , μ n + 1 , ς ϒ ( s ) = w ( M σ , μ n , ς ϒ ( s ) ) ′ + β 1 μ ( M σ , μ n , ς ϒ ( s ) ) , ( s ∈ Δ ∗ ) It should be noted that M σ , μ n , ς ϒ has the following special cases. a) The Komatu Integral operator Ω μ ς should be included if n = 1 , σ = 1 by. 14 b) Add the Dziok-Srivastava operator T σ n if ς = 0 15 , 16 c) Salagean 17 examined the case if μ = 1 , ς = − n , n = 1 , σ = 1 . d) It was examined by Salagean 17 and Flett 18 if μ = 1 , ς = n , n = 1 , σ = 1 . Definition 2.4. If σ ∈ ℕ , 0 ≤ π 0 ; n , ς ≥ 0 , then let R σ , μ n , ς ( π ) denote the class of a function ϒ ∈ Λ that satisfies the given conditions. (8) R ( M σ , μ n , ς ϒ ( s ) ) ′ > π , ( s ∈ Δ ∗ ) . Lemma 2.1. 19 Assume that ϒ is in Λ , if R ( 1 + s ϒ ′ ′ ( s ) ϒ ′ ( s ) ) > − 1 2 , then 2 s ∫ 0 s ϒ ( τ ) dτ , ( s ∈ Δ ∗ and s ≠ 0 ) , is a convex functions. Lemma 2.2. 20 Let ϕ be a convex function in Δ ∗ such that ψ ( s ) = ϕ ( s ) + ηβs ϕ ′ ( s ) , ( s ∈ Δ ∗ ) , in which β > 0 and η ∈ ℕ . If ϖ ( s ) = ϕ ( 0 ) + ϖ η s η + ϖ η + 1 s η + 1 + … , is analytic in Δ ∗ and ϖ ( s ) + βs ϖ ′ ( s ) ≺ ψ ( s ) , then ϖ ( s ) ≺ ϕ ( s ) . Lemma 2.3. 19 Consider the following ψ ( 0 ) = e , ξ is analytic, univalent, and convex in Δ , γ ∈ ℂ ∖ { 0 } is a complex number that produces R { γ } ≥ 0 . Given ϖ ∈ Η [ e , η ] as well as ϖ ( s ) + s ϖ ′ ( s ) γ ≺ ψ ( s ) , then ϖ ( s ) ≺ ξ ( s ) ≺ ψ ( s ) , ( s ∈ Δ ∗ ) , where ξ ( s ) = γ η s γ η ∫ 0 s ψ ( τ ) τ γ η − 1 dτ , ( s ∈ Δ ∗ ) . The optimal prevailing of the subordination is the convex function ξ . 3. Results and Discussion In this study, we will derive multiple characteristics derived from the new linear operator M σ , μ n , ς ϒ ( s ) using differential subordination technique. Theorem 3.1. Let ψ ( 0 ) = 1 , 0 ≤ π 0 ; n , ς ≥ 0 as well as ϒ ∈ A , then (9) ( M σ , μ n , ς ϒ ( s ) ) ′ ≺ ξ ( s ) = ( 3 2 π − 1 ) + 3 / 2 ( β 1 μ + 1 ) ( 1 − π ) η s ( β 1 μ + 1 ) / η ζ ( ( β 1 μ + 1 ) / η ) , where ζ ( x ) = ∫ 0 s τ x − 1 1 + τ dτ , ( s ∈ Δ ∗ ) . Proof. The result of differentiation Eq. (7) is (10) ( β 1 μ + 1 ) ( M σ , μ n + 1 , ς ϒ ( s ) ) ′ = ( β 1 μ + 1 ) ( M σ , μ n , ς ϒ ( s ) ) ′ + s ( M σ , μ n , ς ϒ ( s ) ) ′ ′ ( M σ , μ n + 1 , ς ϒ ( s ) ) ′ = ( β 1 μ + 1 ) ( M σ , μ n , ς ϒ ( s ) ) ′ + s ( M σ , μ n , ς ϒ ( s ) ) ′ ′ ( β 1 μ + 1 ) ( M σ , μ n + 1 , ς ϒ ( s ) ) ′ = ( M σ , μ n , ς ϒ ( s ) ) ′ + s ( M σ , μ n , ς ϒ ( s ) ) ′ ′ ( β 1 μ + 1 ) . When Eq. (10) is used in Eq. (9) , the subordination Eq. (9) is transformed into (11) ( M σ , μ n , ς ϒ ( s ) ) ′ + s ( M σ , μ n , ς ϒ ( s ) ) ′ ′ ( β 1 μ + 1 ) ≺ ψ ( s ) = 1 + ( 3 2 π − 1 ) 1 + s . Let (12) ϖ ( s ) = ( M σ , μ n , ς ϒ ( s ) ) ′ = ( s + ∑ η = 2 ∞ ( β 1 μ ς ) η − 1 ( β 2 ) η − 1 … ( β n ) η − 1 ( π 1 ) η − 1 ( π 2 ) η − 1 … ( π σ ) η − 1 ( 1 μ + η − 1 ) − ς + 1 e η s η ) ′ , ϖ ( s ) = 1 + ϖ 1 s + ϖ 2 s 2 + … , ϖ ∈ Η [ 1 , 1 ] , s ∈ Δ ∗ . Subordination is made possible in Eq. (11) , through the use of Eq. (12) . ϖ ( s ) + s ϖ ′ ( s ) ( β 1 μ + 1 ) ≺ ψ ( s ) = 1 + ( 3 / 2 π − 1 ) 1 + s . Employing Lemma 2.3 , has been ϖ ( s ) ≺ ξ ( s ) = ( β 1 μ + 1 ) η s ( β 1 μ + 1 ) ∕ η ∫ 0 s ψ ( τ ) τ ( ( β 1 μ + 1 ) / η ) − 1 dτ = ( β 1 μ + 1 ) η s ( β 1 μ + 1 ) ∕ η ∫ 0 s [ ( 3 / 2 π − 1 ) + 3 / 2 ( 1 − π ) 1 1 + τ ] τ ( ( β 1 μ + 1 ) / η ) − 1 1 + τ dτ = ( β 1 μ + 1 ) η s ( β 1 μ + 1 ) / η ∫ 0 s ( 3 / 2 π − 1 ) τ ( ( β 1 μ + 1 ) / η ) − 1 dτ + 3 / 2 ( 1 − π ) ( β 1 μ + 1 ) η s ( β 1 μ + 1 ) ∕ η ∫ 0 s τ ( ( β 1 μ + 1 ) / η ) − 1 1 + τ dτ = ( 3 / 2 π − 1 ) + 3 / 2 ( β 1 μ + 1 ) ( 1 − π ) η s ( β 1 μ + 1 ) ∕ η ζ ( ( β 1 μ + 1 ) ∕ η ) , then ( M σ , μ n , ς ϒ ( s ) ) ′ ≺ ξ ( s ) = ( 3 / 2 π − 1 ) + 3 / 2 ( β 1 μ + 1 ) ( 1 − π ) η s ( β 1 μ + 1 ) / η ζ ( ( β 1 μ + 1 ) / η ) . Theorem 3.2. Given that ξ is a convex function in Δ ∗ with ξ ( 0 ) = 1 and ψ ( s ) = ξ ( s ) + s ξ ′ ( s ) . If ϒ ∈ Λ , σ ∈ N , μ > 0 ; n , ς ≥ 0 , 0 ≤ π < 1 satisfies the subordination ( M σ , μ n , ς ϒ ( s ) ) ′ ≺ ψ ( s ) , then (13) M σ , μ n , ς ϒ ( s ) s ≺ ξ ( s ) , ( s ∈ Δ ∗ ) . Proof. Let ϖ ( s ) = M σ , μ n , ς ϒ ( s ) s ϖ ( s ) = s + ∑ η = 2 ∞ ( β 1 μ ς ) η − 1 ( β 2 ) η − 1 … ( β n ) η − 1 ( π 1 ) η − 1 ( π 2 ) η − 1 … ( π σ ) η − 1 ( 1 μ + η − 1 ) − ς + 1 e η s η s , ϖ ( s ) = 1 + ∑ η = 2 ∞ ( β 1 μ ς ) η − 1 ( β 2 ) η − 1 … ( β n ) η − 1 ( π 1 ) η − 1 ( π 2 ) η − 1 … ( π σ ) η − 1 ( 1 μ + η − 1 ) − ς + 1 e η s η − 1 ϖ ( s ) = 1 + ϖ η s η + ϖ η + 1 s η + 1 + … ( M σ , μ n , ς ϒ ( s ) ) ′ = ϖ ( s ) + s ϖ ′ ( s ) . Consequently, with the help of the connection Eq. (13) grows ϖ ( s ) + s ϖ ′ ( s ) ≺ ψ ( s ) = ξ ( s ) + s ξ ′ ( s ) . Utilizing Lemma 2.2 , previously ϖ ( s ) ≺ ξ ( s ) . Through the use of ϖ ( s ) = M σ , μ n , ς ϒ ( s ) s , we obtain M σ , μ n , ς ϒ ( s ) s ≺ ξ ( s ) . Theorem 3.3. If σ ∈ N , μ > 0 ; n , ς ≥ 0 as well as 0 ≤ π < 1 , then R σ , μ n + 1 , ς ( π ) ⊂ R σ , μ n , ς ( β ) such that (14) ε = ( 3 / 2 π − 1 ) + 3 / 2 ( 1 − π ) ( β 1 μ + 1 ) η ζ ( ( β 1 μ + 1 ) / η ) , ζ = ∫ 0 s τ x − 1 1 + τ dτ . Proof. Assume ϒ is in ϒ ∈ R σ , μ n + 1 , ς ( π ) . Next, based on Eq. (8) , there is R ( M σ , μ n , ς ϒ ( s ) ) ′ > π this is equivalent to ( M σ , μ n + 1 , ς ϒ ( s ) ) ′ ≺ ψ ( s ) = 1 + ( 3 / 2 π − 1 ) s 1 + s . Applying Theorem 3.1 , we arrive at ( M σ , μ n , ς ϒ ( s ) ) ′ ≺ ξ ( s ) = ( 3 / 2 π − 1 ) + 3 / 2 ( β 1 μ + 1 ) ( 1 − π ) η s ( β 1 μ + 1 ) / η ζ ( ( β 1 μ + 1 ) / η ) . Since ξ ( Δ ∗ ) is symmetric when compared to the real direction and ξ is convex, we are able to deduce that Re ( M σ , μ n , ς ϒ ( s ) ) ′ > Re ξ ( 1 ) = ε = ε ( π , β 1 , μ , η ) = ( 3 / 2 π − 1 ) + 3 / 2 ( β 1 μ + 1 ) ( 1 − π ) η ζ ( ( β 1 μ + 1 ) / η ) , where ξ ( 1 ) = ( β 1 μ + 1 ) η ∫ 0 1 [ 1 + ( 3 / 2 π − 1 ) τ 1 + τ ] τ ( ( β 1 μ + 1 ) / η ) − 1 dτ = ( β 1 μ + 1 ) η ∫ 0 1 [ ( 3 / 2 π − 1 ) + 3 / 2 ( 1 − π ) 1 1 + τ ] τ ( ( β 1 μ + 1 ) / η ) − 1 1 + τ dτ = ( β 1 μ + 1 ) η ∫ 0 1 ( 3 / 2 π − 1 ) τ ( ( β 1 μ + 1 ) / η ) − 1 dτ + 3 / 2 ( 1 − π ) ( β 1 μ + 1 ) η ∫ 0 1 τ ( ( β 1 μ + 1 ) / η ) − 1 1 + τ dτ = ( 3 / 2 π − 1 ) + 3 / 2 ( β 1 μ + 1 ) ( 1 − π ) η ζ ( ( β 1 μ + 1 ) / η ) It leads us to the conclusion that R σ , μ n + 1 , ς ( π ) ⊂ R σ , μ n , ς ( β ) . The evidence is finished. Theorem 3.4. Let ψ be a convex function in Δ ∗ , ψ ( 0 ) = 1 , 0 ≤ π 0 ; n , ς ≥ 0 , ϒ ∈ Λ satisfies the subordination ( M σ , μ n , ς ϒ ( s ) ) ′ ≺ ψ ( s ) , then M σ , μ n , ς ϒ ( s ) s ≺ ξ ( s ) = ( 3 / 2 π − 1 ) + 3 / 2 ( 1 − π ) ln ( 1 + s ) s , ( s ∈ Δ ∗ ) . The function ξ is convex as well as the most prevailing. Proof. Suppose (15) ϖ ( s ) = M σ , μ n , ς ϒ ( s ) s = s + ∑ η = 2 ∞ ( β 1 μ ς ) η − 1 ( β 2 ) η − 1 … ( β n ) η − 1 ( π 1 ) η − 1 ( π 2 ) η − 1 … ( π σ ) η − 1 ( 1 μ + η − 1 ) − ς + 1 e η s η s , = ϖ ( s ) = 1 + ϖ η s η + ϖ η + 1 s η + 1 + … . By Eq. (15) in relation to ϖ , we get (16) ( M σ , μ n , ς ϒ ( s ) ) ′ = ϖ ( s ) + s ϖ ′ ( s ) , ( s ∈ Δ ∗ ) . By applying Eq. (16) , differential subordination ( M σ , μ n , ς ϒ ( s ) ) ′ ≺ ψ ( s ) , is obtained ϖ ( s ) + s ϖ ′ ( s ) ≺ ψ ( s ) = 1 + ( 3 / 2 π − 1 ) s 1 + s , ( s ∈ Δ ∗ ) . Utilizing Lemma 2.3 , has been ϖ ( s ) ≺ ξ ( s ) = 1 s ∫ 0 s ψ ( τ ) dτ = 1 s ∫ 0 s ( 1 + ( 3 / 2 π − 1 ) τ 1 + τ ) dτ = 1 s ∫ 0 s [ ( 3 / 2 π − 1 ) + 3 / 2 ( 1 − π ) 1 1 + τ ] dτ = 1 s ∫ 0 s ( 3 / 2 π − 1 ) dτ + 3 / 2 ( 1 − π ) s ∫ 0 s 1 1 + τ dτ = ( 3 / 2 π − 1 ) + 3 / 2 ( 1 − π ) ln ( 1 + s ) s . With the help of Eq. (15) , we had M σ , μ n , ς ϒ ( s ) s ≺ ξ ( s ) = ( 3 / 2 π − 1 ) + 3 / 2 ( 1 − π ) ln ( 1 + s ) s . Theorem 3.4 , has been fully proved. Theorem 3.5. If ψ is convex function in Δ ∗ and ψ ( 0 ) = 1 , 0 ≤ π 0 ; n , ς ≥ 0 satisfies the subordination (17) 1 − M σ , μ n , ς ϒ ( s ) ( M σ , μ n , ς ϒ ( s ) ) ′ ′ [ ( M σ , μ n , ς ϒ ( s ) ) ′ ] 2 ≺ ψ ( s ) , then M σ , μ n , ς ϒ ( s ) s ( M σ , μ n , ς ϒ ( s ) ) ′ ≺ ξ ( s ) = ( 3 / 2 π − 1 ) + 3 / 2 ( 1 − π ) ln ( 1 + s ) s . Proof. Suppose ϖ ( s ) = M σ , μ n , ς ϒ ( s ) s ( M σ , μ n , ς ϒ ( s ) ) ′ , ϖ ( s ) = s + ∑ η = 2 ∞ ( β 1 μ ς ) η − 1 ( β 2 ) η − 1 … ( β n ) η − 1 ( π 1 ) η − 1 ( π 2 ) η − 1 … ( π σ ) η − 1 ( 1 μ + η − 1 ) − ς + 1 e η s η s ( 1 + ∑ η = 2 ∞ ( β 1 μ ς ) η − 1 ( β 2 ) η − 1 … ( β n ) η − 1 ( π 1 ) η − 1 ( π 2 ) η − 1 … ( π σ ) η − 1 ( 1 μ + η − 1 ) − ς + 1 e η s η − 1 ) , so that with the help of Eq. (17) becomes ϖ ( s ) + s ϖ ′ ( s ) ≺ ψ ( s ) = 1 + ( 3 / 2 π − 1 ) s 1 + s , ( s ∈ Δ ∗ ) . By Lemma 2.3 allows us to have ϖ ( s ) ≺ ξ ( s ) = 1 s ∫ 0 s ψ ( τ ) dτ = 1 s ∫ 0 s ( 1 + ( 3 / 2 π − 1 ) τ 1 + τ ) dτ = 1 s ∫ 0 s [ ( 3 / 2 π − 1 ) + 3 / 2 ( 1 − π ) 1 1 + τ ] dτ = ( 3 / 2 π − 1 ) + 3 / 2 ( 1 − π ) ln ( 1 + s ) s , then M σ , μ n , ς ϒ ( s ) s ( M σ , μ n , ς ϒ ( s ) ) ′ ≺ ξ ( s ) = ( 3 / 2 π − 1 ) + 3 / 2 ( 1 − π ) ln ( 1 + s ) s . Theorem 3.6. If ϒ ∈ Λ and suppose that ξ is an analytic function that fulfills the next inequality R { 1 + s ψ ′ ′ ( s ) ψ ′ ( s ) } > − 1 2 , and ψ ( 0 ) = 1 , ψ ′ ( 0 ) ≠ 0 , σ ∈ N , μ > 0 ; n , ς ≥ 0 satisfies the subordination (18) ( M σ , μ n , ς ϒ ( s ) ) ′ ≺ ψ ( s ) , ( s ∈ Δ ∗ ) , then M σ , μ n , ς ϒ ( s ) s ≺ ξ ( s ) = 1 s ∫ 0 s ψ ( τ ) dτ . Proof. Suppose (19) ϖ ( s ) = M σ , μ n , ς ϒ ( s ) s = s + ∑ η = 2 ∞ ( β 1 μ ς ) η − 1 ( β 2 ) η − 1 … ( β n ) η − 1 ( π 1 ) η − 1 ( π 2 ) η − 1 … ( π σ ) η − 1 ( 1 μ + η − 1 ) − ς + 1 e η s η s , ϖ ( s ) = 1 + ϖ η s η + ϖ η + 1 s η + 1 + … . By Eq. (19) in relation to w, we get (20) ( M σ , μ n , ς ϒ ( s ) ) ′ = ϖ ( s ) + s ϖ ′ ( s ) , ( s ∈ Δ ∗ ) . Subordination Eq. (18) , when applied to Eq. (20) , grows ϖ ( s ) + s ϖ ′ ( s ) ≺ ψ ( s ) . By Lemma 2.3 allows us to have ϖ ( s ) ≺ ξ ( s ) , in other words, M σ , μ n , δ ϒ ( s ) s ≺ ξ ( s ) = 1 s ∫ 0 s ψ ( τ ) dτ . Theorem 3.7. If ϒ ∈ Λ and suppose ξ be convex function in Δ ∗ , ξ ( 0 ) = 1 and ψ ( s ) = ξ ( s ) + s ξ ′ ( s ) , σ ∈ N , μ > 0 ; n , ς ≥ 0 , 0 ≤ π < 1 satisfies subordination (21) 1 − M σ , μ n , ς ϒ ( s ) ( M σ , μ n , ς ϒ ( s ) ) ′ ′ [ ( M σ , μ n , ς ϒ ( s ) ) ′ ] 2 ≺ ψ ( s ) , ( s ∈ Δ ∗ ) , then (22) M σ , μ n , ς ϒ ( s ) s ( M σ , μ n , ς ϒ ( s ) ) ′ ≺ ξ ( s ) . Proof. Suppose ϖ ( s ) = M σ , μ n , ς ϒ ( s ) s ( M σ , μ n , ς ϒ ( s ) ) ′ , p ∈ H [ 1 , 1 ] . Taking the product of both perspectives gives us ϖ ( s ) + s ϖ ′ ( s ) = 1 − M σ , μ n , ς ϒ ( s ) ( M σ , μ n , ς ϒ ( s ) ) ′ ′ [ ( M σ , μ n , ς ϒ ( s ) ) ′ ] 2 , so that with the help of Eq. (21) becomes ϖ ( s ) + s ϖ ′ ( s ) ≺ ψ ( s ) = ξ ( s ) + s ξ ′ ( s ) , ( s ∈ Δ ∗ ) . By Lemma 2.2 , we obtain ϖ ( s ) ≺ ξ ( s ) , then M σ , μ n , ς ϒ ( s ) s ( M σ , μ n , ς ϒ ( s ) ) ′ ≺ ξ ( s ) . 4. Conclusions There are many fascinating discoveries regarding harmonic multivalent functions derived from differential operators. The study concentrated on a subclass of analytical univalent functions related to the concept of differential subordination. Everyone looked at a few differential subordination and superordination outcomes, such as a class determined by a dimension for univalent meromorphic functions inside the open unit disc. Learn about geometrical characteristics such as coefficient border, coefficient disparities, distortion theorem, closing theorem, severe points, starlikeness radii, convexity, near-perfect convexity, and combining principles. An analysis is conducted on the concept of the differential subordination subclass of analytical univalent functions. Utilizing a particular class of univalent functions stated on a particular space of univalent functions stated on the open unit disc, we examined certain findings on differential subordination as well as superordination. We found several properties of subordinations as well as superordinations connected to the notion of Hadamard product by using characteristics of the operator. We examined various facets of subordinations as well as superordinations through a novel operator, M σ , μ n , ς ϒ ( s ) . Data availability Data sharing is not applicable to this article, as no data were created or analyzed in this study. References 1. Littlewood JE: Lectures on the Theory of Functions. UK: Oxford University Press; 1944; 244. 