# More accurate numerical radius inequalities

## TL;DR

This paper introduces new, refined inequalities for the numerical radius of Hilbert space operators, extending existing bounds and providing tighter estimates for operator norms.

## Contribution

It presents novel general forms of numerical radius inequalities that improve upon and extend known results in operator theory.

## Key findings

- Derived an upper bound for the numerical radius involving an integral of operator expressions.
- Established a refined inequality connecting the numerical radius with the operator norm.
- Provided a tighter inequality that generalizes previous bounds in the literature.

## Abstract

In this article, we present some new general forms of numerical radius inequalities for Hilbert space operators. The significance of these inequalities follow from the way they extend and refine some known results in this field. Among other inequalities, it is shown that if $A$ is a bounded linear operator on a complex Hilbert space, then \[{{w}^{2}}\left( A \right)\le \left\| \int_{0}^{1}{{{\left( t\left| A \right|+\left( 1-t \right)\left| {{A}^{*}} \right| \right)}^{2}}dt} \right\|\le \frac{1}{2}\left\| \;{{\left| A \right|}^{2}}+{{\left| {{A}^{*}} \right|}^{2}} \right\|\] where $w\left( A \right)$ and $\left\| A \right\|$ are the numerical radius and the usual operator norm of $A$, respectively.