Sharp double inequality for complete elliptic integral of the first kind

TL;DR

This paper proves the absolute monotonicity of functions involving the complete elliptic integral of the first kind, leading to improved inequalities and bounds for this special function.

Contribution

It establishes new monotonicity properties of elliptic integrals, resulting in sharper inequalities compared to previous results.

Findings

Proved absolute monotonicity of specific elliptic integral functions

Derived improved inequalities for the complete elliptic integral of the first kind

Enhanced existing bounds and inequalities in elliptic integral theory

Abstract

For $r\in(0,1)$, the function $\K(r)=\int_0^{\pi/2}(1-r^2\sin^2t)^{-1/2}dt$ is known as the complete elliptic integral of the first kind. In this paper, we prove the absolute monotonicity of two functions involving $\K(r)$. As a consequence, we improve Alzer and Richards' result.