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*} Gₚ( x) =eK( √x )-{p}{√1-x} {equation*} for p∈ R and x∈ ( 0,1). In this paper we find the necessary and sufficient conditions for the functions ± G ₚ( 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 Gₚ( x).