  In this work, the mixed Schwarz inequality for semi-Hilbertian space
operators is proved. Namely, for every positive Hilbert space operator $A$. If
$f$ and $g$ are nonnegative continuous functions on $\left[0,\infty\right)$
satisfying $f(t)g(t) =t$ $(t\ge0)$, then \begin{align*} \left| {\left\langle {T
x,y} \right\rangle_A } \right| \le \left\| {f\left( {\left| T \right|_A x}
\right)} \right\|_A \left\| {g\left( {\left| {T^{\sharp_A } } \right|_A y}
\right)} \right\|_A \end{align*} for every Hilbert space operator $T$ such that
the range of $T^* A$ is a subset in the range of $A$, such that $A$ commutes
with $T$, and for all vectors $x,y\in \mathscr{H}$, where $\left| T \right|_A =
\left(AT^{\sharp_A}T\right)^{1/2}$ such that $T^{\sharp_A}=A^\dagger T^*A$,
where $A^\dagger$ is the Moore-Penrose inverse of $A$. Based on that, some
inequalities for the $A$-numerical radius are introduced.
