The function $(b^x-a^x)/x$: Logarithmic convexity and applications to extended mean values

TL;DR

This paper proves the logarithmic convexity of a specific elementary function and applies this to establish properties of extended mean values, including Schur-convexity and other convexity relations.

Contribution

It introduces a new proof of the logarithmic convexity of (b^x - a^x)/x and applies it to analyze convexity properties of extended mean values.

Findings

Logarithmic convexity of (b^x - a^x)/x established

Simplified proof of Schur-convexity for extended mean values

New convexity relations related to extended mean values discovered

Abstract

In the present paper, we first prove the logarithmic convexity of the elementary function $\frac{b^x-a^x}x$, where $x\ne0$ and $b>a>0$. Basing on this, we then provide a simple proof for Schur-convex properties of the extended mean values, and, finally, discover some convexity related to the extended mean values.