On the monotonicity and convexity for generalized elliptic integral of the first kind

In this paper, we investigate the monotonicity and convexity of the function x bar right arrow K-a(root x)/log(1 + c/root 1 - x) on (0, 1) for (a, c) is an element of (0, 1/2] x (0, infinity), and the log-concavity of the function x bar right arrow (1 - x)K-lambda(a)(root x) on (0, 1) for lambda is an element of R, where K-a(r) is the generalized elliptic integral of the first kind. These results are the generalization of [1, Theorem 2] and [2, Theorems 1.7 and 1.8], also give an affirmative answer of [3, Problem 3.1].