Bounds for the generalized elliptic integral of the second kind

Miskolc Mathematical Notes

article

Xiaohui Zhang¹  Zhixia Xing¹ 

[1] School of Science, Zhejiang Sci-Tech University

关键词: generalized elliptic integrals;    Ramanujan’s constant;    Gaussian hypergeometric function;    Hersch-Pfluger distortion function;   

DOI  :  10.18514/MMN.2022.3828

学科分类：数学（综合）

来源: Miskolci Egyetem

【 摘 要 】

For $a\in(0,1)$ and $r\in(0,1)$, let $\mathcal{E}_{a}(r)$ be the generalized elliptic integral of the second kind and $R(a)$ Ramanujan's constant. In this paper, we prove the following inequalities \begin{align*} \frac{\sin(\pi a)}{2(1-a)}+r'^2\left((1-a)\sin(\pi a)\log \left(\frac{e^{R(a)/2}}{r'}\right)-\gamma\right)<\mathcal{E}_{a}(r)\\ <\frac{\sin(\pi a)}{2(1-a)}+r'^2\left((1-a)\sin(\pi a)\log \left(\frac{e^{R(a)/2}}{r'}\right)-\delta\right) \end{align*} with the best possible constants $\gamma=\dfrac{1}{4}\sin(\pi a)$ and $\delta=\dfrac{\sin(\pi a)}{2(1-a)}+(1-a)\sin(\pi a)\dfrac{R(a)}{2}-\dfrac{\pi}{2}$.

【 授权许可】

CC BY