Sharp Power Mean Bounds for Sándor Mean

Abstract

We prove that the double inequality holds for all $a,b>0$ with $a\ne b$ if and only if $p\le 1/3$ and $q\ge \text{l}\text{o}\text{g}\mathrm{\hspace{0.17em}2}/(1+\text{l}\text{o}\text{g}\mathrm{\hspace{0.17em}2})=0.4093$…, where $X(a,b)$ and $M_r(a,b)$ are the Sándor and $r$th power means of $a$ and $b$, respectively.