2. Rogosinski W: On subordination functions. Proc. Camb. Philos. Soc. 1939; 35 : 1–26. Publisher Full Text 3. Srivastava HM, Owa S: Some Applications of the Generalized Hypergeometric Function Involving Certain Subclasses of Analytic Functions. Publ. Math. Debr. 1987; 34 (3-4): 299–306. Publisher Full Text 4. Miller SS, Mocanu PT: Second-order Differential Inequalities in the Complex Plane. J. Math. Anal. Appl. 1978; 65 (2): 289–305. Publisher Full Text 5. 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Kumar SS, Banga S: On Convex Dominants of Exact Differential Subordination. arXiv preprint arXiv. 2020; 11404. 2011. Comments on this article Comments (0) Version 1 VERSION 1 PUBLISHED 31 Dec 2025 ADD YOUR COMMENT Comment Author details Author details 1 University of Anbar, Ramadi, Al Anbar Governorate, Iraq 2 University of Anbar, Ramadi, Al Anbar Governorate, Iraq Maryam S. Majel Roles: Data Curation, Formal Analysis, Investigation, Methodology, Writing – Original Draft Preparation Mustafa I. Hameed Roles: Project Administration, Supervision, Writing – Original Draft Preparation, 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: 31 Dec 2025, 14:1479 https://doi.org/10.12688/f1000research.174492.1 Copyright © 2025 Majel MS and Hameed MI. 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F1000Research 2025, 14 :1479 ( https://doi.org/10.5256/f1000research.192398.r447074 ) The direct URL for this report is: https://f1000research.com/articles/14-1479/v1#referee-response-447074 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 16 Jan 2026 Kassim A. Jassim , University of Baghdad, Baghdad, Baghdad Governorate, Iraq Approved VIEWS 0 https://doi.org/10.5256/f1000research.192398.r447074 The paper is studied some results on differential subordination and superordination using a class of univalent functions defined on the open unit disc. Many properties of the of subordinations and superordinations are investigated , using new operator and it's identity ... Continue reading READ ALL The paper is studied some results on differential subordination and superordination using a class of univalent functions defined on the open unit disc. Many properties of the of subordinations and superordinations are investigated , using new operator and it's identity .Also , cited references is really used in paper and many of these are modern which make paper more specified and important. 