  In this paper, we establish some upper bounds for numerical radius
inequalities including of $2\times 2$ operator matrices and their off-diagonal
parts. Among other inequalities, it is shown that if $T=\left[\begin{array}{cc}
  0&X,
  Y&0
  \end{array}\right]$, then
  \begin{align*}
  \omega^{r}(T)\leq
2^{r-2}\left\|f^{2r}(|X|)+g^{2r}(|Y^*|)\right\|^\frac{1}{2}\left\|f^{2r}(|Y|)+g^{2r}(|X^*|)\right\|^\frac{1}{2}
  \end{align*} and
  \begin{align*}
  \omega^{r}(T)\leq
2^{r-2}\left\|f^{2r}(|X|)+f^{2r}(|Y^*|)\right\|^\frac{1}{2}\left\|g^{2r}(|Y|)+g^{2r}(|X^*|)\right\|^\frac{1}{2},
  \end{align*} where $X, Y$ are bounded linear operators on a Hilbert space
${\mathscr H}$, $r\geq 1$ and $f$, $g$ are nonnegative continuous functions on
$[0, \infty)$ satisfying the relation $f(t)g(t)=t\,(t\in[0, \infty))$.
Moreover, we present some inequalities involving the generalized Euclidean
operator radius of operators $T_{1},\cdots,T_{n}$.
