nonnegative.

Equivalently, we have shown, in all the three cases, the following inequality holds, $$\begin{aligned}
    \di\frac{n-1}{n}\int_{\O}f^2V^\tau  \ge \int_{\O}V^\tau Ric^{D}(\bn^D \phi, \bn^D \phi).
    \label{Neumann Poincare equ 1}\end{aligned}$$

On the other hand, by integration by parts and recall Definition \[notation\], also noting the divergence property , we have $$\begin{aligned}
    \begin{array}[]{rll}
    \di\int_{\O}f^2V^\tau  =&\di\int_{\O}fV^\tau \bd^D \phi=\int_{\O} f\bar\n(V^{\tau}\bar\n^D\phi )\\
\\
    =&\displaystyle\int_{\S} fV^{\tau}\bar\n^D_\nu\phi   -\int_{\O}V^{\tau}\<\bn^D \phi, \bar \n  f\>\\
    \\
    =&\displaystyle-\int_{\O}V^{\tau}\<\bn^D \phi, \bar \n  f\>,\\
    \end{array}
    \label{Neumann Poincare equ 1.2}\end{aligned}$$ where the last identity holds due to the following observations: case(i), $\S=\emptyset$; case (ii), $f\equiv 0$ on $\S$; case (iii), $\phi_\nu\equiv 0$ on $\S$.

Applying Hölder’s inequality to (\[Neumann Poincare equ 1.2\]), we obtain $$\begin{aligned}
    \label{Neumann Poincare equ 2}
    \di&&\left(\int_{\O}f^2V^\tau \right)^2 \le \left(\di\int_{\O}V^\tau Ric^{D}(\bn^D \phi,\bn^D\phi)\right)\left(\int_{\O}V^\tau \left<{(Ric^{D})}^{-1}\nb f,\nb f\right>\right).\end{aligned}$$ Combining (\[Neumann Poincare equ 1\]) and (\[Neumann Poincare equ 2\]), we proved .

As a consequence, we get the Lichnerowicz type inequality for the first eigenvalue of $D$-Laplacian $\bd^D$. [*Proof of Theorem \[thm ineq\] (iii).*]{} By using the divergence property , One sees directly that the first eigenvalues have the following variational representation $$\begin{aligned}
             \lambda_1^{Dir} =\inf_{\{f\in C^1(\overline \O); f\not\equiv 0, f|_{\S}=0\}}\di\frac{\int_\O|\bar \n f|^2V^{\tau+\g-\a} d\O}{\int_M f^2V^\tau d\O},
             \label{eigenvalue}
         \end{aligned}$$ $$\begin{aligned}
             \lambda_1^{Neu} =\inf_{\{f\in C^1(\overline \O); f\not\equiv 0, \int_\O f V^\tau=0\}}\di\frac{\int_M|\bar \n 