Best Possible Bounds for Neuman-Sándor Mean by the Identric, Quadratic and Contraharmonic Means

Abstract

We prove that the double inequalities hold for all $a,b>0$ with $a\ne b$ if and only if ${\alpha }_1\ge 1/2$, ${\beta }_1\le \log [\sqrt{2}\log (1+\sqrt{2})]/(1-\log \sqrt{2})$, ${\alpha }_2\ge 5/7$, and ${\beta }_2\le \log [2\log (1+\sqrt{2})]$, where $I(a,b)$, $M(a,b)$, $Q(a,b)$, and $C(a,b)$ are the identric, Neuman-Sándor, quadratic, and contraharmonic means of $a$ and $b$, respectively.