Benaissa, BouharketBudak, Huseyin2025-10-112025-10-1120251687-2770https://doi.org/10.1186/s13661-024-01984-7https://hdl.handle.net/20.500.12684/21728This paper develops a novel Milne inequality for third-differentiable and h-convex functions using Riemann integrals. Furthermore, new Milne inequalities are proposed utilizing a summation parameter p >= 1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$p\geq 1$\end{document} for s-convexity, convexity, and P-functions class. We examine cases when the third derivative functions are also bounded and Lipschitzian.en10.1186/s13661-024-01984-7info:eu-repo/semantics/openAccessh-convex functionMilne's inequalityH & ouml;lder's inequalityRiemann's integralArticle202512-s2.0-85214258737WOS:001391665800001Q1Q1