  New inequalities for the $A$-numerical radius of the products and sums of
operators acting on a semi-Hilbert space, i.e. a space generated by a positive
semidefinite operator $A$, are established. In particular, it is proved for
operators $T$ and $S,$ having $A$-adjoint, that $$ \omega_A(TS) \leq
\frac{1}{2}\omega_A(ST)+\frac{1}{4}\Big(\|T\|_A\|S\|_A+\|TS\|_A\Big),$$ where
$\omega_A(T)$ and $\|T\|_A$ denote the $A$-numerical radius and the
$A$-operator seminorm of an operator $T$.
