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We give several inequalities involving numerical radii $\omega \left( \cdot \right)$ and the usual operator norm $\|\cdot\|$ of Hilbert space operators. These inequalities lead to a considerable improvement in the well-known inequalities\begin{equation*}\frac{1}{2}\left\| T \right\|\le \omega \left( T \right)\le \left\| T \right\|.\end{equation*}