In the first part of the work, the authors establish a sharp geometric inequality involving the \(k\)-th boundary momentum \(\int_\Sigma \lambda(r)^k \,d\mu\), area \(|\Sigma|\), and weighted volume\(\int_\Omega \lambda^{k-1} \lambda' \,dv\) of a smooth bounded domain \(\Omega\) enclosed by a hypersurface \(\Sigma\) (with some additional convexity assumptions) in a warped product manifold \(\overline{M}^{n+1}\), with \(\mathbb{R}^{n+1}\), \(\mathbb{H}^{n+1}\), \(\mathbb{S}^{n+1}\) as explicit examples. The obtained inequality is used to prove the Weinstock inequality for the first nonzero Steklov eigenvalue for bounded domains with star-shaped mean convex boundary. This solves a problem posed in [D. Bucur et al., J. Differ. Geom. 118, No. 1, 1–21 (2021; Zbl 1468.35038)].
In the second part of the work, the authors establish a sharp geometric inequality involving the weighted curvature integral \(\int_\Sigma \Phi E_k \, d\mu\) and two quermassintegrals \(W_{k-1}(\Omega)\), \(W_{k}(\Omega)\), under related assumptions on \(\Omega\), \(\Sigma\), where, in the particular case of \(\mathbb{R}^{n+1}\), \(\Phi(r)=r^2/2\).