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

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