Sharp bounds for generalized elliptic integrals of the first kind

Wang Miao-Kun, Chu Yu-Ming, Qiu Song-Liang

In this paper, we prove that the double inequality \begin{equation*} 1+αr'^2<\frac{\mathcal{K}_{a}(r)}{\sin(πa)\log(e^{R(a)/2}/r')}<1+βr'^2 \end{equation*} holds for all $a\in (0,…