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? Yes 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 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 A. Jassim K. Reviewer Report For: Study analytical function subordination properties by applying a novel linear operator [version 1; peer review: 2 approved] . F1000Research 2025, 14 :1479 ( https://doi.org/10.5256/f1000research.192398.r447074 ) The direct URL for this report is: https://f1000research.com/articles/14-1479/v1#referee-response-447074 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 Views 0 Cite How to cite this report: Lupas AA. Reviewer Report For: Study analytical function subordination properties by applying a novel linear operator [version 1; peer review: 2 approved] . F1000Research 2025, 14 :1479 ( https://doi.org/10.5256/f1000research.192398.r447076 ) The direct URL for this report is: https://f1000research.com/articles/14-1479/v1#referee-response-447076 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 07 Jan 2026 Alina Alb Lupas , University of Oradea, Oradea, Romania Approved VIEWS 0 https://doi.org/10.5256/f1000research.192398.r447076 REVIEWER’s REPORT On the paper Study analytical function subordination properties by applying a novel linear operator by Maryam S. Majel and Mustafa I. Hameed In this paper the authors define ... Continue reading READ ALL REVIEWER’s REPORT On the paper Study analytical function subordination properties by applying a novel linear operator by Maryam S. Majel and Mustafa I. Hameed In this paper the authors define and study a new operator by using the Dziok-Srivastava operator and the Hadamard product with the Komatu integral operator. They establish several properties of subordinations and superordinations regarding the new defined operator. The results are new, correct and detailed. The paper is original and doesn’t contradict to ethical or policy issues, the question posed by authors is new and well defined, the methods used by authors are appropriate and well described, the data are sound and well controlled, the discussion and conclusions are well balanced, the title and abstract convey the obtained results, the writing is acceptable, the paper contains good scientific results. The paper doesn’t require a revision. Taking the above into consideration, I recommend the paper for indexing. "No competing interests were disclosed" 4.01.2026 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 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 Lupas AA. Reviewer Report For: Study analytical function subordination properties by applying a novel linear operator [version 1; peer review: 2 approved] . F1000Research 2025, 14 :1479 ( https://doi.org/10.5256/f1000research.192398.r447076 ) The direct URL for this report is: https://f1000research.com/articles/14-1479/v1#referee-response-447076 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 31 Dec 2025 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 31 Dec 25 read read Alina Alb Lupas , University of Oradea, Oradea, Romania Kassim A. Jassim , University of Baghdad, Baghdad, Iraq 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 A. Jassim K. 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. 16 Jan 2026 | for Version 1 Kassim A. Jassim , University of Baghdad, Baghdad, Baghdad Governorate, Iraq 0 Views copyright © 2026 A. Jassim K. 