Operator revision of a Ky Fan type inequality

Abstract

Let $\mathcal{H}$ be a complex Hilbert space and $A,B\in \mathbb{B}(\mathcal{H})$ such that $0<A,B\le \frac{1}{2}I$. Setting $A^{\prime }:=I-A$ and $B^{\prime }:=I-B$, we prove $$A^{\prime }{\nabla }_{\lambda }{B}^{\prime }-A^{\prime }{!}_{\lambda }{B}^{\prime }\le A{\nabla }_{\lambda }B-A{!}_{\lambda }B,$$ 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.

References in corpus (1)

• Alzer Inequality for Hilbert Spaces Operators

Cited by in corpus (1)

• Operator Ky Fan type inequalities