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.
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机译：Zhang Da-wei Zhang [J.M.A.A.，237（1999）：721-725]获得了Hermitian矩阵和之间的不等式，其中是自然数。在此证明，当厄米矩阵的乘方的幂指数为非负偶数时，这些结果成立。同时指出，当幂指数为非负奇数时，与之间的关系比较复杂，因此上述不等式不能推广到所有非负整数。作为应用，我们不仅改善了杨晓静[J.M.A.A.，250（2000），372-374]，杨新民[J.M.A.A.，263（2001）：327-333]和Fozi M. Dannan [J.Ineq。纯和应用。 Math。，2（3）Art。34（2001）]，并且为焦正明提出的关于Hermitian和偏斜Hermitian矩阵的幂的迹不等式的问题提供了完整的解决方案。
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