Rational Goldberg extremals

Definitions and mathematical statement

Fix an integer $N\geq2$. A rational function of degree $N$ means $f=P/Q$ with relatively prime complex polynomials and $\max(\deg P,\deg Q)=N$. Require $Q$ to have no zero in $\mathbb D$. Only zeros and $1$-points inside this open disk enter the following problems; poles outside it are allowed.

First consider functions with exactly one simple zero $z_0$ and two $1$-points $z_1,z_2$, counting multiplicity. Minimize their hyperbolic enclosing radius $\inf_{c\in\mathbb D}\max_jd_{\mathbb D}(c,z_j)$, where $d_{\mathbb D}(z,w)=2\operatorname{artanh} |(z-w)/(1-\overline w z)|$.

Next impose $z_0=0$ and determine both the infimum of $\max(|z_1|,|z_2|)$ and the supremum of $|f'(0)|$. Determine the same two quantities in the real subclass $f(\overline z)=\overline{f(z)}$. Finally, within that real subclass, require two distinct simple $1$-points precisely at $\pm ia$, $0<a<1$, and minimize $a$.

Solve these extremal problems for each fixed $N$, including attainment and the corresponding extremal rational functions. Except in the final variant, a double $1$-point is permitted. The sole zero is simple in every variant, and there are no additional zeros or $1$-points in the disk. Degree is the global rational-map degree, not the number of preimages counted within the disk. A limit in which numerator and denominator acquire a common factor can lower degree and therefore need not attain an extremum in the prescribed class.

Short English statement

Solve the Goldberg and related disk extremal problems among rational functions of each fixed degree.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 3700028 (AMR-036-0028), 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 4. Original source indexed by AMR-036-0028. https://www.math.purdue.edu/~eremenko/uns1.html.
• S3 A. Eremenko, Goldberg's constant and its relatives, Problem 4. Author-maintained primary problem note. https://www.math.purdue.edu/~eremenko/dvi/goldconst.pdf.