  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) $.
