Research Problems in Function Theory — Problem 9.4

Definitions and mathematical statement

Let $H^\infty=H^\infty(\mathbb D)$ be the algebra of bounded holomorphic functions on the unit disk. For $f,g\in H^\infty$, the desired conclusion is \[ \exists a,b\in H^\infty\quad h(z)=a(z)f(z)+g(z)b(z)\ne0\quad\text{for every }z\in\mathbb D. \] Equivalently, the ideal generated by $f,g$ should contain a zero-free bounded holomorphic function. Zero-free does not require $1/h$ to be bounded; no uniform positive lower bound for $|h|$ is requested.

The first sentence literally asks this for every pair $f,g$. It omits any hypothesis excluding a common zero: at a common zero every such combination vanishes. Thus a nontrivial universal statement needs a qualification; the source's subsequent proposed harmonic-minorant condition may be intended to supply one.

For a real extended-valued function $\psi$ on $\mathbb D$, a harmonic minorant means a finite real harmonic function $u$ on $\mathbb D$ with $u(z)\le\psi(z)$ everywhere. The source claims as a necessary condition that \[ \psi(z)=\log\left(|f(z)|-|g(z)|\right) \] have a harmonic minorant, and asks whether this condition is also sufficient. This expression is retained exactly as printed. It is not a real logarithm where $|f|<|g|$, and its definition at equality would require the convention $\log0=-\infty$. No assumption $|f|>|g|$ is stated, and the expression is asymmetric although the desired combination property is symmetric in $f,g$.

A possible intended correction is the symmetric quantity $\log(|f|+|g|)$, but no such correction is adopted without source confirmation. Under that candidate reading the precise question would ask whether existence of a harmonic minorant of $\log(|f|+|g|)$ forces the displayed zero-free ideal element for every $f,g\in H^\infty$. It would still concern a zero-free element rather than the stronger equation $af+gb=1$. The extract alone does not establish which condition is intended or validate its necessity.

The source relates the problem to Problem 9.3, whose statement is not supplied. The triage labels the item open as of the 2018 edition and reports no definitive later resolution. This is an attributed source assessment, not an independently verified status for either the literal or candidate-corrected question.

Short English statement

When does the ideal generated by two bounded disk functions contain a bounded function with no zeros? The source's universal wording lacks a common-zero qualification, and its proposed logarithmic condition has a subtraction that may not define a real function. A symmetric sum is only a possible correction, and zero-free combinations must not be confused with a bounded inverse or a Bezout identity equal to one.

Sources

• Local source registry: data/unsolvedmath/problems.json, permanent UnsolvedMath ID $2309004$, source code catalogue entry, including the supplied OPEN-TRIAGE assessment. Dataset attribution: Ulam AI and UnsolvedMath Contributors; original-source attribution and rights are retained.
• Hayman, Research Problems in Function Theory, source-list date 2018, Problem 9.4, proposed by B. A. Taylor and communicated by L. A. Rubel: https://arxiv.org/abs/1809.07200. The desired combination has internal source label C9.1. The metadata reports source-TeX extraction and July 29, 2026 access, with status and rights review flags.
• The statement refers to Problem 9.3 but supplies neither that statement nor a separate authoritative citation for the claimed necessary condition. No linked source was inspected for this draft.