Problem 1117

Erdős Problem #1117: declared status 'open'. Formalized: no. 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$$\liminf \nu(r)=\infty?$$ Prize: no. Tags: analysis.