Research Problems in Function Theory — Problem 7.2

Definitions and mathematical statement

The source lets $D$ be simply connected and writes a Faber expansion \[ f(z)=\sum_{n=0}^{\infty}a_n(f)P_n(z). \] A standard Faber construction, when applicable to a bounded domain, uses a normalized conformal map $\Phi$ from the exterior of $\overline D$ to the exterior unit disc, with $\Phi(\infty)=\infty$ and positive leading coefficient, and defines $P_n$ as the polynomial part of $\Phi(z)^n$. The source does not supply the geometric assumptions or normalization needed for its expansion; the displayed contextual construction does not extend that claim to every simply connected domain.

Variability problem (W. C. Royster). For a fixed index $n$, determine the set \[ V_n(D)=\{a_n(f):f\text{ analytic in }D,\ \operatorname{Re}f(z)>0\text{ for all }z\in D\}, \] with the intended Faber basis and any omitted normalization recovered from the source. The statement reports that $V_n(D)$ is a circle when $D$ is a circle, and an ellipse when $D$ is an ellipse, referring to Royster.

As printed, there is no normalization of $f$ to fix scale. Positive real part is preserved by multiplication by any positive constant, so a nonzero attainable coefficient can be scaled arbitrarily. For the unit disc with the usual basis $P_n(z)=z^n$ and $n\geq1$, every $b\in{\mathbb{C}}$ occurs in $f(z)=c+bz^n$ with $c>|b|$. Consequently the unnormalized reading is incompatible with a bounded circular variability region. A condition such as a prescribed value or real part at a basepoint may be missing, but none is selected here. The index range and whether "circle/ellipse" refers to the boundary or filled region are likewise not stated.

The supplied summary records OPEN-TRIAGE, says the problem was open in the 2018 edition, and reports no definitive resolution located. These are status claims about the intended normalized problem, whose full formulation is absent.

Short English statement

Determine the possible Faber coefficients of analytic functions with positive real part on a simply connected domain. The reported circular and elliptical cases indicate missing normalization: as printed, positive scaling makes the coefficient regions unbounded. The basis assumptions and coefficient index also need clarification.

Sources

• Local UnsolvedMath registry, permanent ID 2307002, source code catalogue entry, "Research Problems in Function Theory --- Problem 7.2"; attribution: W. C. Royster. Supplied status open, classification OPEN-TRIAGE, updated 2026-08-06.
• Hayman, Research Problems in Function Theory (2018), Problem 7.2, https://arxiv.org/abs/1809.07200. The record reports source-TeX extraction and access on 2026-07-29; the 2018 update was retained for status triage. Current status and rights require review; the record describes publicly available arXiv source TeX.
• The statement refers to Royster for the circular and elliptical cases, without a publication title, date, or theorem reference.