Trace inequalities for positive semidefinite block matrices

If a 2×2 block matrix [ABB⁎C] is positive semidefinite, where each block is square, then the following trace inequality holds|trAC−trB⁎B|≤trAtrC−|trB|2. This improves a result of Besenyei [2]. Moreover, we show that|trBm|≤trAm/2Cm/2,m=2,3,….