Variants of Ando--Hiai type inequalities for deformed means and applications

Abstract

For an n-tuple of positive invertible operators on a Hilbert space, we present some variants of Ando-Hiai type inequalities for deformed means from an n-variable operator mean by an operator mean, which is related to the information monotonicity of a certain unital positive linear map. As an application, we investigate the monotonicity of the power mean from the deformed mean in terms of the generalized Kantorovich constants under the operator order. Moreover, we improve the norm inequality for the operator power means related to the Log-Euclidean mean in terms of the Specht ratio.

Mixture and interpolation of the parameterized ordered means

“…for any ρ, σ ∈ P ∞ (X). Thus, the parameterized ordered mean G µ = {G µ n } with invariance under permutation and repetition can be extended to a d W ∞ -contractive barycentric map β G µ by Proposition 3.1 (6). It is also an interesting problem to generalize results in Section…”