Some refinements of numerical radius inequalities

In this paper, we give some refinements for the second inequality in \(\frac{1}{2}\|A\| \leq w(A) \leq \|A\|\), where \(A\in B(H)\). In particular, if \(A\) is hyponormal by refining the Young inequality with the Kantorovich constant \(K(\cdot, \cdot)\), we show that \(w(A)\leq \dfrac{1}{\displaystyle {2\inf_{\| x \|=1}}\zeta(x)}\| |A|+|A^{*}|\|\leq \dfrac{1}{2}\| |A|+|A^*|\|\), where \(\zeta(x)=K(\frac{\langle |A|x,x \rangle}{\langle |A^{*}|x,x \rangle},2)^{r},~~~r=\min\{\lambda,1-\lambda\}\) and \(0\leq \lambda \leq 1\) . We also give a reverse for the classical numerical radius power inequality \(w(A^{n})\leq w^{n}(A)\) for any operator \(A \in B(H)\) in the case when \(n=2\).