Power numerical radius inequalities from an extension of Buzano's inequality

Several numerical radius inequalities are studied by developing an extension of the Buzano’s inequality. It is shown that if $T$ is a bounded linear operator on a complex Hilbert space, then \begin{eqnarray*} w^n(T) &\leq& \frac{1}{2^{n-1}} w(T^n)+ \sum_{k=1}^{n-1} \frac{1}{2^{k}} \left|T^k \right| \left|T \right|^{n-k}, \end{eqnarray*} for every positive integer $n\ge 2.$ This is a non-trivial improvement of the classical inequality $w(T)\le \mathrm{\parallel }T\mathrm{\parallel }\mathrm{.}$ The above inequality gives an estimation for the numerical radius of the nilpotent operators, i.e., if $T^n=0$ for some least positive integer $n\ge 2$, then \begin{eqnarray*} w(T) &\leq& \left(\sum_{k=1}^{n-1} \frac{1}{2^{k}} \left|T^k \right| \left|T \right|^{n-k}\right)^{1/n} \leq \left( 1- \frac{1}{2^{n-1}}\right)^{1/n} |T|. \end{eqnarray*} Also, we deduce a reverse inequality for the numerical radius power inequality $w(T^n)\le w^n(T)$. We show that if $\mathrm{\parallel }T\mathrm{\parallel }\le 1$, then \begin{eqnarray*} w^n(T) &\leq& \frac{1}{2^{n-1}} w(T^n)+ 1- \frac{1}{2^{n-1}}, \end{eqnarray*} for every positive integer $n\ge 2.$ This inequality is sharp.