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

Abstract

In the present paper, we first prove the logarithmic convexity of the elementary function $\frac{b^x-a^x}{x}$, where $x\ne 0$ 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.

8 pages

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