Universal Holomorphic Functions under Composition Operators

Definitions and mathematical statement

Let $\Omega\subseteq{\mathbb{C}}$ be a domain and let $H(\Omega)$ have the compact-open Fréchet topology, defined by locally uniform convergence. Let $(\varphi_n)_{n\geq1}$ be arbitrary holomorphic self-maps of $\Omega$. For the chosen function class $X$, a universal $f$ (in the usual $f\in X$ convention) satisfies \[ \forall g\in X\ \forall K\subset\Omega\text{ compact}\ \forall\varepsilon>0\ \exists n\geq1:\quad \sup_{z\in K}|f(\varphi_n(z))-g(z)|<\varepsilon. \] This is density of the composition sequence, not of all iterates of one map unless the sequence is specified to consist of such iterates.

The source gives two choices for $X$: $H(\Omega)$ or "its unit ball". A general Fréchet space $H(\Omega)$ has no distinguished single norm and hence no canonical unit ball. A common intended bounded-holomorphic choice is $\{f:\sup_\Omega|f|\leq1\}$ with the locally uniform topology; another is holomorphic maps into the open unit disk. These must be distinguished from supremum-norm density and from a ball for one compact seminorm. The excerpt does not choose the precise ball or explicitly say whether universal $f$ must belong to it, so those conventions are not silently imposed.

The main question asks for necessary and sufficient conditions on the given sequence for such an $f$ to exist on an arbitrary domain. Neither automorphisms, eventual injectivity, simple connectivity nor a single-generator semigroup assumption is added. The annulus question is a special existence/classification case; for a usual bounded annulus $A(r,R)=\{r<|z|<R\}$, $0<r<R<\infty$, the intended quantifier over moduli and permitted sequences also needs confirmation. It is not an assertion that every self-map sequence is universal.

The literature triage checked 21 August 2026 reports criteria for automorphism sequences and several simply connected or eventually injective settings, but says the arbitrary-domain/self-map formulation is broader. Its recent classification reference must be applied with its hypotheses, not promoted to a result for all the source's sequences. The dated partial-status claim is retained without independent verification.

Short English statement

Classify self-map sequences for which one holomorphic function has compositions locally uniformly dense in the chosen function class, and resolve the intended annulus case. The unit-ball convention is unspecified, and criteria for automorphisms or injective sequences do not automatically answer the arbitrary-self-map problem.

Sources

• Local UnsolvedMath registry, permanent ID 30000702, exact source code catalogue entry; Universal Holomorphic Functions under Composition Operators. Proposed year 2007, status partially_solved; corrected_verified extraction retains the two function classes, full sequence, classification and annulus questions.
• Normal Families and Complex Dynamics (2007), Oberwolfach Reports, DOI 10.4171/owr/2007/09, pp. 487--548, https://doi.org/10.4171/owr/2007/09.
• Supplied literature triage checked 2026-08-21: "original problem report", https://doi.org/10.4171/owr/2007/09; "universal functions for composition operators", https://publications.mfo.de/bitstream/handle/mfo/2966/OWR_2006_37.pdf?isAllowed=y&sequence=1; "recent universality classification", https://doi.org/10.1112/jlms.70262. The dated OWR-derived curation requests independent verification of time-sensitive status and cited claims; no linked PDF was accessed here.