Goldberg's constant

Definitions and mathematical statement

Let $\mathbb D=\{z:|z|<1\}$ carry the hyperbolic metric of curvature $-1$, with line element $2|dz|/(1-|z|^2)$. Its distance is \[ d_{\mathbb D}(z,w)=2\operatorname{artanh} \left|\frac{z-w}{1-\overline w z}\right|. \] Consider all holomorphic functions $f$ on $\mathbb D$ having exactly one zero $z_0$, of multiplicity one, and exactly two $1$-points $z_1,z_2$, counted with multiplicity. The two $1$-points may coincide, in which case that single point is a zero of multiplicity two of $f-1$.

Define the hyperbolic enclosing radius of these marked points by \[ R(f)=\inf_{c\in\mathbb D} \max_{j=0,1,2}d_{\mathbb D}(c,z_j), \qquad A_2=\inf_f R(f). \] Determine $A_2$, including sharpness and the functions attaining the infimum if it is attained. Equivalently, find the largest universal lower bound for the radius of a hyperbolic disk containing all the zero and $1$-point locations.

The associated structural question is whether every extremal function must have a double $1$-point. Disk automorphisms may move the locations without changing their hyperbolic enclosing radius. No boundedness or univalence of $f$ is assumed, but no additional zeros or $1$-points anywhere in $\mathbb D$ are allowed. The metric normalization is fixed here so that numerical values of a radius constant are unambiguous.

Short English statement

Find the smallest hyperbolic disk that can contain the unique simple zero and two one-points of a holomorphic disk function.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 3700024 (AMR-036-0024), 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's constant. Original source indexed by AMR-036-0024. https://www.math.purdue.edu/~eremenko/uns1.html.
• S3 A. Eremenko, Goldberg's constant and its relatives, introductory problem. Author-maintained primary problem note. https://www.math.purdue.edu/~eremenko/dvi/goldconst.pdf.