Norm inequalities related to operator monotone functions

TL;DR

This paper investigates inequalities involving unitarily invariant norms of derivatives of operator monotone functions, providing bounds and convexity properties that extend classical inequalities to the operator setting.

Contribution

It establishes new norm inequalities and convexity results for derivatives of operator monotone functions, with applications to bounds on differences of operator functions.

Findings

Derived bounds for $|||f(B)-f(A)|||$ in terms of $|||B-A|||$.

Proved quasi-convexity of $ orm{f^{(n)}(ullet)}$ on positive definite operators.

Extended Hermite-Hadamard type inequalities to operator functions.

Abstract

Let $A$ be a positive definite operator on a Hilbert space $H$, and $|||.|||$ be a unitarily invariant norm on $B(H)$. We show that if $f$ is an operator monotone function on $(0,\infty)$ and $n\in \mathbb{N}$, then $|||D^n f(A)|||\leq\|f^{(n)}(A)\|$ and $\|f^{(n)}(\cdot)\|$ is a quasi-convex function on the set of all positive definite operators in $B(H)$. We establish some estimates of the right hand side of some Hermite-Hadamard type inequalities in which differentiable functions are involved, and norms of the maps induced by them on the set of self adjoint operators are convex, quasi-convex or $s$-convex. As applications, we obtain some of bounds for $|||f(B)-f(A)|||$ in term of $|||B-A|||$. For instance, Let $f,g$ be two operator monotone functions on $(0,\infty)$. Then, for every unitarily invariant norm $|||.|||$ and every positive definite operators $A,B$, \begin{align*} &\left|\left|\left|f(A)g(A)-f(B)g(B)\right|\right|\right|\notag\\ &\leq|||B-A|||\Big[\max\left\{\|f'(A)\|,\|f'(B)\|\right\}\times\max\left\{\|g(A)\|,\|g(B)\|\right\}\notag\\ &+\max\left\{\|f(A)\|,\|f(B)\|\right\}\times \max\left\{\|g'(A)\|,\|g'(B)\|\right\}\Big]. \end{align*}