Numerical Radius Inequalities via Orlicz function

TL;DR

This paper introduces new inequalities for the numerical radius of bounded linear operators using Orlicz functions, extending classical inequalities and providing improved upper bounds for the numerical radius.

Contribution

It extends Buzano's inequality via Orlicz functions and derives novel upper bounds for the numerical radius of operators and their products.

Findings

Derived new upper bounds for the numerical radius.

Extended Buzano's inequality using Orlicz functions.

Established inequalities involving operator norms and powers.

Abstract

Employing the Orlicz functions we extend the Buzano's inequality which is a refinement of the Cauchy-Schwarz inequality. Also using the Orlicz functions we obtain several numerical radius inequalities for a bounded linear operator as well as the products of operators. We deduce different new upper bounds for the numerical radius. It is shown that \begin{eqnarray*} {w(T)} \leq \sqrt[n]{ \log \left[ \frac{1}{2^{n-1}} e^{w(T^n)} + \left( 1-\frac{1}{2^{n-1}}\right) e^{\|T\|^n}\right]} &\leq& \|T\| \quad \forall n=2,3,4, \ldots \end{eqnarray*} where $w(T)$ and $\|T\|$ denote the numerical radius and the operator norm of a bounded linear operator $T$, respectively.