Affiliation:
1. Department of Mathematics, Faculty of Science P.O.Box(7), Mu'tah university, Al-Karak, Jordan
2. Department of Mathematics, Faculty of Science, The Hashemite University P.O.Box 330127, Zarqa 13133, Jordan
Abstract
<abstract><p>We provide a number of sharp inequalities involving the usual operator norms of Hilbert space operators and powers of the numerical radii. Based on the traditional convexity inequalities for nonnegative real numbers and some generalize earlier numerical radius inequalities, operator. Precisely, we prove that if $ {\bf A}_i, {\bf B}_i, {\bf X}_i\in \mathcal{B}(\mathcal{H}) $ ($ i = 1, 2, \cdots, n $), $ m\in \mathbb N $, $ p, q > 1 $ with $ \frac{1}{p}+\frac{1}{q} = 1 $ and $ \phi $ and $ \psi $ are non-negative functions on $ [0, \infty) $ which are continuous such that $ \phi(t)\psi(t) = t $ for all $ t \in [0, \infty) $, then</p>
<p><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \begin{equation*} w^{2r}\left({\sum\limits_{i = 1}^{n} {\bf X}_i {\bf A}_i^m {\bf B}_i}\right)\leq \frac{n^{2r-1}}{m}\sum\limits_{j = 1}^{m}\left\Vert{\sum\limits_{i = 1}^{n}\frac{1}{p}S_{i, j}^{pr}+\frac{1}{q}T_{i, j}^{qr}}\right\Vert-r_0\inf\limits_{\left\Vert{\xi}\right\Vert = 1}\rho(\xi), \end{equation*} $\end{document} </tex-math></disp-formula></p>
<p>where $ r_0 = \min\{\frac{1}{p}, \frac{1}{q}\} $, $ S_{i, j} = {\bf X}_i\phi^2\left({\left\vert{ {\bf A}_i^{j*}}\right\vert}\right) {\bf X}_i^* $, $ T_{i, j} = \left({ {\bf A}_i^{m-j} {\bf B}_i}\right)^*\psi^2\left({\left\vert{ {\bf A}_i^j}\right\vert}\right) {\bf A}_i^{m-j} {\bf B}_i $ and</p>
<p><disp-formula> <label/> <tex-math id="FE2"> \begin{document}$ \rho(\xi) = \frac{n^{2r-1}}{m}\sum\limits_{j = 1}^{m}\sum\limits_{i = 1}^{n}\left({\left<{S_{i, j}^r\xi, \xi}\right>^{\frac{p}{2}}-\left<{T_{i, j}^r\xi, \xi}\right>^{\frac{q}{2}}}\right)^2. $\end{document} </tex-math></disp-formula></p>
</abstract>
Publisher
American Institute of Mathematical Sciences (AIMS)