Some extensions from famous theorems for $h$-mid-convex function

TL;DR

This paper extends classical theorems to $h$-mid-convex functions, showing that under certain conditions, continuous $h$-mid-convex functions are $h$-convex and generalizing key convexity theorems.

Contribution

It introduces new extensions of well-known theorems for $h$-mid-convex functions, broadening the understanding of their properties.

Findings

Continuous $h$-mid-convex functions are $h$-convex under certain conditions.

Extended Ostrowski, Blumberg-Sierpinski, Bernstein-Doetsch, and Mehdi theorems.

Established new links between $h$-mid-convexity and classical convexity results.

Abstract

In this paper, we prove that every continuous $h$-mid-convex with suitable conditions on $h$ is $h$-convex function. Also, we extend Ostrowski theorem, Blumberg-Sierpinski theorem, Bernstein-Doetsch theorem, Mehdi theorem.