On the first Dirichlet Laplacian eigenvalue of regular polygons (Q487280)

In \({\mathbb{R}}^{2}\), let \(N\geq3\), \(r>0\), and let \(P_{N}^{r}\) be a regular polygon with \(N\) sides and circumradius \(r\), where \(l_{N}^{r}\) and \(\rho_{N}^{r}\) are the side length and the inradius of \(P_{N}^{r}\), respectively. Let also \(\lambda(P_{N}^{r})\) be the least positive number such that there exists \(u\in H_{0}^{1}(P_{N}^{r})\setminus\{0\}\) for which \(-\Delta u=\lambda(P_{N}^{r})u\). Finally, let \(j_{0}\) be the first zero of the Bessel function of the first kind and of order \(0\). The author proves that \[ \lambda(P_{N+1}^{r})<\lambda(P_{N}^{r}){{\cos{{\pi}\over{N}}}\over{\cos{{\pi}\over{N+1}}}} \] and \[ l_{N+1}^{r} \rho_{N+1}^{r} \lambda(P_{N+1}^{r})<l_{N}^{r}\rho_{N}^{r}\lambda(P_{N}^{r})-{{2\pi j_{0}^{2}}\over{N(N+1)}}. \]