  In this paper, we establish two families of sharp geometric inequalities for
closed hypersurfaces in space forms or other warped product manifolds. Both
families of inequalities compare three distinct geometric quantities. The first
family concerns the $k$-th boundary momentum, area, and weighted volume, and
has applications to Weinstock-type inequalities for Steklov or Wentzell
eigenvalues on star-shaped mean convex domains. This generalizes the main
results of [12]. The second family involves a weighted $k$-th mean curvature
integral and two distinct quermassintegrals and extends the authors' recent
work [33] with G. Wheeler and V.-M. Wheeler.
