Refined Heinz operator inequalities and norm inequalities

TL;DR

This paper presents new refined inequalities related to Heinz and Hermite-Hadamard inequalities involving unitarily invariant norms, along with operator inequalities for positive definite matrices and operator monotone functions, improving existing results.

Contribution

It introduces a series of refinements for Heinz and Hermite-Hadamard inequalities and establishes new operator inequalities involving positive definite matrices and operator monotone functions.

Findings

Refined Heinz and Hermite-Hadamard inequalities with unitarily invariant norms.

Operator inequality involving positive definite matrices and operator monotone functions.

Series of improved Heinz operator inequalities.

Abstract

In this article we study the Heinz and Hermite-Hadamard inequalities. We derive the whole series of refinements of these inequalities involving unitarily invariant norms, which improve some recent results, known from the literature. We also prove that if $A , B, X\in M_n(\mathbb{C})$ such that $A$ and $B$ are positive definite and $f$ is an operator monotone function on $(0,\infty)$. Then \begin{equation*} |||f(A)X-Xf(B)|||\leq \max\{||f'(A)||, ||f'(B)||\} |||AX-XB|||. \end{equation*} Finally we obtain a series of refinements of the Heinz operator inequalities, which were proved by Kittaneh and Krni\'c.