New Estimates on Numerical Radius and Operator Norm of Hilbert Space Operators

The main goal of this article is to present a new approach, made up of integrals, to refine some numerical radius inequalities. Let $A$ be a bounded linear operator on a complex Hilbert space. If $1\le r\le 2$, it is shown that \[{{\omega }^{2r}}\left( A \right)\le \left\| \int_{0}^{1}{{{\left( \left( 1-t \right)\left( \frac{{{\left| A \right|}^{2}}+{{\left| {{A}^{*}} \right|}^{2}}}{2} \right)+t\omega \left( {{A}^{2}} \right)I \right)}^{r}}dt} \right\|\,.\] Here $\omega \left( \cdot \right)$, $\left\| \cdot \right\|$ are the numerical radius and the usual operator norm, $\left| A \right|={{\left( {{A}^{*}}A \right)}^{{1}/{2}\;}}$, and $I$ is the identity operator.