  In the present paper, we provide several inequalities for the generalized
numerical radius of operator matrices as introduced by A. Abu-omar and F.
Kittaneh in [3]. We generalize the convexity and the log-convexity results
obtained by M. Sababheh in [12] for the case of the numerical radius to the
case of the generalized numerical radius. We illustrate our work by providing a
positive answer for the question addressed in [12] for the convexity of a
certain matrix operator function. Moreover, and motivated by the results of A.
Aldalabih and F. Kittaneh in [2] for the case of Hilbert-Schmidt numerical
radius norm, we use some Schatten $p$-norm inequalities for partitioned
$2\times 2$ block-matrices to provide several Schatten $p$-norm numerical
radius inequalities.
