Numerical radius inequalities for products and sums of semi-Hilbertian space operators

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 $${\omega }_A(TS)\le \frac{1}{2}{\omega }_A(ST)+\frac{1}{4}(\mathrm{\parallel }T{\mathrm{\parallel }}_A\mathrm{\parallel }S{\mathrm{\parallel }}_A+\mathrm{\parallel }TS{\mathrm{\parallel }}_A),$$ where ${\omega }_A(T)$ and $\mathrm{\parallel }T{\mathrm{\parallel }}_A$ denote the $A$-numerical radius and the $A$-operator seminorm of an operator $T$.