Problem Statement
Let $f(z)$ be an entire function which is not a monomial. Let $\nu(r)$ count the number of $z$ with $\lvert z\rvert=r$ such that $\lvert f(z)\rvert=\max_{\lvert z\rvert=r}\lvert f(z)\rvert$. (This is a finite quantity if $f$ is not a monomial.)
Is it possible for\[\limsup \nu(r)=\infty?\]Is it possible for\[\liminf \nu(r)=\infty?\]
Is it possible for\[\limsup \nu(r)=\infty?\]Is it possible for\[\liminf \nu(r)=\infty?\]
Categories:
Analysis
Progress
This is Problem 2.16 in [Ha74], where it is attributed to Erdős.The answer to the first question is yes, as shown by Herzog and Piranian [HePi68]. The second question is still open, although an 'approximate' affirmative answer is given by Glücksam and Pardo-Simón [GlPa24].
Source: erdosproblems.com/1117 | Last verified: January 19, 2026