Further Inequalities for the Numerical Radius of Hilbert Space Operators (1907.06003v1)

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.