Pexider invariance equation for embeddable mean-type mappings

Pexider invariance equation for embeddable mean-type mappings | Research Square window.SnipcartSettings = { analytics: { enabled: false } }; (function() { var accessVector = localStorage.getItem('access_vector') || ''; window.dataLayer = window.dataLayer || []; if (accessVector) { window.dataLayer.push({ user: { profile: { profileInfo: { snid: accessVector } } } }); } })(); (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;f.parentNode.insertBefore(j,f);})(window,document,'script','dataLayer','GTM-K279D39R'); Browse Preprints In Review Journals COVID-19 Preprints AJE Video Bytes Research Tools Research Promotion AJE Professional Editing AJE Rubriq About Preprint Platform In Review Editorial Policies Our Team Advisory Board Help Center Sign In Submit a Preprint Cite Share Download PDF Research Article Pexider invariance equation for embeddable mean-type mappings Paweł Pasteczka This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-3847885/v1 This work is licensed under a CC BY 4.0 License Status: Published Journal Publication published 03 Jan, 2025 Read the published version in Aequationes mathematicae → Version 1 posted 9 You are reading this latest preprint version Abstract We prove that whenever $M_1,\dots,M_n\colon I^k \to I$, ($n,k \in \mathbb{N}$) are symmetric, continuous means on the interval $I$ and $S_1,\dots,S_m\colon I^k \to I$ ($m <n$) satisfies a sort of embeddability assumptions then for every continuous function $\mu \colon I^n \to \R$ which is strictly monotone in each coordinate, the functional equation has the unique solution $F=F_\mu \colon I^k \to I$ which is a mean. We deliver some sufficient conditions so that $F_\mu$ is well-defined (in particular uniquely determined) and study its properties. The background of this research is to provide a broad overview of the family of Beta-type means introduced in (Himmel and Matkowski, 2018). functional equations solvability means invariant means uniqueness Beta-type means Full Text Additional Declarations No competing interests reported. Cite Share Download PDF Status: Published Journal Publication published 03 Jan, 2025 Read the published version in Aequationes mathematicae → Version 1 posted Editorial decision: Revision requested 30 Nov, 2024 Reviews received at journal 27 Nov, 2024 Reviews received at journal 27 Sep, 2024 Reviewers agreed at journal 15 Jan, 2024 Reviewers agreed at journal 15 Jan, 2024 Reviewers invited by journal 14 Jan, 2024 Editor assigned by journal 11 Jan, 2024 Submission checks completed at journal 09 Jan, 2024 First submitted to journal 09 Jan, 2024 You are reading this latest preprint version Research Square lets you share your work early, gain feedback from the community, and start making changes to your manuscript prior to peer review in a journal. As a division of Research Square Company, we’re committed to making research communication faster, fairer, and more useful. We do this by developing innovative software and high quality services for the global research community. 