# Further Inequalities for the Numerical Radius of Hilbert Space Operators

## TL;DR

This paper introduces new inequalities for the numerical radius of Hilbert space operators using convex functions, extending previous results and providing sharper bounds for operator analysis.

## Contribution

It generalizes and improves existing inequalities for the numerical radius, offering new bounds involving convex functions and operator norms.

## Key findings

- Derived inequalities for numerical radius involving convex functions.
- Extended bounds for the numerical radius when r ≥ 2.
- Improved previous inequalities by El-Haddad and Kittaneh.

## Abstract

In this article, we present some new inequalities for numerical radius of Hilbert space operators via convex functions. Our results generalize and improve earlier results by El-Haddad and Kittaneh. Among several results, we show that if $A\in \mathbb{B}\left( \mathcal{H} \right)$ and $r\ge 2$, then \[{{w}^{r}}\left( A \right)\le {{\left\| A \right\|}^{r}}-\underset{\left\| x \right\|=1}{\mathop{\inf }}\,{{\left\| {{\left| \left| A \right|-w\left( A \right) \right|}^{\frac{r}{2}}}x \right\|}^{2}}\] where $w\left( \cdot \right)$ and $\left\| \cdot \right\|$ denote the numerical radius and usual operator norm, respectively.