Research Problems in Function Theory — Problem 6.72

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 $f(z)=z+\sum_{n=2}^{\infty}a_nz^n$. Equip $H(\mathbb D)$, the vector space of holomorphic functions on $\mathbb D$, with uniform convergence on compact subsets. A support point of $S$ maximizes $\operatorname{Re}L$ over $S$ for a continuous complex-linear functional $L:H(\mathbb D)\to\mathbb C$ whose real part is not constant on $S$. For a support point, the source describes $\mathbb C\setminus f(\mathbb D)$ as an analytic arc $\Gamma$. Parametrize its regular part continuously from its finite endpoint toward infinity by $w=\gamma(s)$. Since $0\in f(\mathbb D)$, the arc avoids $0$. Use continuous real lifts of $\arg\gamma(s)$ and of $\arg\gamma'(s)-\arg\gamma(s)$, the angle from the radial to the tangent vector. Monotonicity means nondecreasing or nonincreasing along the oriented arc; adding a constant multiple of $2\pi$ to a lift does not change it.

Statement recorded in the supplied source.

Must $\arg\gamma(s)$ be monotone along every omitted support-point arc? Must the radial-to-tangent angle $\arg\gamma'(s)-\arg\gamma(s)$ be monotone? Both questions concern all support points, rather than only point-evaluation extremals. [S2]

Short English statement

Must the omitted arc of every support univalent map turn monotonically both in argument and relative to its radial direction?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 2306072 (AMR-022-6072). 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 6.72 and its complete update; Chapter 6 initial notation. https://arxiv.org/abs/1809.07200.