Optimal Inequalities for Power Means

Abstract

We present the best possible power mean bounds for the product $M_p^{\alpha }(a,b)M_{-p}^{1-\alpha }(a,b)$ for any $p>0$, $\alpha \in \left(\mathrm{0,1}\right)$, and all $a,b>0$ with $a\ne b$. Here, $M_p(a,b)$ is the $p$th power mean of two positive numbers $a$ and $b$.