Research Problems in Function Theory — Problem 2.34

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 $\lambda\ge0$ define the best universal minimum-modulus constant by \[C(\lambda)=\inf_{f:\,\text{lower order}(f)=\lambda} \limsup_{r\to\infty}\frac{\log m_0(r,f)}{\log M(r,f)},\] where $f$ ranges over nonconstant entire functions. The preceding notation paragraph gives $C(\lambda)=\cos(\pi\lambda)$ for $0\le\lambda\le1$.

Statement recorded in the supplied source.

Determine the behaviour of $C(\lambda)$ for $\lambda$ just above $1$. Is there an $\varepsilon>0$ such that $C(\lambda)=-1$ for $1<\lambda<1+\varepsilon$? Alternatively, is $C(\lambda)$ strictly decreasing for $\lambda>1$? [S2] Update 2.34 reports Fryntov's construction with a limiting minimum-to-maximum logarithmic ratio below $-1$ at each order greater than one. [S2]

Short English statement

Find the optimal minimum-modulus bound immediately beyond lower order one, including whether it has a plateau or strictly decreases.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 2302034 (AMR-022-2034). 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 2.34 and Update 2.34; minimum-modulus notation before Problem 2.33. https://arxiv.org/abs/1809.07200.