A monotonicity theorem for the generalized elliptic integral of the first kind

For a ? (0,1/2] and r (0,1), let Ka(r) (K (r)) denote the generalized elliptic integral (complete integral, respectively) of first kind. In this article, we mainly present sufficient necessary condition under which function [K(r)-Ka(r)]=(1-2a)?(?? R) is monotone on (0,1/2) for each fixed (0,1). The obtained result leads to conclusion that inequality K (r)- (1-2a)? [K(r)- ?/2] (r)-(1-2a)? [K(r)-?/2] holds all (0,1) with best possible constants = ?/2 2.