Some refinements of numerical radius inequalities

Abstract: 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 $n=2$.