@article{Heydarbeygi_Amyari_Khanehgir_2020, title={Some refinements of numerical radius inequalities}, volume={72}, url={https://umj.imath.kiev.ua/index.php/umj/article/view/6027}, DOI={10.37863/umzh.v72i10.6027}, abstractNote={<p>UDC 517.5</p> <div class="page" title="Page 1"> <div class="layoutArea"> <div class="column"> <p>In this paper, we give some refinements for the second inequality in $\dfrac{1}{2}\|A\| \leq w(A) \leq \|A\|,$&nbsp; where $A\in B(H).$&nbsp;&nbsp;In particular, if $A$ is hyponormal by refining the Young inequality with the Kantorovich constant $K(\cdot, \cdot),$&nbsp;&nbsp;we show that $w(A)\leq \dfrac{1}{\displaystyle {2\inf olimits_{\| x \|=1 }\zeta(x)}\| |A|+|A^{*}|\|\leq \dfrac{1}{2}\| |A|+|A^*|\|,$&nbsp;&nbsp;where $\zeta(x)=K\left(\dfrac{\langle |A|x,x \rangle}{\langle |A^{*}|x,x \rangle},2\right)^{r},$ $r=\min\{\lambda,1-\lambda\}$ and $0\leq \lambda \leq 1$ .&nbsp;We also give a reverse for the classical numerical radius power inequality $w(A^{n})\leq w^{n}(A)$&nbsp;for any operator $A \in B(H)$ in the case when $n=2.$&nbsp;</p> </div> </div> </div&gt;}, number={10}, journal={Ukrains’kyi Matematychnyi Zhurnal}, author={Heydarbeygi, Z. and Amyari, M. and Khanehgir, M.}, year={2020}, month={Oct.}, pages={1443 - 1451} }