Concavity and Bounds Involving Generalized Elliptic Integral of the First Kind

Abstract

In the article, we provide a sufficient condition for value range of the constant c such that the function (Formula presented) is strictly concave on (0,1) for a ∈ (0,1/2], which generalize a very recently obtained result that the function (Formula presented) is strictly concave on (0,1) if and only if c = e4/3. As applications, we present new bounds for (Formula presented), where (Formula presented)a(x) is the generalized elliptic integral of the first (Formula presented)ind and (Formula presented) (x) =(Formula presented)1/2(x).