Research Problems in Function Theory — Problem 2.66

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 $f$, put $M(r,f)=\max_{|z|=r}|f(z)|$, $T(r,f)=(2\pi)^{-1}\int_0^{2\pi}\log^+|f(re^{i\theta})|\,d\theta$, and $\rho(f)=\limsup_{r\to\infty}\log\log M(r,f)/\log r$, where $\log^+t=\max(0,\log t)$. Let $\omega_1$ be the first uncountable ordinal. Work in ZFC, without assuming the continuum hypothesis.

Statement recorded in the supplied source.

Does there exist a family of entire functions $(f_\alpha)_{\alpha<\omega_1}$ such that \[ \lim_{r\to\infty}\frac{M(r,f_\alpha)}{M(r,f_\beta)}=0\quad(\alpha<\beta<\omega_1) \] and, for every entire function $f$, there is $\gamma<\omega_1$ for which \[\lim_{r\to\infty}\frac{M(r,f)}{M(r,f_\gamma)}=0?\] [S2]

Short English statement

Without the continuum hypothesis, can an increasing family of length the first uncountable ordinal eventually dominate the growth of every entire function?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 2302066 (AMR-022-2066). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 W. K. Hayman and E. F. Lingham, Research Problems in Function Theory, arXiv:1809.07200v2 (2018), Problem 2.66. https://arxiv.org/abs/1809.07200.