Sharp inequalities involving the power mean and complete elliptic integral of the first kind (Q380195)

For \(0<r<1\), define the complete elliptic integral of the first kind by NEWLINE\[NEWLINEK(r)=\int_{0}^{\pi/2} \frac{d\theta}{\sqrt{1 - r^{2} \sin^{2} \theta}}.NEWLINE\]NEWLINE NEWLINEFor each real number \(p\), denote the power mean of order \(p\) of two positive numbers \(x\) and \(y\) by NEWLINENEWLINE\[NEWLINEM_{p}(x,y) = (x^{p} + y^{p}/2)^{1/p}\quad \text{for }\;p \neq 0NEWLINE\]NEWLINE NEWLINEand NEWLINENEWLINE\[NEWLINEM_{p}(x,y)=\sqrt{xy}\quad \text{for }\;p =0.NEWLINE\]NEWLINE NEWLINEThe main results of this paper are the inequalities NEWLINE\[NEWLINEM_{p}(K(r), K(r')) \geq K(\sqrt{2}/2)NEWLINE\]NEWLINE and NEWLINE\[NEWLINEM_{q}(K(r), K(r'))\leq K(\sqrt{2}/2).NEWLINE\]NEWLINE These are shown to hold for all \(r\in (0,1)\) and \(r' = \sqrt{1 - r^{2}}\) if and only if NEWLINE\[NEWLINEp\geq \frac{1-4[K(\sqrt{2}/2)]^4}{\pi^{2}}NEWLINE\]NEWLINE and NEWLINE\[NEWLINEq\leq \frac{\log 2}{\log(\pi /2) - \log K(\sqrt{2}/2)}.NEWLINE\]