A MONOTONICITY THEOREM FOR THE GENERALIZED ELLIPTIC INTEGRAL OF THE FIRST KIND

For a E (0, 1/2] and r E (0, 1), let Xa(r) (X (r)) denote the generalized elliptic integral (complete elliptic integral, respectively) of the first kind. In this article, we mainly present a sufficient and necessary condition under which the function a 7 -> [X (r) - Xa(r)]/(1 - 2a)lambda(lambda E R) is monotone on (0, 1/2) for each fixed r E (0, 1). The obtained result leads to the conclusion that inequalityX (r) - (1 - 2a)alpha [X (r) - pi ] < Xa(r) < X (r) - (1 - 2a)beta [X (r) - pi ] 2 2holds for all a E (0, 1/2] and r E (0, 1) with the best possible constants alpha = pi /2 and beta = 2.