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

TL;DR

This paper develops variants of Ando--Hiai inequalities for deformed operator means, explores their monotonicity properties, and improves related norm inequalities, advancing the understanding of operator inequalities and means.

Contribution

It introduces new variants of Ando--Hiai inequalities for deformed means and analyzes their monotonicity and norm inequalities, extending existing operator inequality theory.

Findings

Variants of Ando--Hiai inequalities for deformed means are established.

Monotonicity of the power mean from deformed means is analyzed using generalized Kantorovich constants.

Norm inequalities for operator power means related to the Log-Euclidean mean are improved using the Specht ratio.

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.