Maximum-modulus points on concentric circles

Let $f$ be a nonzero, non-monomial entire function, and let ${\nu }_f(r)$ count the points at which $f$ attains its maximum modulus on $\mathrm{\mid }z\mathrm{\mid }=r$. Write $f(z)=z^m(a_0+a_k{z}^k+O(z^{k+1}))$, where $a_0{a}_k\mathrm{\ne }0$. We prove that ${\nu }_f(r)\le 2k$ outside a locally finite subset of $(0,\mathrm{\infty })$. The constant $2k$ is sharp. In particular, ${\mathrm{lim\hspace{0.17em}inf}}_{r\to \mathrm{\infty }}{\nu }_f(r)<\mathrm{\infty }$, so the number of maximum-modulus points of a fixed non-monomial entire function cannot tend to infinity on all sufficiently large circles. The proof associates the logarithmic derivative $q=z{f}^{\mathrm{\prime }}\mathrm{/}f$ with its reflection $q^{\ast }(w)=\overline{q(\stackrel{ˉ}{w})}$ through the correspondence $q(z)=q^{\ast }(w)$ and the map $(z,w)\mapsto (zw,q(z))$. Local valence at the origin bounds the number of inverse pairs that can escape. A local elimination argument then bounds the generic degree of each nonvertical image component. The residue structure of $f^{\mathrm{\prime }}\mathrm{/}f$ excludes vertical components.