Several Absolutely Monotonic Functions Related to the Complete Elliptic Integral of the First Kind

Abstract

Let ${\mathcal {K}}\left( r\right)$ ($0<r<1$) be the complete elliptic integral of the first kind. In this paper, we investigate the absolute monotonicity of the functions

$$\begin{aligned} f_{p}\left( x\right) =\frac{{\mathcal {K}}\left( \sqrt{x}\right) }{p-\ln \sqrt{ 1-x}}(p>0) \end{aligned}$$

and $g_{q}\left( x\right) =1/f_{q}\left( x\right)$ on $\left( 0,1\right)$ by recursive method, and prove that $-f_{p}^{\prime \prime }\left( x\right)$, $f_{p}^{\prime }\left( x\right)$, $-g_{q}^{\prime \prime }\left( x\right)$ and $-g_{q}^{\prime }\left( x\right)$ are absolutely monotonic on $\left( 0,1\right)$ if and only if $p=4/3$, $p\ge 2$, $q\ge 8/5$ and $q\ge 2$, respectively. These results improve and extend some existing results, and yield some new functional inequalities involving ${\mathcal {K}} \left( r\right)$ and ${\mathcal {K}}\left( r^{\prime }\right)$, where $r^{\prime }=\sqrt{1-r^{2}}$.