Absolutely monotonic functions related to the asymptotic formula for the complete elliptic integral of the first kind

TL;DR

This paper characterizes when certain functions involving the complete elliptic integral of the first kind are absolutely monotonic, extending previous results and introducing a new method based on power series coefficients.

Contribution

It provides necessary and sufficient conditions for absolute monotonicity of functions related to elliptic integrals, extending prior work and proposing a novel approach using coefficient sequence monotonicity.

Findings

Established conditions for absolute monotonicity of functions involving elliptic integrals.

Extended known results on functional inequalities related to elliptic integrals.

Introduced a new method using power series coefficients to analyze monotonicity.

Abstract

Let $\mathcal{K}\left( x\right) $ be the complete elliptic integral of the first kind and \begin{equation*} \mathcal{G}_{p}\left( x\right) =e^{\mathcal{K}\left( \sqrt{x} \right) }-\frac{p}{\sqrt{1-x}} \end{equation*} for $p\in \mathbb{R}$ and $x\in \left( 0,1\right) $. In this paper we find the necessary and sufficient conditions for the functions $\pm \mathcal{G} _{p}^{\left( k\right) }\left( x\right) $ ($k\in \mathbb{N\cup }\left\{ 0\right\} $) to be absolutely monotonic on $\left( 0,1\right) $, which extend previous known results and yield several new functional inequalities involving the complete elliptic integral of the first kind. More importantly, we provide a new method to deal with those absolute monotonicity problem by proving the monotonicity of a sequence generated by the coefficients of the power series of $\mathcal{G}_{p}\left( x\right) $.