Very accurate approximations for the elliptic integrals of the second kind in terms of Stolarsky means

Abstract

For $a,b>0$ with $a\neq b$, the Stolarsky means are defined by% \begin{equation*} S_{p,q}\left(a,b\right) =\left({\dfrac{q(a^{p}-b^{p})}{p(a^{q}-b^{q})}}% \right) ^{1/(p-q)}\text{if}pq\left(p-q\right) \neq 0 \end{equation*}% and $S_{p,q}\left(a,b\right) $ is defined as its limits at $p=0$ or $q=0$ or $p=q$ if $pq\left(p-q\right) =0$. The complete elliptic integrals of the second kind $E$ is defined on $\left(0,1\right) $ by% \begin{equation*} E\left(r\right) =\int_{0}^{\pi /2}\sqrt{1-r^{2}\sin ^{2}t}dt. \end{equation*}% We prove that the functions% \begin{equation*} F\left(r\right) =\frac{1-\left(2/\pi \right) E\left(r\right)}{% 1-S_{11/4,7/4}\left(1,r^{\prime}\right)}\text{and}G\left(r\right) =% \frac{1-\left(2/\pi \right) E\left(r\right)}{1-S_{5/2,2}\left(1,r^{\prime}\right)} \end{equation*}% are strictly decreasing and increasing on $\left(0,1\right) $, respectively, where $r^{\prime}=\sqrt{1-r^{2}}$. These yield some very accurate approximations for the complete elliptic integrals of the second kind, which greatly improve some known results.