Sharp Stolarsky mean bounds for the complete elliptic integral of the second kind


Authors

Zhen-Hang Yang - School of Mathematics and Computation Sciences, Hunan City University, Yiyang 413000, China. - Customer Service Center, State Grid Zhejiang Electric Power Research Institute, Hangzhou 310009, China. Yu-Ming Chu - School of Mathematics and Computation Sciences, Hunan City University, Yiyang 413000, China. Xiao-Hui Zhang - Department of Mathematics, Zhejiang Sci-Tech University, Hangzhou 310018, China.


Abstract

In the article, we prove that the double inequality \[25/16<\varepsilon(r)/S_{5/2,2}(1,\acute{r})<\pi/2,\] holds for all \(r \in (0, 1)\) with the best possible constants \(25/16\) and \(\pi/2\), where \(\acute{r}=(1-r^2)^{1/2}, \varepsilon(r)=\int^{\pi/2}_0\sqrt{1-r^2\sin^2(t)}dt\) , is the complete elliptic integral of the second kind and \(S_{p,q}(a,b)=[q(a^p-b^p)/(p(a^q-b^q))]^{1/(p-q)}\), is the Stolarsky mean of a and b.


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