On the absolute monotonicity of generalized elliptic integral of the first kind
The purpose of this paper is to prove that the second derivative of the generalized elliptic integral of the first kind \(K_a(r)\), is absolutely monotonic on (0, 1). The generalized elliptic integrals of the first kind, which can be expressed as a hypergeometric series, plays an important role in different branches of modern mathematics. Starting with the inequality for \(K_{\tfrac 12}(r)\), several functions are shown to be absolutely monotonic on (0, 1). Then several abbreviations are introduced for brevity. The proofs use generalized Wallis' ratio, long, long differentiations, partial differentiations, numerical experiment, induction. Finally, the main theorems are proved by using Maclaurin series with hypergeometric series. Several new functional inequalities are proved in the end of paper by using formulas for the derivative of hypergeometric series. Inequalities with the best possible constants are proved and open problems are given.
scientific article; zbMATH DE number 7733899
scientific article