On the P\'olya-Szeg\"o operator inequality

TL;DR

This paper extends classical Pólya-Szegő inequalities to positive invertible operators on Hilbert spaces using various operator means, and applies these results to derive related inequalities such as Grüss, Diaz–Metcalf, and Klamkin–McLenaghan.

Contribution

It introduces generalized Pólya-Szegő inequalities for operator means between arithmetic and harmonic means, broadening the scope of existing operator inequalities.

Findings

Established new operator inequalities of Pólya-Szegő type.

Derived applications including operator Grüss, Diaz–Metcalf, and Klamkin–McLenaghan inequalities.

Abstract

In this paper, we present generalized P\'olya-Szeg\"o type inequalities for positive invertible operators on a Hilbert space for arbitrary operator means between the arithmetic and the harmonic means. As applications, we present Operator Gr\"uss, Diaz--Metcalf and Klamkin--McLenaghan inequalities.