a “weight”, to get the divergent-free property. Second, we choose a suitable vector field $X$ which satisfies $DX$ is symmetric. It turns out that we choose $X=\bn^D \phi$ and the volume form $V^\tau d\O$, see Lemmata \[sym vf\] and \[weight\].

With the integral formula in Theorem \[theo RF0\] at hand, we are able to prove Heintze-Karcher type, Minkowski type, and Lichnerowicz type inequalities.

\[thm ineq\] Let $(M^n, \bar g, V=e^u)$ be an $n$-dimensional Riemannian triple and $\a,\g\in \mathbb{R}$. Let $D=D^{\a,\g}$ be the affine connection defined as in (\[defi D\]) and $\tau=(n+1)\a+\g$. Then we have the following results.

-   [**Heintze-Karcher type inequality:**]{} if $Ric^{D}\ge 0$ and $H^D>0$, then $$n\di\int_{\O}V^\tau d\O\le\di (n-1)\int_{\S}\frac{V^\tau}{H^D} dA.
             \label{Heintze-Karcher}$$ Equality in holds only if $\S$ is umbilical.

-   [**Minkowski inequality:**]{} If $Ric^{D}\ge 0$ and $h-\g u_\nu g> 0$, then $$\begin{aligned}
    \label{rm2}
            \left(\int_\S V^{\tau-\a} dA\right)^2\geq \frac{n}{n-1}\int_\Omega V^\tau d\O \int_\S H^DV^{\tau-2\a} dA,
        \end{aligned}$$ Equality in holds only if $\S$ is umbilical.

-   [**Lichnerowicz inequality:**]{} If $Ric^{D}\ge (n-1)V^{\a-\g} \bar g$ and\
    [**a)** ]{}$\S=\emptyset$, then $\lambda_1(\bd^D)\ge n;$\
    [**b)** ]{}$\S \neq\emptyset$ and $\S$ satisfies $H^D\ge 0$, then $\lambda_1^{Dir}(\bd^D)\ge n;$\
    [**c)** ]{}$\S\neq\emptyset$ and $h-\g u_\nu g\ge 0$, then $\lambda_1^{Neu}(\bd^D)\ge n.$

Here $\lambda_1, \lambda_1^{Dir}$ and $\lambda_1^{Neu}$ indicate the closed, the Dirichlet and the Neumann 