Research Problems in Function Theory — Problem 6.41

Definitions and mathematical statement

Let $S$ be the normalized univalent holomorphic functions on $\mathbb D$: $f(0)=0$, $f'(0)=1$. Define \[ K(\alpha)=\left\{f\in S:\operatorname{Re}\left(1+\frac{zf''(z)}{f'(z)}\right)\ge\alpha\quad(z\in\mathbb D)\right\}, \] \[ S^*(\alpha)=\left\{f\in S:\operatorname{Re}\frac{zf'(z)}{f(z)}\ge\alpha\quad(z\in\mathbb D)\right\}. \] The quotient at $0$ is interpreted by its analytic extension, equal to $1$. The source omits the range of $\alpha$; the customary $0\le\alpha<1$ range is a possible intended domain, not an unqualified source hypothesis. For $0<r<1$, part (a) asks to prove \[ \min_{f\in K(\alpha)}\min_{|z|=r}\left|\frac{zf'(z)}{f(z)}\right| =\min_{f\in K(\alpha)}\min_{|z|=r}\operatorname{Re}\frac{zf'(z)}{f(z)}. \] The source uses minima; existence of extremizers or an intended infimum convention requires checking.

Part (b) asks to prove specified starlikeness orders for \[ h_\alpha(z)=\frac{1-(1-z)^{2\alpha-1}}{2\alpha-1}\quad(\alpha\ne\tfrac12), \qquad h_{1/2}(z)=-\log(1-z), \] where the analytic branch has $\log(1)=0$. The respective claimed orders are printed as \[ \frac{4^\alpha(2\alpha-1)}{4-4.2^\alpha},\qquad \frac1{\log4}. \] The denominator $4-4.2^\alpha$ is reproduced literally and is ambiguous. Interpreting the dot as multiplication gives $4-4\cdot2^\alpha$, which vanishes at $\alpha=0$ and does not yield the stated half-order limit. No corrected formula is guessed. Once that expression and parameter range are fixed, the requested membership is $h_\alpha\in S^*(\beta_\alpha)$.

The source says either part, together with I. S. Jack's work, would solve F. R. Keogh's problem, printed as \[ \max_{f\in K(\alpha)}\{\beta:f\in S^*(\beta)\}. \] This is not a precise scalar optimization as written. Moreover, maximizing individual starlikeness order is different from finding the largest $\beta$ with $K(\alpha)\subseteq S^*(\beta)$; in the customary range the identity $f(z)=z$ already has individual order $1$. The intended uniform-inclusion constant may instead be \[ \sup\{\beta:K(\alpha)\subseteq S^*(\beta)\}, \] but that is explicitly a possible interpretation, not a replacement of the source. The supplied OPEN-TRIAGE summary reports open status in the 2018 edition and no definitive resolution located.

Short English statement

Determine extremal starlikeness information for convex functions of order $\alpha$. The record includes two proposed routes, but its parameter range, explicit extremal-order formula, and final maximum-versus-uniform-inclusion target need checking.

Sources

• Local UnsolvedMath registry, permanent ID $2306041$, exact source code catalogue entry, "Research Problems in Function Theory --- Problem 6.41."
• Hayman, Research Problems in Function Theory (2018), Problem 6.41, attributed to D. Benjamin and referencing F. R. Keogh's problem; https://arxiv.org/abs/1809.07200. Source-TeX extraction; recorded access 2026-07-29.
• Source references Goel for part (a), MacGregor for part (b), and I. S. Jack for the proposed implication, without complete citations. The supplied OPEN-TRIAGE summary gives no separate resolution reference.