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

Yang, Zhen-Hang¹ ; Chu, Yu-Ming² ; Zhang, Xiao-Hui³

¹ 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
² School of Mathematics and Computation Sciences, Hunan City University, Yiyang 413000, China
³ Department of Mathematics, Zhejiang Sci-Tech University, Hangzhou 310018, China

Journal of nonlinear sciences and its applications, Tome 10 (2017) no. 3, p. 929-936.

Résumé

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.

Détail

DOI : 10.22436/jnsa.010.03.06

Classification : 33E05, 26D15, 26E60
Keywords: Gaussian hypergeometric function, complete elliptic integral, Stolarsky mean.

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