Maximum-modulus points on concentric circles

Let $f$ be a nonzero, non-monomial entire function, and let $ν_f(r)$ count the points at which $f$ attains its maximum modulus on $|z|=r$. Write $f(z)=z^m(a_0+a_kz^k+O(z^{k+1}))$, where $a_0a_k\ne0$. We prove that $ν_f(r)\le 2k$ outside a locally finite subset of $(0,\infty)$. The constant $2k$ is s