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

Let K (x) be the complete elliptic integral of the first kind and equation* G₏ (x) =e^K (x) -p1-x equation* for p R and x (0, 1). In this paper we find the necessary and sufficient conditions for the functions G ₏^ (k) (x) (k N \ 0\) to be absolutely monotonic on (0, 1), 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 G₏ (x).