A power mean inequality involving the complete elliptic integrals
Let \(\mathcal K(r)\) for \(r\in(0,1)\) be a complete elliptic integrals of the first kind \[ \mathcal K(r)=\int_0^{\pi/2}\frac{d\theta} {\sqrt{1-r^2\sin^2\theta}} \] and \(\mathcal K'(r)=\mathcal K(r')\), where \(r'=\sqrt{1-r^2}\). Let \(m(r)\) be a special function defined as \(m(r)=\frac2{\pi}{r'}^2\mathcal K(r)\mathcal K'(r)\) for \(r\in(0,1)\) and \(M_\lambda(x,y)=\left(\frac{x^\lambda+y^\lambda}2\right)^{1/\lambda}\) be the power mean. In this paper one power mean inequality for the special function \(m(r)\) is obtained. The main result consists in the following statement: For \(\lambda\in\mathbb R\) the inequality \[ M_\lambda(m(x),m(y))\leq m(M_\lambda(x,y))\tag{\(\ast\)} \] holds for all \(x,y\in(0,1)\) if and only if \(\lambda\leq0\). The reverse of \((\ast)\) holds for all \(x,y\in(0,1)\) if and only if \(\lambda\geq C>0\), where \(C\) is some constant. Equality holds in \((\ast)\) if and only if \(x=y\). An open problem is formulated: What is the value of \(C\)?
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