Research Problems in Function Theory — Problem 6.113

Definitions and mathematical statement

Let $x\in{\mathbb{C}}$ and $\rho\in[0,1]$ satisfy $|x|\leq\rho$, and set \[ g_x(z)=\frac{z}{1-xz},\qquad \gamma=(1+\rho)^{-2}. \] The rational function $g_x$ is holomorphic in $\mathbb D$. Retaining the notation imported by the source from Problems 6.111 and 6.112, the known assertion is \[ N_\gamma(g_x)\subseteq S^*. \] Sheil-Small and Silvia ask whether the corresponding inclusion \[ \Sigma_\gamma(g_x)\subseteq S^* \] holds for every such $x,\rho$ with this exact value of $\gamma$. Inclusion is meant, without an additional claim that it is proper.

The supplied record does not define the neighborhood or transform constructions $N_\gamma$ and $\Sigma_\gamma$, their ambient function class, or $S^*$. Consequently the displayed source-specific inclusion is the precise available target, but is not a self-contained function-theoretic statement until the preceding definitions are supplied.

The source also says that not all convex functions belong to $S^*$ and concludes that the known $N_\gamma(g_x)$ inclusion cannot be replaced by $N_\gamma(g)$ for arbitrary convex $g$. This qualification must be retained, but it conflicts with reading $S^*$ as the usual normalized starlike class, since normalized convex functions are starlike about their image of $0$. Thus the meaning of $S^*$, the normalization, or the extracted assertion needs checking; a familiar notation must not silently be substituted for the undefined source class. The supplied summary reports openness as of 2018 and no definitive resolution located.

Short English statement

The question asks whether a known inclusion for $N_\gamma$-neighborhoods of $z/(1-xz)$ also holds for the source's $\Sigma_\gamma$ construction, at the same radius $(1+\rho)^{-2}$. Both constructions and the target class are missing from the extraction, and a convexity remark prevents assuming the usual meaning of $S^*$.

Sources

• Local UnsolvedMath registry, permanent ID 2306113, exact source code catalogue entry, "Research Problems in Function Theory --- Problem 6.113"; status open, classification OPEN-TRIAGE.
• Hayman, Research Problems in Function Theory (2018), Problem 6.113, following the notation of Problems 6.111 and 6.112, attributed to T. Sheil-Small and E. M. Silvia, https://arxiv.org/abs/1809.07200. The record identifies extraction from public arXiv source TeX, accessed July 29, 2026; the 2018 update was retained for status triage.
• Sheil-Small and Silvia, cited in the source for the convex-function qualification, without a complete bibliographic entry. The prior $N_\gamma$ inclusion is reported as known but no proof or precise separate citation is included.
• The supplied research summary reports openness as of the 2018 edition and no definitive resolution located, with no additional authoritative reference or precise search date.