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In this paper, we shall give a concrete relation between the generalizedKarcher equation and operator means as its solution. Next, we shall show twotypes of the Ando-Hiai inequalities for the solution of the generalized Karcherequation. In this discussion, we also give a property of relative operatorentropy.

The Ando-Hiai inequalities for the solution of the generalized Karcher equation and related results