An optimal power mean inequality for the complete elliptic integrals

Abstract

In this work, we prove that $M_p(\mathcal{K}\left(r\right),\mathcal{E}\left(r\right))>\pi /2$ for all $r\in (0,1)$ if and only if $p\ge -1/2$, where $M_p(x,y)$ denotes the power mean of order $p$ of two positive numbers $x$ and $y$, and $\mathcal{K}\left(r\right)$ and $\mathcal{E}\left(r\right)$ denote the complete elliptic integrals of the first and second kinds, respectively.