  We prove two anisotropic type weighted geometric inequalities that hold for
star-shaped and $F$-mean convex hypersurfaces in $\mathbb{R}^{n+1}$. These
inequalities involve the anisotropic $p$-momentum, the anisotropic perimeter
and the volume of the region enclosed by the hypersurface. We show that the
Wulff shape of $F$ is the unique minimizer of the corresponding functionals
among all star-shaped and $F$-mean convex sets. We also consider their
quantitative versions characterized by the Hausdorff distance between the
hypersurface and a rescaled Wulff shape. As a corollary, we obtain the
stability of the Weinstock inequality for star-shaped and strictly mean convex
domains, which requires weaker convexity compared to \cite{Gavitone}.