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Also discoverable on Platform About Our Team In Review Editorial Policies Advisory Board Help Center Resources Author Services Accessibility API Access RSS feed Manage Cookie Preferences © Research Square 2026 | ISSN 2693-5015 (online) Privacy Policy Terms of Service Do Not Sell My Personal Information {"props":{"pageProps":{"initialData":{"identity":"rs-3847885","acceptedTermsAndConditions":true,"allowDirectSubmit":false,"archivedVersions":[],"articleType":"Research Article","associatedPublications":[],"authors":[{"id":266206382,"identity":"dc80dc65-149c-41d9-8124-2fc64db3df3d","order_by":0,"name":"Paweł Pasteczka","email":"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAZAAAAAyAQMAAABI0h/eAAAABlBMVEX///8AAABVwtN+AAAACXBIWXMAAA7EAAAOxAGVKw4bAAABK0lEQVRIie3RwUrDMBgH8JRAvdTtml7aV0gpOHbQvUpKobus94GCGYV5Uc966ivkVK8pgXopeC2sh3rZTewYjJ7EbtMxIVOPgvlDwpeEH+RLAFBR+aupdhUHVjvDdpD1UuOHyOZY3xJ3n4BfEY/+RHpHuVMRUNp2HKWLJj8bshmF7vgisDA/5jLSvx65mIC5wzLdR6jwQ1Zy6OVZu807REYwDxNEgNCYbpwAXMOQFQSmVB97CTewlDy9PDQtGcTT7gqQ+nKIN+TtG1KESduq8Ghm6IAXgqyJN5mODpL+3esSETz3WRa4Js0fnfsyjZzJbeAOhLyXXjfw6npcnsaReF402bndmUXCpCvfMq9u0kp2sY9H2NtCGt0W0JCAT/Lly9CukhMVFRWVf5d3R7h0FI2IIZkAAAAASUVORK5CYII=","orcid":"","institution":"University of the National Education Commission, Krakow","correspondingAuthor":true,"prefix":"","firstName":"Paweł","middleName":"","lastName":"Pasteczka","suffix":""}],"badges":[],"createdAt":"2024-01-09 09:59:16","currentVersionCode":1,"declarations":"","doi":"10.21203/rs.3.rs-3847885/v1","doiUrl":"https://doi.org/10.21203/rs.3.rs-3847885/v1","draftVersion":[],"editorialEvents":[{"content":"https://doi.org/10.1007/s00010-024-01139-0","type":"published","date":"2025-01-03T15:57:33+00:00"}],"editorialNote":"","failedWorkflow":false,"files":[{"id":73093315,"identity":"8c458e99-a3ee-4763-94e2-9118911bd23c","added_by":"auto","created_at":"2025-01-06 16:13:28","extension":"pdf","order_by":1,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":251003,"visible":true,"origin":"","legend":"","description":"","filename":"AeQ.pdf","url":"https://assets-eu.researchsquare.com/files/rs-3847885/v1_covered_bc4c162d-7e02-4533-b738-0468de29c554.pdf"}],"financialInterests":"No competing interests reported.","formattedTitle":"Pexider invariance equation for embeddable mean-type mappings","fulltext":[],"fulltextSource":"","fullText":"","funders":[],"hasAdminPriorityOnWorkflow":false,"hasManuscriptDocX":false,"hasOptedInToPreprint":true,"hasPassedJournalQc":"","hasAnyPriority":false,"hideJournal":false,"highlight":"","institution":"","isAcceptedByJournal":true,"isAuthorSuppliedPdf":true,"isDeskRejected":"","isHiddenFromSearch":false,"isInQc":false,"isInWorkflow":false,"isPdf":true,"isPdfUpToDate":true,"isWithdrawnOrRetracted":false,"journal":{"display":true,"email":"[email protected]","identity":"aequationes-mathematicae","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":false,"externalIdentity":"aeqm","sideBox":"Learn more about [Aequationes mathematicae](http://link.springer.com/journal/10)","snPcode":"10","submissionUrl":"https://submission.nature.com/new-submission/10/3","title":"Aequationes mathematicae","twitterHandle":"","acdcEnabled":true,"dfaEnabled":true,"editorialSystem":"em","reportingPortfolio":"Springer Hybrid","inReviewEnabled":true,"inReviewRevisionsEnabled":false},"keywords":"functional equations, solvability, means, invariant means, uniqueness, Beta-type means","lastPublishedDoi":"10.21203/rs.3.rs-3847885/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-3847885/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"We prove that whenever $M_1,\\dots,M_n\\colon I^k \\to I$, ($n,k \\in \\mathbb{N}$) are symmetric, continuous means on the interval $I$ and\n$S_1,\\dots,S_m\\colon I^k \\to I$ ($m \u003cn$) satisfies a sort of embeddability assumptions then for every continuous function $\\mu \\colon I^n \\to \\R$ which is strictly monotone in each coordinate, the functional equation\nhas the unique solution $F=F_\\mu \\colon I^k \\to I$ which is a mean. 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