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

Volume 10, Issue 3, pp 929--936 Publication Date: March 20, 2017       Article History
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### 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.

### Keywords

• Gaussian hypergeometric function
• complete elliptic integral
• Stolarsky mean.

•  33E05
•  26D15
•  26E60

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