Research Problems in Function Theory — Problem 2.35

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 $\log0=-\infty$. Let $f$ be nonconstant entire and let $\Gamma\subset\mathbb C$ be an unbounded continuum, meaning a closed connected unbounded subset of the plane. Write \[L(f,\Gamma)=\limsup_{\substack{z\in\Gamma\\|z|\to\infty}} \frac{\log|f(z)|}{\log M(|z|,f)}.\] The limit is along points of the continuum going to infinity, not a separate radial parameter.

Statement recorded in the supplied source.

The source gives an absolute constant $A$ with $L(f,\Gamma)\ge -A$ for all such $f,\Gamma$. Can $A=1$ always be used? If not, find a good numerical estimate for the optimal universal $A$. [S2] Update 2.35 records Hayman and Kjellberg's affirmative answer to the $A=1$ question. [S2]

Short English statement

How small can an entire function remain along an unbounded continuum compared with its maximum size on the same circles? Is the universal logarithmic factor one?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 2302035 (AMR-022-2035). 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 2.35 and Update 2.35. https://arxiv.org/abs/1809.07200.