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.

In this article, the finite-time H _∞ control problem for singular positive Markovian jump systems with time-varying delay and saturation constraint was studied. Firstly, considering the discontinuities caused by the mode-dependent singular matrix and Markov jump switching behavior, a state feedback controller is designed to guarantee the positivity and mean-square locally finite-time H _∞ stability of the considered system. Secondly, the maximum finite-time domain of attraction of the considered system subject to input saturation is estimated. Finally, to show the effectiveness of our control strategy, the simulation results are given.