Research Problems in Function Theory — Problem 6.112

Definitions and mathematical statement

Let $A$ conventionally be the normalized analytic functions $f(z)=z+\sum_{n\geq2}a_nz^n$ on $\mathbb D$, let $S\subset A$ be the univalent class, and let $St$ be the starlike class. For two analytic series, $f*g$ denotes coefficientwise Hadamard multiplication. The source defines the dual of a normal family $\mathcal F\subset A$ by \[ \mathcal F^*=\{g\in A:(g*f)(z)\ne0\text{ for every }f\in\mathcal F \text{ and }0<|z|<1\}. \] In this problem $S^*$ denotes this dual of the univalent class, not the common alternative notation for starlike functions; the source uses $St$ for the latter.

The printed definition of the $\Sigma_\delta(f)$ neighborhood has unmatched absolute-value/parenthesis markup. Its apparently intended symmetric version, with $h=g-f$, is \[ \Sigma_\delta(f)=\left\{g\in A: \left|h'(z)-\frac{h(z)}z\right|+ \left|h'(z)+\frac{h(z)}z\right|<2\delta \text{ for every }z\in\mathbb D\right\}. \tag{1} \] At $0$ both expressions have removable value $0$. Formula (1) is conditional on recovering the damaged second absolute value as shown; it is not an authenticated silent correction. The source also alternates $\Sigma$ and $\sum$ for the neighborhood symbol.

Question (T. Sheil-Small and E. M. Silvia). Is \[ \Sigma_1(\operatorname{id})\subset S^*,\qquad \operatorname{id}(z)=z? \] Under (1), this asks whether every $g\in A$ obeying the displayed symmetric inequality with $f(z)=z$ satisfies $(g*u)(z)\ne0$ for every $u\in S$ and $0<|z|<1$.

The source reports $N_\delta(f)\subset\Sigma_\delta(f)$, $\Sigma_1(\operatorname{id})\subset St$, and $\Sigma_{1/4}(f)\subset St$ for convex $f$, attributing the latter neighborhood results to Sheil-Small and Silvia. The neighborhood $N_\delta$ and additional notation are referred to Problem 6.111, which is absent. It also states that the Bieberbach conjecture is equivalent to $N_1(\operatorname{id})\subset S^*$ and recalls $S^*\subset St$, referring to de Branges and Ruscheweyh without complete citations. These contextual claims do not identify the larger $\Sigma$-neighborhood with $N$, and no conclusion about the proposed inclusion is inferred from them.

The supplied summary labels the question OPEN-TRIAGE, says it was open in the 2018 edition, and reports no definitive resolution located.

Short English statement

Does the indicated symmetric derivative neighborhood of the identity lie in the convolution dual of the normalized univalent class? Here $S^*$ means a dual class and $St$ means starlike. The neighborhood formula has damaged absolute-value markup, and its repaired symmetric reading needs source confirmation.

Sources

• Local UnsolvedMath registry, permanent ID 2306112, source code catalogue entry, "Research Problems in Function Theory --- Problem 6.112"; attribution: T. Sheil-Small and E. M. Silvia. Supplied status open, classification OPEN-TRIAGE, updated 2026-08-06.
• Hayman, Research Problems in Function Theory (2018), Problem 6.112, with cross-reference to Problem 6.111, 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 source names Sheil-Small and Silvia for neighborhood inclusions and refers to Ruscheweyh for convolution duality and de Branges for the Bieberbach conjecture. Complete citations are absent.