Buzano, Kre\u{i}n and Cauchy-Schwarz inequalities

TL;DR

This paper presents refinements of the Buzano and Cauchy-Schwarz inequalities, along with a new proof of a Kren-type inequality, and explores applications to operator inequalities in Hilbert spaces.

Contribution

It introduces new refinements and a novel proof for existing inequalities, expanding their applicability to operator theory in Hilbert spaces.

Findings

Refined Buzano and Cauchy-Schwarz inequalities

A new proof of a Kren-type inequality

Applications to operator norm and numerical radius inequalities

Abstract

The Cauchy-Schwarz, Buzano and Kre\u{\i}n inequalities are three inequalities about inner product. The main goal of this article is to present refinements of Buzano and Cauchy-Schwarz inequalities, and to present a new proof of a refined version of a Kre\u{\i}n-type inequality. Applications that include Buzano-type inequalities for certain operators, operator norm and numerical radius inequalities of Hilbert space operators will be presented.