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

TL;DR

This paper introduces highly accurate approximation formulas for the elliptic integral of the second kind using Stolarsky means, improving upon existing methods by analyzing the monotonicity of specific functions.

Contribution

The authors establish new bounds for elliptic integrals based on Stolarsky means, providing more precise approximation formulas and demonstrating their effectiveness through monotonicity analysis.

Findings

Derived new bounds for elliptic integrals of the second kind.

Proved monotonicity properties of key functions involving Stolarsky means.

Achieved significantly improved approximation accuracy over previous results.

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.