Hermite-Hadamard Inequalities via Gateaux Derivatives in Banach Spaces

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dc.contributor.authorSarikaya, Mehmet Zeki
dc.date.accessioned2026-07-01T11:40:04Z
dc.date.available2026-07-01T11:40:04Z
dc.date.issued2026
dc.departmentDüzce Üniversitesi
dc.description.abstractThis paper establishes a new framework for Hermite-Hadamard type inequalities by employing Gateaux derivatives and the Bochner integral. By extending the classical scalar results to real Banach spaces, we provide sharp error bounds for trapezoidal and midpoint type functional approximations. Under the assumption that the magnitude of the directional curvature |phi '|\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$|\varphi <^>{\prime }|$$\end{document} is convex, we derive deterministic error estimates with an optimal constant of 1/8. Furthermore, we demonstrate the practical utility of these results through applications in nonlinear optimization search paths, quadratic growth characterization, and the stability analysis of energy potentials in nonlinear elasticity. These results remain valid in infinite-dimensional Banach spaces and therefore extend the classical Hermite-Hadamard framework beyond the usual real-variable setting. An illustrative example on the space & ell;2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\ell <^>2$$\end{document} demonstrates how the proposed approach provides effective error bounds in infinite-dimensional optimization problems.
dc.identifier.doi10.1007/s41980-026-01057-8
dc.identifier.issn1017-060X
dc.identifier.issn1735-8515
dc.identifier.issue4
dc.identifier.scopus2-s2.0-105041299443
dc.identifier.scopusqualityQ2
dc.identifier.urihttps://doi.org/10.1007/s41980-026-01057-8
dc.identifier.urihttps://hdl.handle.net/20.500.12684/23629
dc.identifier.volume52
dc.identifier.wosWOS:001786954500001
dc.identifier.wosqualityQ2
dc.indekslendigikaynakWeb of Science
dc.indekslendigikaynakScopus
dc.institutionauthorSarikaya, Mehmet Zeki
dc.language.isoen
dc.publisherSpringer Singapore Pte Ltd
dc.relation.ispartofBulletin of the Iranian Mathematical Society
dc.relation.publicationcategoryMakale - Uluslararası Hakemli Dergi - Kurum Öğretim Elemanı
dc.rightsinfo:eu-repo/semantics/openAccess
dc.snmzKA_WOS_20260623
dc.subject[Keyword Not Available]
dc.titleHermite-Hadamard Inequalities via Gateaux Derivatives in Banach Spaces
dc.typeArticle

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