Assume $\lambda(P)$ is the first Dirichlet eigenvalue of a regular polygon $P$. Let $u$ be the corresponding eigenfunction, normalized by $\|u\|\_{L^2(P)}=1$, and $\partial\_{\nu}u$ be its normal derivative on the boundary. Is the following estimate correct:
\begin{eqnarray\*}
\lambda(P)\geq \|\partial\_{\nu} u\|^2\_{L^{\infty}(\partial P)}\vert P\vert\quad?
\end{eqnarray\*}
Here $\vert P\vert$ denotes the area of $P$.