Denote by $N_{ \cal N} (\Omega,\lambda)$ the counting function of the spectrum of the Neumann problem in the domain $\Omega$ on the plane. G. P\'olya conjectured that $N_{ \cal N} (\Omega,\lambda) \ge (4\pi)^{ -1} |\Omega| \lambda$. We prove that for convex domains $N_{ \cal N} (\Omega,\lambda) \ge (2 \sqrt 3 \,j_0^2)^{ -1} |\Omega| \lambda$. Here $j_0$ is the first zero of the Bessel function $J_0$.
Categories: math.SP math.AP