Inequalities for generalized matrix function and inner product

We present inequalities related to generalized matrix function for positive semidefinite block matrices. We introduce partial generalized matrix functions corresponding to partial traces and then provide an unified extension of the recent inequalities due to Choi [6], Lin [14] and Zhang et al. [5,19]. We demonstrate the applications of a positive semidefinite $3times 3$ block matrix, which motivates us to give a simple alternative proof of Dragomirs inequality and Kreins inequality.