Sharp bounds for generalized elliptic integrals of the first kind

Abstract

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 $a\in (0, 1/2]$ and $r\in (0, 1)$ if and only if $α\leq π/[R(a)\sin(πa)]-1$ and $β\geq a(1-a)$, where $r'=\sqrt{1-r^2}$, $\mathcal{K}_{a}(r)$ is the generalized elliptic integral of the first kind and $R(x)$ is the Ramanujan constant function. Besides, as the key tool, the series expression for the Ramanujan constant function $R(x)$ is given.