An estimate for the numerical radius of the Hilbert space operators and a numerical radius inequality

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

$ \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*} $

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

$ \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. $