Affine and Functional Form of Jensen's Inequalitiy for $3$-convex Functions at a Point

TL;DR

This paper refines Jensen's inequality for affine combinations and extends its functional form to continuous 3-convex functions at a point, providing new insights into inequalities for higher-order convex functions.

Contribution

It introduces a refined version of Jensen's inequality for affine combinations and establishes its functional form for continuous 3-convex functions at a point, advancing the theory of convex inequalities.

Findings

Refinement of Jensen's inequality for affine combinations.

Functional form of Jensen's inequality for 3-convex functions.

Extension of Jensen's inequality to higher-order convex functions.

Abstract

In this paper, we give the refinement of an extension of Jensen's inequality to affine combinations. Furthermore, we present the functional form of Jensen's inequality for continuous 3-convex functions of one variable at a point.