Maximum-modulus points on concentric circles

Abstract

We give a negative answer to the second question of Erdős Problem #1117: can the number ${\nu }_f(r)$ of maximum-modulus points of a nonzero, non-monomial entire function on $|z|=r$ tend to infinity as $r\to \infty$? Write $f(z)=z^{m}g(z)$ with $g(0)\ne 0$, and let $k$ be the first positive exponent with a nonzero coefficient in the Taylor expansion of $g$. We prove that $\{r>0:{\nu }_f(r)>2k\}$ is locally finite in $(0,\infty )$, and that the constant $2k$ is sharp. In particular, ${\mathrm{liminf}}_{r\to \infty }{\nu }_f(r)\le 2k<\infty$, without a growth assumption. The proof combines finite local parametrizations of a logarithmic-derivative correspondence with analytic continuation and a local description of stationary points.