Research Problems in Function Theory — Problem 2.49

Definitions and mathematical statement

Shared notation.

$\mathbb N=\{1,2,\ldots\}$, and $\mathbb Z,\mathbb Q,\mathbb R,\mathbb C$ denote the integers, rationals, reals and complex numbers. Any other conventions are specified locally.

For an entire function $f$, write \[ M(r,f)=\max_{|z|=r}|f(z)|,\qquad m_0(r,f)=\min_{|z|=r}|f(z)|. \] For nonconstant $f$, its order is $\rho(f)=\limsup_{r\to\infty}\log\log M(r,f)/\log r$; the lower order is defined using $\liminf$ instead. Every constant entire function, including the zero function, has order and lower order zero. In logarithmic modulus ratios, use the extended-real convention $\log0=-\infty$. For a transcendental entire function, let \[\mathcal M(f)=\{z:|f(z)|=M(|z|,f)\},\qquad \nu_f(r)=\#\left(\mathcal M(f)\cap\{z:|z|=r\}\right). \] Points are counted without multiplicity.

Statement recorded in the supplied source.

Must every transcendental entire function satisfy \[\liminf_{r\to\infty}\nu_f(r)<\infty?\] [S2] The opening hypothesis is transcendental entire, excluding the degenerate monomial case. The original question cites examples with unbounded $\limsup\nu_f(r)$, and Update 2.49 records no further progress. [S2]

Short English statement

Must there be arbitrarily large circles on which a transcendental entire function attains its maximum modulus at only a bounded number of points?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 2302049 (AMR-022-2049). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 W. K. Hayman and E. F. Lingham, Research Problems in Function Theory (2018), Chapter 2, Problem 2.49 and Update 2.49. https://arxiv.org/abs/1809.07200.