Two one-points in the unit disk

Definitions and mathematical statement

Let $\mathcal F$ be the class of holomorphic functions $f$ on the unit disk $\mathbb D$ satisfying $f(0)=0$, $f'(0)\ne0$, and $f(z)\ne0$ for every $z\in\mathbb D\setminus\{0\}$. Require that $f-1$ have exactly two zeros in $\mathbb D$, counted with multiplicity, and denote them by $z_1,z_2$. A single double zero of $f-1$ is allowed and is represented by $z_1=z_2$.

Define the two extremal quantities \[ R_*=\inf_{f\in\mathcal F}\max\{|z_1|,|z_2|\}, \qquad D_*=\sup_{f\in\mathcal F}|f'(0)|. \] Determine their sharp values, decide whether the infimum and supremum are attained, and characterize the corresponding extremal functions when they exist. If attainment fails, describe admissible functions approaching the sharp value.

The zero at the origin is fixed and simple, so arbitrary precomposition by a disk automorphism is not a permitted normalization. Rotations of the variable remain allowed and preserve both quantities. The radius objective is Euclidean and centered at the origin, unlike a freely centered hyperbolic enclosing radius. The derivative objective is a separate optimization; the formulation does not assume that one function must extremize both. There are no other restrictions on the range of $f$, its behavior at the boundary, or the multiplicity of its critical points.

Short English statement

For a disk function with one simple zero at the origin and two one-points, find the smallest enclosing centered radius and the largest derivative.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 3700025 (AMR-036-0025), AMR Open Problem Lists. Catalogue identity and source transcription; CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Eremenko - My favorite unsolved problems, Goldberg relatives, Problem 1. Original source indexed by AMR-036-0025. https://www.math.purdue.edu/~eremenko/uns1.html.
• S3 A. Eremenko, Goldberg's constant and its relatives, Problem 1. Author-maintained primary problem note. https://www.math.purdue.edu/~eremenko/dvi/goldconst.pdf.