Superquadratic functions and refinements of inequalities between averages

TL;DR

This paper establishes upper bounds for differences between averages involving superquadratic functions, refining existing inequalities for convex functions and extending results to subquadratic functions.

Contribution

It introduces new bounds for differences of averages with superquadratic functions, improving upon known convex function inequalities and generalizing to more diverse function classes.

Findings

Bounds for differences of averages with superquadratic functions

Refinement of inequalities between averages for convex functions

Extensions to subquadratic functions

Abstract

In this paper upper bounds are given for the successive differences $A_{n+1}-A_{n}$ and B$_{n}-B_{n-1}$ where $A_{n}=1/(n-1) \tsum_{r=1}^{n-1}f(r/n)$, $B_{n}=1/(n+1) \tsum_{r=0}^{n}f(r/n)$ and $f$ is superquadratic function. We obtain bounds for the successive differences of the more general sequence $1/c_{n}\tsum_{r=1}^{n}f(a_{r}/b_{n})$ when $f$ is superquadratic, which refine known results for convex functions. We also obtain bounds for various successive differences when $f$ is an increasing subquadratic function.