A new treatment of convex functions

TL;DR

This paper introduces a new generalized concept called g-convexity, which enhances estimates in inequalities involving convex functions and explores its applications in various mathematical contexts.

Contribution

It proposes g-convexity as a broader framework than log-convexity and defines an index of convexity to quantify the degree of convexity of functions.

Findings

g-convex functions provide tighter inequality estimates

The index of convexity measures how much a function is convex

Applications extend to Hilbert space operators, matrices, and entropies

Abstract

Convex functions have played a major role in the field of Mathematical inequalities. In this paper, we introduce a new concept related to convexity, which proves better estimates when the function is somehow more convex than another. In particular, we define what we called $g-$convexity as a generalization of $\log-$convexity. Then we prove that $g-$convex functions have better estimates in certain known inequalities like the Hermite-Hadard inequality, super additivity of convex functions, the Majorization inequality and some means inequalities. Strongly related to this, we define the index of convexity as a measure of ``how much the function is convex". Applications including Hilbert space operators, matrices and entropies will be presented in the end.