On the Pólya conjecture for the Neumann problem in planar convex domains

Denote by $N_{\mathcal{N}}(\mathrm{\Omega },\lambda )$ the counting function of the spectrum of the Neumann problem in the domain $\mathrm{\Omega }$ on the plane. G. Pólya conjectured that $N_{\mathcal{N}}(\mathrm{\Omega },\lambda )\ge (4\pi {)}^{-1}\mathrm{\mid }\mathrm{\Omega }\mathrm{\mid }\lambda$. We prove that for convex domains $N_{\mathcal{N}}(\mathrm{\Omega },\lambda )\ge (2\sqrt{3}{j}_0^2{)}^{-1}\mathrm{\mid }\mathrm{\Omega }\mathrm{\mid }\lambda$. Here $j_0$ is the first zero of the Bessel function $J_0$.