On relations between some types of convex functions

TL;DR

This paper explores how superquadratic functions can be utilized to analyze and derive new properties of various convex functions, including $\,\phi$-convexity, strong-convexity, and uniform convexity, through inequalities and examples.

Contribution

It introduces methods to connect superquadratic functions with other convex functions, providing new inequalities and insights into their relationships.

Findings

Superquadratic functions can be used to derive properties of other convex functions.

New inequalities relating superquadratic and uniformly convex functions.

Examples illustrating the connections between superquadracity and other convexity types.

Abstract

In this paper we show how the superquadratic functions can be used as a tool for researching other types of convex functions like $\phi $-convexity, strong-convexity and uniform convexity. We show how to use inequalities satisfied by superquadratic functions and how to adapt the technique used to get them in order to obtain new results satisfied by uniformly convex functions and to $\phi $-convex functions. Also, we show examples that emphasize relations between superquadracity and some other types of convex functions