Research Problems in Function Theory — Problem 6.99

Definitions and mathematical statement

Consider normalized holomorphic univalent functions $f(z)=z+\sum_{n\ge2}a_nz^n$ on $\mathbb D$, with convex image. The source defines $CV(R_1,R_2)$ by requiring a boundary quantity called curvature $\rho$ to satisfy \[ R_1\le\rho\le R_2\quad\text{on }f(\{|z|=1\}). \] Its concrete extremal problem is to determine the sharp coefficient bound \[ B(R_1,R_2)=\sup\{|a_2|:f\in CV(R_1,R_2)\} =\tfrac12\sup\{|f''(0)|:f\in CV(R_1,R_2)\}, \] for the intended parameter range in which the class is nonempty. Sharpness means the least valid upper bound; attainment is a separate issue.

The boundary and curvature conventions are missing. For a regular $C^2$ boundary parametrization $h(t)=f(e^{it})$, geometric curvature is \[ \kappa(t)=\frac{|\operatorname{Im}(\overline{h'(t)}h''(t))|}{|h'(t)|^3}. \] If $\rho$ means radius of curvature instead, then $\rho=1/\kappa$, giving different parameter inequalities and a different extremal class. The source explicitly says curvature, but its notation and intended convention must be checked. Neither a smooth boundary extension nor a generalized curvature notion is supplied, so the expression $f(\{|z|=1\})$ is not automatically defined for an arbitrary disk map. The values and admissible range of $R_1,R_2$ are also unstated.

Goodman's source request includes broader study of this class, with the sharp $|a_2|$ bound as the explicit unresolved target. The supplied OPEN-TRIAGE summary reports open status in the 2018 edition and no definitive resolution located; no new bound or source correction is asserted.

Short English statement

Find the best possible second Taylor-coefficient bound for normalized convex disk maps with two-sided boundary-curvature constraints. The curvature-versus-radius convention, boundary regularity, and admissible parameters require review.

Sources

• Local UnsolvedMath registry, permanent ID $2306099$, exact source code catalogue entry, "Research Problems in Function Theory --- Problem 6.99."
• Hayman, Research Problems in Function Theory (2018), Problem 6.99, attributed to A. W. Goodman; https://arxiv.org/abs/1809.07200. Source-TeX extraction; recorded access 2026-07-29.
• The statement says "see Goodman and" with incomplete/missing references. The supplied OPEN-TRIAGE summary gives no separate current-resolution citation.