Belgian Chocolate constant

Definitions and mathematical statement

For $a\in(0,1)$, let $\mathcal B_a$ consist of the holomorphic functions $f:\mathbb D\to\mathbb C$ satisfying the reality condition $f(\overline z)=\overline{f(z)}$ and the following exact value constraints: \[ f^{-1}(0)=\{0\},\quad f'(0)\ne0,\qquad f^{-1}(1)=\{ia,-ia\},\quad f'(ia)f'(-ia)\ne0. \] All inverse images here are taken within $\mathbb D$. Thus there is one simple zero and precisely two distinct simple $1$-points, placed symmetrically on the imaginary axis.

Define \[ a_{\mathrm B}= \inf\{a\in(0,1):\mathcal B_a\ne\varnothing\}. \] Determine this sharp constant, and decide whether an admissible function realizes it. If it does, characterize the extremal functions; if it does not, describe a sequence of admissible parameters and functions approaching the infimum.

This is the disk formulation of the Belgian Chocolate problem. The symmetry of the two prescribed $1$-points does not by itself replace the separate reality condition on the whole function. Holomorphic functions of arbitrary growth toward the boundary are allowed, and neither polynomial nor rational realizations are required. The strict inequalities $0<a<1$ keep both $1$-points distinct from each other, from the zero, and from the boundary. Numerical constructions yield admissible upper bounds only when their global zero and $1$-point counts are also established.

Short English statement

How close to the origin can the two symmetric imaginary one-points of a real holomorphic disk function lie?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 3700027 (AMR-036-0027), 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 3. Original source indexed by AMR-036-0027. https://www.math.purdue.edu/~eremenko/uns1.html.
• S3 A. Eremenko, Goldberg's constant and its relatives, Problem 3. Author-maintained primary problem note. https://www.math.purdue.edu/~eremenko/dvi/goldconst.pdf.