  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.