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 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 The paper is studied some results on differential subordination and superordination using a class of univalent functions defined on the open unit disc. Many properties of the of subordinations and superordinations are investigated , using new operator and it's identity .Also , cited references is really used in paper and many of these are modern which make paper more specified and important. 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? Yes 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 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 (0) A. Jassim K. Peer Review Report For: Study analytical function subordination properties by applying a novel linear operator [version 1; peer review: 2 approved] . F1000Research 2025, 14 :1479 ( https://doi.org/10.5256/f1000research.192398.r447074) 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/14-1479/v1#referee-response-447074 keyboard_arrow_left Back to all reports Reviewer Report 0 Views copyright © 2026 Lupas A. 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. 07 Jan 2026 | for Version 1 Alina Alb Lupas , University of Oradea, Oradea, Romania 0 Views copyright © 2026 Lupas A. 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 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’s REPORT On the paper Study analytical function subordination properties by applying a novel linear operator by Maryam S. Majel and Mustafa I. Hameed In this paper the authors define and study a new operator by using the Dziok-Srivastava operator and the Hadamard product with the Komatu integral operator. They establish several properties of subordinations and superordinations regarding the new defined operator. The results are new, correct and detailed. The paper is original and doesn’t contradict to ethical or policy issues, the question posed by authors is new and well defined, the methods used by authors are appropriate and well described, the data are sound and well controlled, the discussion and conclusions are well balanced, the title and abstract convey the obtained results, the writing is acceptable, the paper contains good scientific results. The paper doesn’t require a revision. Taking the above into consideration, I recommend the paper for indexing. "No competing interests were disclosed" 4.01.2026 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 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 (0) Lupas AA. Peer Review Report For: Study analytical function subordination properties by applying a novel linear operator [version 1; peer review: 2 approved] . F1000Research 2025, 14 :1479 ( https://doi.org/10.5256/f1000research.192398.r447076) 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/14-1479/v1#referee-response-447076 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. 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