Operator revision of a Ky Fan type inequality

TL;DR

This paper extends a Ky Fan type inequality involving operator means on Hilbert space operators, establishing a new inequality that generalizes previous results and exploring related operator inequalities.

Contribution

It introduces a novel operator inequality extending Ky Fan type inequalities using weighted operator means on Hilbert space operators.

Findings

Proves a new inequality involving weighted arithmetic and harmonic operator means.

Extends a classical Ky Fan inequality to the operator setting.

Provides related parallel results in operator inequalities.

Abstract

Let $\mathscr{H}$ be a complex Hilbert space and $A,B\in \mathbb{B}(\mathscr{H})$ such that $0<A,B\leq\frac{1}{2}I$. Setting $A':=I-A$ and $B':=I-B$, we prove $$ A'\nabla_\lambda B'-A'!_\lambda B' \leq A\nabla_\lambda B-A!_\lambda B, $$ where $\nabla_\lambda$ and $!_\lambda$ denote the weighted arithmetic and harmonic operator means, respectively. This inequality is the natural extension of a Ky Fan type inequality due to H. Alzer. Some parallel and related results are also obtained.