Research Problems in Function Theory — Problem 2.52

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 $\sigma\ge0$, let $\mathcal H_\sigma$ be the class of nonconstant entire functions such that \[\limsup_{r\to\infty}\frac{\log M(r,f)}{(\log r)^2}\le\sigma.\] Define the best universal lower bound \[g_*(\sigma)=\inf_{f\in\mathcal H_\sigma}\limsup_{r\to\infty}\frac{m_0(r,f)}{M(r,f)}.\]

Statement recorded in the supplied source.

Determine $g_*(\sigma)$ for every $\sigma\ge0$. [S2] Update 2.52 records the sharp formula \[ g_*(\sigma)=\left(\prod_{n=1}^{\infty}\frac{1-q^{2n-1}}{1+q^{2n-1}}\right)^2, \qquad q=e^{-1/(4\sigma)}\quad(\sigma>0), \] with the continuous value $g_*(0)=1$. It attributes the stronger meromorphic-function theorem to Gol'dberg and a different proof of the original entire-function problem to Fenton. [S2]

Short English statement

Find the sharp minimum-to-maximum modulus ratio forced by a bound on the squared-logarithmic growth of an entire function.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 2302052 (AMR-022-2052). 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 2.52 and Update 2.52. https://arxiv.org/abs/1809.07200.