on $\S$, we see that $\S$ is umbilical.

Poincaré type and Lichnerowicz type inequalities {#sec Lich}
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Along the same line of the above results, we now prove a Poincare type inequality.

\[Poincare\] Let $(M^n, \bar g, V=e^u)$ be an $n$-dimensional Riemannian triple and $\a,\g\in \mathbb{R}$. Let $\O$ be a bounded domain in $M$ with smooth boundary $\S$. Assume $Ric^{D}$ of $D=D^{\a, \g}$ is positive definite. For any $f\in C^\infty(M)$ and $\tau=(n+1)\a+\g$, if one of the following alternatives holds,

-   $\S=\emptyset$ and $\di\int_{\O}f V^\tau d\O=0$;

-   $\S\neq\emptyset$, $f\equiv 0$ on $\S$ and $H^D\ge0$;

-   $\S\neq \emptyset$, $\di\int_{\O}f V^\tau d\O=0$ and $\S$ satisfies $h-\g u_\nu g\ge 0$.

Then we have $$\begin{aligned}
\frac{n}{n-1}\di\int_{\O}f^2V^\tau d\O\le \di\int_{\O}\left<{(\mathrm{Ric}^{D})}^{-1}\nb f,\nb f\right>V^\tau d\O.
            \label{Poincare case 1}
        \end{aligned}$$

The proof is similar as in [@Mi] while we use $Ric^D$ in this paper. In case (i), we solve PDE $$\begin{aligned}
\label{closed Poincare}
\bd^D \phi=f \hbox{ in }\O.\end{aligned}$$ In case (ii), we solve the Dirichlet boundary value problem below, $$\label{Dirichlet Poincare}
\left\{
    \begin{array}[]{rlll}
        \di \bd^D \phi&=&f&\mathrm{ in }\ \O,\\
        \phi&=& 0 &\mathrm{ on }\ \S.\\
    \end{array}
    \right.$$ In case (iii), we solve the Neumann boundary value problem $$\label{Neumann Poincare}
\left\{
    \begin{array}[]{rlll}
        \di \bd^D \phi&=&f&\mathrm{ in }\ \O,\\
        \phi_{\nu}&=& 0 &\mathrm{ 