Research Problems in Function Theory — Problem 2.50

Definitions and mathematical statement

For a nonconstant entire function $f$, set \[ M(r,f)=\max_{|w|=r}|f(w)|, \qquad \mathcal M(f)=\{z\in\mathbb C:|f(z)|=M(|z|,f)\}. \] Thus $\mathcal M(f)$ is the locus of all points achieving the maximum modulus on their respective centred circles, including the origin. Every circle meets this closed set, but those intersections need not fit together into one unbounded continuous curve. Characterize the functions $f$ for which there is a continuous path $\gamma:[0,\infty)\to\mathbb C$ satisfying \[ \gamma(0)=0,\qquad |\gamma(t)|\longrightarrow\infty, \qquad\gamma(t)\in\mathcal M(f)\quad(t\ge0). \] The path need not follow a straight ray, and its modulus is not required to be monotone. It must remain in the maximum-modulus locus at every point, not merely stay close to it or attain comparable values of $|f|$. A classification should give conditions on $f$, for example on its coefficients or on the geometry of $\mathcal M(f)$, that are both necessary and sufficient. Local maximum-modulus arcs near zero by themselves do not establish the existence of a single path continuing to infinity.

Short English statement

Which entire functions admit a continuous path from zero to infinity along which they always attain their circular maximum modulus?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 2302050 (AMR-022-2050), AMR Open Problem Lists. Catalogue identity and source transcription; CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Hayman - Research Problems in Function Theory (2018), Problem 2.50. Original source indexed by AMR-022-2050. https://arxiv.org/abs/1809.07200.