For a ? (0,1/2] and r ? (0,1), let Ka(r) (K (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 ? [K(r)-Ka(r)]=(1-2a)?(?? R) is monotone
on (0,1/2) for each fixed r ? (0,1). The obtained result leads to the
conclusion that inequality K (r)- (1-2a)? [K(r)- ?/2] ? Ka(r) ? K
(r)-(1-2a)? [K(r)-?/2] holds for all a ? (0,1/2] and r ? (0,1)
with the best possible constants ? = ?/2 and ? = 2.