Symmetric one-points in the unit disk

Definitions and mathematical statement

Let $\mathcal S$ consist of pairs $(f,b)$ with $f$ holomorphic on $\mathbb D$ and $0<|b|<1$, satisfying \[ f^{-1}(0)=\{0\},\quad f'(0)\ne0,\qquad f^{-1}(1)=\{b,-b\},\quad f'(b)f'(-b)\ne0. \] These are exact preimage sets within the unit disk. The zero and both $1$-points are simple, and no additional zeros or $1$-points are allowed. Define $b_0=\inf_{(f,b)\in\mathcal S}|b|$.

Define $c$ as the infimum of $a\in(0,1)$ for which a holomorphic disk function has one simple zero at $0$ and exactly one $1$-point at the real point $a$, of multiplicity two. This is the source's double-point covering constant, approximately $0.0505468$.

Determine $b_0$, whether it is attained, and the extremal functions or limiting extremal sequences. In particular, prove or disprove the strict inequality $b_0>c$. The two prescribed $1$-points in this problem are opposite about the origin; they cannot merge into one nonzero point while retaining that symmetry. Rotation of the variable allows one to take $b$ positive real without changing the infimum. No reality condition on the whole function is imposed, so this class differs from the real-function version with imaginary $1$-points. The comparison asks whether the opposite-point geometry forces a strictly larger radius.

Short English statement

Find the least radius for two opposite simple one-points, and decide whether it exceeds the double-one-point covering constant.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 3700046 (AMR-036-0046), 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, Simultaneous stabilization, Problem 2. Original source indexed by AMR-036-0046. https://www.math.purdue.edu/~eremenko/uns1.html.
• S3 A. Eremenko, Simultaneous stabilization, Problem 2. Author-maintained primary problem note. https://www.math.purdue.edu/~eremenko/dvi/simulstab.pdf.