Research Problems in Function Theory — Problem 2.47

Definitions and mathematical statement

Fix $\rho>0$. Let $E_\rho$ consist of entire functions $f$ for which $|f(z)|\le B\exp(A|z|^\rho)$ with some constants $A,B>0$ depending on $f$. Let $K_\rho$ consist of positive continuous functions $k$ on $\mathbb C$ such that $\exp(A|z|^\rho)/k(z)\to0$ as $|z|\to\infty$ for every $A>0$. For $S\subset\mathbb C$ define seminorms \[ p_{k,S}(f)=\sup_{z\in S}\frac{|f(z)|}{k(z)},\qquad p_k(f)=\sup_{z\in\mathbb C}\frac{|f(z)|}{k(z)}. \] Call $S$ sufficient when the families $(p_{k,S})_{k\in K_\rho}$ and $(p_k)_{k\in K_\rho}$ induce the same locally convex topology on $E_\rho$. The first question asks whether the implication \[ f\in E_\rho,\quad\sup_S|f|<\infty \quad\Longrightarrow\quad \sup_{\mathbb C}|f|<\infty \] forces $S$ to be sufficient. The second asks whether sufficiency forces this implication, or at least a precise smaller-growth conclusion for functions bounded on $S$. By Liouville's theorem global boundedness here means constancy. The topology compares entire families of weighted seminorms and is stronger information than a single pointwise uniqueness condition.

Short English statement

For entire functions of bounded order and finite type, how does a sufficient sampling set relate to a set on which boundedness forces global boundedness?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 2302047 (AMR-022-2047), AMR Open Problem Lists. Catalogue identity and source transcription; CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Hayman - Research Problems in Function Theory (2018), Problem 2.47. Original source indexed by AMR-022-2047. https://arxiv.org/abs/1809.07200.