we obtain $$\begin{aligned}
\label{equa1}
    \left(\int_\O V^\tau d\O\right)^2&=& \left(\int_\S V^{\tau}\<\bn^D \phi,\nu\> dA\right)^2\\
    &\leq& \int_\S V^{\tau}H^D\<\bn^D \phi,\nu\>^2 dA\int_\S \frac{V^\tau}{H^D}dA\nonumber
\\&\leq &\frac{n-1}{n}\int_\O V^\tau d\O\int_\S \frac{V^\tau}{H^D}dA.\nonumber\end{aligned}$$ The assertion for the inequality follows. If the equality holds, we have $$\begin{aligned}
 \label{equa2}\bn^{2,D}\phi=\frac1n \bar g.\end{aligned}$$ Restricting on $\S$, using $\phi=0$ on $\S$ and Gauss formula, we conclude that $h_{\a\b}=\l g_{\a\b}$ for some smooth function $\l$, i.e., $\S$ is umbilic.

[*Proof of Theorem \[thm ineq\] (ii).*]{} Consider the Neumann boundary value problem $$\left\{
    \begin{array}[]{rlll}
        \di  \bd^D \phi&=&1&\mathrm{ in }\ \O,\\
       V^\g \phi_\nu&=& c &\mathrm{ on }\ \S,\\
    \end{array}
    \right.  
       \label{Neumann}$$ where $c=\frac{\int_{\O}V^\tau}{\int_{\S}V^{\tau-\a}}$. The existence and uniqueness follows from the Fredholm alternative as in standard elliptic PDE theory. We will apply the solution $\phi$ of (\[Neumann\]) to the integral formula (\[formula RF 1\]). By using the Cauchy-Schwarz inequality, the equation and boundary condition in (\[Neumann\]) and the curvature assumptions, we get $$\begin{aligned}
\frac{n-1}{n}\int_\O V^\tau d\O&\geq &\int_\S V^\tau H^D \<\bn^D\phi,\nu\>^2dA
\\&=& \int_\S V^\tau H^D V^{2\g-2\a}\phi_\nu^2dA
\\&=& c^2\int_\S V^{\tau-2\a} H^D dA.\end{aligned}$$ Inserting the value of $c$ we get the assertion.

If the equality holds, we have $$\begin{aligned}
 \label{equa3}\bn^{2,D}\phi=\frac1n \bar g,\end{aligned}$$ and $$\begin{aligned}
 \label{equa4}(h-\g u_\nu g)(\nabla \phi,\nabla \phi)=0.\end{aligned}$$ Since by assumption $(h_{\a\b}-\g\frac{V_{\nu}}{V}g_{\a\b})>0$ on $\S$, it follows from that $\phi=0$ on $\S$. Restricting 