Research Problems in Function Theory — Problem 6.85

Definitions and mathematical statement

Shared notation.

$\mathbb N=\{1,2,\ldots\}$, and $\mathbb Z,\mathbb Q,\mathbb R,\mathbb C$ denote the integers, rationals, reals and complex numbers. Any other conventions are specified locally.

Write $\mathbb D=\{z\in\mathbb C:|z|<1\}$ and $\mathbb T=\partial\mathbb D$. The class $S$ consists of holomorphic injective functions $f:\mathbb D\to\mathbb C$ normalized by $f(0)=0$ and $f'(0)=1$, so $f(z)=z+\sum_{n=2}^{\infty}a_nz^n$. For $h\in H(\mathbb D)$, let $\Lambda_n(h)$ be the coefficient of $z^n$ in its Taylor expansion. In particular, $\Lambda_n(f)=a_n$, and $f^2,f^3$ mean pointwise powers, not iterates.

Statement recorded in the supplied source.

For every $n\ge2$ and every $f\in S$ maximizing $\operatorname{Re}\Lambda_n$ on $S$, is the quotient \[\frac{\Lambda_n(f^3)}{[\Lambda_n(f^2)]^2}\] real? In the source's support-map description this says the asymptotic line of the omitted arc, \[w=\frac{\Lambda_n(f^3)}{3\Lambda_n(f^2)}-\Lambda_n(f^2)t,\qquad t\ge0,\] has a supporting line through the origin. [S2] Update 6.85 points to Update 6.1, which records a positive answer as a consequence of de Branges' proof. [S2]

Short English statement

Must every coefficient-extremal omitted slit approach a radial line at infinity?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 2306085 (AMR-022-6085). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 W. K. Hayman and E. F. Lingham, Research Problems in Function Theory (2018), Chapter 6, Problem 6.85 and Update 6.85; full Problem 6.1 and Update 6.1. https://arxiv.org/abs/1809.07200.