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Denote by $N_{\cal N} (\Omega,\lambda)$ the counting function of the spectrumof the Neumann problem in the domain $\Omega$ on the plane. G. P\'olyaconjectured 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 theBessel function $J_0$.

On the P\'olya conjecture for the Neumann problem in planar convex domains