  In this paper, we present several sharp upper bounds for the numerical radii
of the diagonal and off-diagonal parts of the $2\times2$ block operator matrix
$\begin{bmatrix}A&B\\ C&D\end{bmatrix}$. Among extensions of some results of
Kittaneh et al., it is shown that if $T=\begin{bmatrix}A&0\\ 0&D\end{bmatrix}$,
and $f$ and $g$ are non-negative continuous functions on $[0,\infty)$ such that
$f(t)g(t)=t\,\,(t\geq 0)$, then for all nonnegative nondecreasing convex
functions $h$ on $[0,\infty)$ , we obtain that
\begin{align*}h\left(w^r(T)\right)\leq
\max\left(\left\|\frac{1}{p}h\left(f^{pr}(\left|A\right|)\right)+
\frac{1}{q}h\left(g^{qr}(\left|A^*\right|)\right)\right\|,
\left\|\frac{1}{p}h\left(f^{pr}(\left|D\right|)\right)+
\frac{1}{q}h\left(g^{qr}(\left|D^*\right|)\right)\right\|\right), \end{align*}
where $p, q>1$ with $\frac{1}{p}+\frac{1}{q}=1$ and $r\min(p,q)\geq 2$.
