Real two-one-point extremals

Definitions and mathematical statement

Call a holomorphic function on $\mathbb D$ real if $f(\overline z)=\overline{f(z)}$ for every $z\in\mathbb D$, equivalently if its Taylor coefficients at the origin are real. Let $\mathcal F_{\mathbb R}$ consist of these functions with one simple zero at $0$, no other zeros in the disk, and exactly two $1$-points $z_1,z_2$, counted with multiplicity.

Determine \[ R_{\mathbb R}=\inf_{f\in\mathcal F_{\mathbb R}} \max\{|z_1|,|z_2|\}, \qquad D_{\mathbb R}=\sup_{f\in\mathcal F_{\mathbb R}}|f'(0)|. \] For each optimization, decide whether its sharp value is attained and describe every extremal function, or appropriate extremizing sequences if necessary. The derivative is a nonzero real number, but its absolute value is the objective.

Reality forces the $1$-point multiset to be invariant under conjugation. It may therefore consist of two real points, a nonreal conjugate pair, or a single real point of multiplicity two. All these possibilities are included; requiring simple or nonreal $1$-points would give a different extremal problem. No univalence or bounded-range hypothesis is made, and the functions need not extend across the unit circle. The zero remains fixed at the center and the radius is Euclidean. The aim is the exact effect of the reality constraint on both sharp quantities and their equality cases.

Short English statement

Determine the two-one-point radius and derivative extremals when the disk function has real Taylor coefficients.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 3700026 (AMR-036-0026), 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 2. Original source indexed by AMR-036-0026. https://www.math.purdue.edu/~eremenko/uns1.html.
• S3 A. Eremenko, Goldberg's constant and its relatives, Problem 2. Author-maintained primary problem note. https://www.math.purdue.edu/~eremenko/dvi/goldconst.pdf.