Sharp Stolarsky mean bounds for the complete elliptic integral of the second kind
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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.
Share and Cite
ISRP Style
Zhen-Hang Yang, Yu-Ming Chu, Xiao-Hui Zhang, Sharp Stolarsky mean bounds for the complete elliptic integral of the second kind, Journal of Nonlinear Sciences and Applications, 10 (2017), no. 3, 929--936
AMA Style
Yang Zhen-Hang, Chu Yu-Ming, Zhang Xiao-Hui, Sharp Stolarsky mean bounds for the complete elliptic integral of the second kind. J. Nonlinear Sci. Appl. (2017); 10(3):929--936
Chicago/Turabian Style
Yang, Zhen-Hang, Chu, Yu-Ming, Zhang, Xiao-Hui. "Sharp Stolarsky mean bounds for the complete elliptic integral of the second kind." Journal of Nonlinear Sciences and Applications, 10, no. 3 (2017): 929--936
Keywords
- Gaussian hypergeometric function
- complete elliptic integral
- Stolarsky mean.
MSC
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