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 $P$.