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For a probability measure of compact support  μ  on the setof all positive definite matrices and, letbe the unique positive solution of. In this paper, we show that if  μ  and  ν  are probability measures onand, respectively, andis a unital positive bilinear map, thenfor all, where  λ  is the push-forward of the product measureby the mapping. In particular,. Moreover, we show thatfor every, whereand  I  denotes the identity matrix.

Matrix inequalities related to power means of probability measures