  Let $A$ be a positive operator on a complex Hilbert space $\mathcal{H}.$ We
present inequalities concerning upper and lower bounds for $A$-numerical radius
of operators, which improve on and generalize the existing ones, studied
recently in [A. Zamani, A-Numerical radius inequalities for semi-Hilbertian
space operators, Linear Algebra Appl. 578 (2019) 159-183]. We also obtain some
inequalities for $B$-numerical radius of $2\times 2$ operator matrices where
$B$ is the $2\times 2$ diagonal operator matrix whose diagonal entries are $A$.
Further we obtain upper bounds for $A$-numerical radius for product of
operators which improve on the existing bounds.
