Conjugate one-points in the unit disk

Definitions and mathematical statement

Let $\mathcal A$ consist of pairs $(f,a)$, where $f$ is holomorphic on $\mathbb D$, $a\in\mathbb D\setminus\mathbb R$, and \[ f^{-1}(0)=\{0\},\quad f'(0)\ne0,\qquad f^{-1}(1)=\{a,\overline a\},\quad f'(a)f'(\overline a)\ne0. \] All preimages are taken inside the disk. Thus both $1$-points are simple and distinct. There is no assumption that $f$ itself has real Taylor coefficients. Put $a_0=\inf_{(f,a)\in\mathcal A}|a|$.

For an exact comparison constant, let $c$ be the infimum of the numbers $b\in(0,1)$ for which a holomorphic disk function has its only zero simple at $0$ and its only $1$-point at $b$, of multiplicity two. This is the double-$1$-point covering extremal constant described in the source, approximately $0.0505468$; the approximation is not its definition.

Determine $a_0$ and the extremal behavior. In particular, is $a_0=c$? Decide whether the infimum is realized with distinct conjugate $1$-points or only approached by a sequence in which they coalesce into a double real $1$-point. Describe any extremizers or limiting extremal functions. The zero count and $1$-point count apply globally throughout $\mathbb D$, so satisfying the values at the displayed points alone does not constitute an admissible construction.

Short English statement

Can two distinct conjugate one-points approach the origin more closely than the double-one-point covering extremal permits?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 3700045 (AMR-036-0045), 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 1. Original source indexed by AMR-036-0045. https://www.math.purdue.edu/~eremenko/uns1.html.
• S3 A. Eremenko, Simultaneous stabilization, Problem 1. Author-maintained primary problem note. https://www.math.purdue.edu/~eremenko/dvi/simulstab.pdf.