Research Problems in Function Theory — Problem 2.51

Definitions and mathematical statement

For a nonconstant entire function $f$, let $M(r,f)=\max_{|z|=r}|f(z)|$ and $\mathcal M(f)=\{z:|f(z)|=M(|z|,f)\}$. Suppose this entire maximum-modulus locus is a single curve $\Gamma$ joining the origin to infinity: all its points are maximum-modulus points, it is connected, and there are no maximum-modulus points outside it. Determine how prescribed geometric information about $\Gamma$ constrains the minimum possible growth of $M(r,f)$. The main example in the source is an infinitely winding spiral. For a concrete radial parametrization, let $\vartheta:(0,\infty)\to\mathbb R$ be continuous and prescribe \[ \Gamma=\{0\}\cup\{re^{i\vartheta(r)}:r>0\}, \qquad |\vartheta(r)|\to\infty\quad(r\to\infty). \] Among entire functions having exactly this maximum locus, find sharp lower growth bounds in terms of the spiral's winding, and determine when such a locus is realizable. More general prescribed curves are also included in the question. A useful answer must distinguish geometries forcing large growth from the straight-ray case, which permits very slowly growing examples. The hypothesis is about the complete maximum locus, not one selected path inside a larger locus.

Short English statement

How fast must an entire function grow when its unique maximum-modulus curve has prescribed geometry, particularly an endlessly winding spiral?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 2302051 (AMR-022-2051), 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 2.51. Original source indexed by AMR-022-2051. https://arxiv.org/abs/1809.07200.