Sharp bounds for generalized elliptic integrals of the first kind

TL;DR

This paper establishes sharp bounds for generalized elliptic integrals of the first kind using properties of the Ramanujan constant function, providing new inequalities and series representations.

Contribution

The paper introduces precise bounds for generalized elliptic integrals of the first kind involving the Ramanujan constant function, with conditions for equality and a series expansion for the function.

Findings

Established necessary and sufficient conditions for the bounds to hold.

Derived a series expression for the Ramanujan constant function.

Provided explicit inequalities involving elliptic integrals and the Ramanujan constant.

Abstract

In this paper, we prove that the double inequality \begin{equation*} 1+\alpha r'^2<\frac{\mathcal{K}_{a}(r)}{\sin(\pi a)\log(e^{R(a)/2}/r')}<1+\beta r'^2 \end{equation*} holds for all $a\in (0, 1/2]$ and $r\in (0, 1)$ if and only if $\alpha\leq \pi/[R(a)\sin(\pi a)]-1$ and $\beta\geq a(1-a)$, 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.