  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 $\A_i,\B_i,\X_i\in\bh$ ($i=1,2,\cdots,n$), $m\in\N$, $p,q>1$ 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}\bra{\sum_{i=1}^{n}\X_i\A_i^m\B_i}\leq
\frac{n^{2r-1}}{m}\sum_{j=1}^{m}\norm{\sum_{i=1}^{n}\frac{1}{p}S_{i,j}^{pr}+\frac{1}{q}T_{i,j}^{qr}}-r_0\inf_{\norm{x}=1}\rho(\xi),
  \end{equation*}
  where $r_0=\min\{\frac{1}{p},\frac{1}{q}\}$,
$S_{i,j}=\X_i\phi^2\bra{\abs{\A_i^{j*}}}\X_i^*$,
$T_{i,j}=\bra{\A_i^{m-j}\B_i}^*\psi^2\bra{\abs{\A_i^j}}\A_i^{m-j}\B_i$ and
  $$\rho(x)=\frac{n^{2r-1}}{m}\sum_{j=1}^{m}\sum_{i=1}^{n}\bra{\seq{S_{i,j}^r\xi,\xi}^{\frac{p}{2}}-\seq{T_{i,j}^r\xi,\xi}^{\frac{q}{2}}}^2.$$
