  In this paper we first establish an optimal Sobolev type inequality for
hypersurfaces in $\H^n$(see Theorem \ref{mainthm1}). As an application we
obtain hyperbolic Alexandrov-Fenchel inequalities for curvature integrals and
quermassintegrals. Precisely, we prove a following geometric inequality in the
hyperbolic space $\H^n$, which is a hyperbolic Alexandrov-Fenchel inequality,
\begin{equation*} \begin{array}{rcl} \ds \int_\Sigma \s_{2k}\ge \ds\vs
C_{n-1}^{2k}\omega_{n-1}\left\{\left(\frac{|\Sigma|}{\omega_{n-1}}
\right)^\frac 1k + \left(\frac{|\Sigma|}{\omega_{n-1}} \right)^{\frac 1k\frac
{n-1-2k}{n-1}} \right\}^k, \end{array} \end{equation*} provided that $\Sigma$
is a horospherical convex, where $2k\leq n-1$. Equality holds if and only if
$\Sigma$ is a geodesic sphere in $\H^n$. Here $\sigma_{j}=\s_{j}(\kappa)$ is
the $j$-th mean curvature and $\kappa=(\kappa_1,\kappa_2,\cdots, \kappa_{n-1})$
is the set of the principal curvatures of $\Sigma$. Also, an optimal inequality
for quermassintegrals in $\H^n$ is as following: $$ W_{2k+1}(\Omega)\geq\frac
{\omega_{n-1}}{n}\sum_{i=0}^k\frac{n-1-2k}{n-1-2k+2i}\,C_k^i\bigg(\frac{nW_1(\Omega)}{\omega_{n-1}}\bigg)^{\frac{n-1-2k+2i}{n-1}},
$$ provided that $\Omega\subset\H^n$ is a domain with $\Sigma=\partial\Omega$
horospherical convex, where $2k\leq n-1$. Equality holds if and only if
$\Sigma$ is a geodesic sphere in $\H^n$. Here $W_r(\Omega)$ is
quermassintegrals in integral geometry.
