On generalized Seiffert means (Q2510420)

In his previous paper [Aequationes Math. 83, No. 1--2, 191--197 (2012; Zbl 1238.33011)] the author has defined a mean \(X_{k}\) \((0\leq k\leq 1)\) as follows \[ X_{k}\left( x,y\right) =\left[ R_{F}\left( x^{2},y^{2},z^{2}\right) \right] ^{-1}, \] where \[ z=\begin{cases} \sqrt{\left( kx\right) ^{2}+\left( k^{\prime }y\right) ^{2}}\quad \text{if } x<y, \\ \sqrt{\left( k^{\prime }x\right) ^{2}+\left( ky\right) ^{2}}\quad \text{if } y<x, \end{cases} \] \(k^{\prime }=\sqrt{1-k^{2}},\) and \[ R_{F}(x,y,z)=\frac{1}{2}\int_{0}^{\infty }\frac{dt}{\sqrt{\left( t+x\right) \left( t+y\right) \left( t+z\right) }} \] is the completely symmetric elliptic integral of the first kind. Let us denote by \(G\), \(A\), \(Q\) the geometric mean, the arithmetic mean, and the root-square mean, respectively. It is known that \[ SB(x,y)=\left[ R_{F}(x^{2},y^{2},y^{2})\right] ^{-1} \] represents the Schwab-Borchard mean and \[ P=SB(G,A),\quad T=SB(A,Q) \] gives the Seiffert means \(P\) and \(T.\) In this paper two generalized Seiffert means \[ \mathcal{P}_{k}=X_{k}(A,G),\quad \mathcal{T}_{k}=X_{k}(A,Q) \] are defined and some lower and upper bounds are established.