The Function $(b^x-a^x)/x$ Logarithmic Convexity and Applications to Extended Mean Values

Bai-Ni Guo, Feng Qi

We first prove the logarithmic convexity of the elementary function $\frac{b^x-a^x}{x}$, where $x\neq0$ 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.