Hyperbolic Alexandrov-Fenchel quermassintegral inequalities I

In this paper we prove the following geometric inequality in the hyperbolic space \H^n ($n\ge 5)$, which is a hyperbolic Alexandrov-Fenchel inequality, $$\begin{array}{r}\text{ds}{\int }_{\mathrm{\Sigma }}{\text{s}}_{4}d\mu \ge \text{ds}\text{vs}{C}_{n-1}^4{\omega }_{n-1}{\{{\left(\frac{\mathrm{\mid }\mathrm{\Sigma }\mathrm{\mid }}{{\omega }_{n-1}}\right)}^{\frac{1}{2}}+{\left(\frac{\mathrm{\mid }\mathrm{\Sigma }\mathrm{\mid }}{{\omega }_{n-1}}\right)}^{\frac{1}{2}\frac{n-5}{n-1}}\}}^2,\end{array}$$ provided that $\mathrm{\Sigma }$ is a horospherical convex hypersurface. Equality holds if and only if $\mathrm{\Sigma }$ is a geodesic sphere in \H^n.