Maximum-modulus points on concentric circles

Abstract

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 sharp. In particular, $\liminf_{r\to\infty}ν_f(r)<\infty$, 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 $q=zf'/f$ with its reflection $q^*(w)=\overline{q(\bar w)}$ through the correspondence $q(z)=q^*(w)$ 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 $f'/f$ excludes vertical components.