15A42,15A57

Da-wei Zhang [J.M.A.A., 237 (1999):721-725] obtained the inequality between and for Hermitian matrices and , where is natural number. Here it is proved that these results hold when the power index of the product of Hermitian matrices and is nonnegative even number. In the meantime, it is pointed out that the relation between and is complicated when the power index is a nonnegative odd number, therefore the above inequality can't be generalized to all nonnegative integers. As an application, we not only improve the results of Xiaojing Yang [J.M.A.A., 250 (2000), 372-374], Xinmin Yang [J.M.A.A., 263 (2001):327-333] and Fozi M. Dannan [J.Ineq. Pure and Appl. Math., 2(3) Art.34 (2001)], moreover give the complete resolution for the question of the trace inequality about the powers of Hermitian and skew Hermitian matrices that is proposed by Zhengming Jiao.;