Research Problems in Function Theory — Problem 2.58

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 $\log0=-\infty$. A finite Picard exceptional value $\alpha$ is taken only finitely often. A natural asymptotic path for a nonzero such $\alpha$ is a continuous curve $\gamma:[0,\infty)\to\mathbb C$ with \[|\gamma(t)|\to\infty,\qquad |f(\gamma(t))|=|\alpha|,\qquad f(\gamma(t))\to\alpha.\]

Statement recorded in the supplied source.

Does there exist an entire function $f$ of infinite order with a nonzero finite Picard exceptional value $\alpha$ and no natural asymptotic path for $\alpha$? [S2] The original paragraph states the finite-order positive result. Update 2.58 records Eremenko's infinite-order example, answering the existence question affirmatively. [S2]

Short English statement

Can an infinite-order entire function have a nonzero exceptional value without approaching it along any path on the corresponding constant-modulus level set?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 2302058 (AMR-022-2058). 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 2.58 and Update 2.58. https://arxiv.org/abs/1809.07200.