  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 $J_0$.
