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In this work, the mixed Schwarz inequality for semi-Hilbertian spaceoperators 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 {Tx,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 thatthe range of $T^* A$ is a subset in the range of $A$, such that $A$ commuteswith $T$, 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, someinequalities for the $A$-numerical radius are introduced.

The generalized Schwarz inequality for semi-Hilbertian space operators and Some $A$-numerical radius inequalities