Research Problems in Function Theory — Problem 6.87

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 $H(\mathbb D)$ with its compact-convergence topology. Let $L_1,L_2:H(\mathbb D)\to\mathbb C$ be continuous complex-linear functionals, neither constant on $S$, and assume $L_1\ne tL_2$ as functionals on $H(\mathbb D)$ for every real $t>0$. A normalized rotation of the Koebe function is $k_\theta(z)=z/(1-e^{i\theta}z)^2$.

Statement recorded in the supplied source.

If $f\in S$ satisfies $\operatorname{Re}L_j(f)=\max_{g\in S}\operatorname{Re}L_j(g)$ for both $j=1,2$, must $f=k_\theta$ for some real $\theta$? The nonproportionality condition is exactly that on the ambient functionals, with no extra independence modulo constants on $S$. [S2] Update 6.87 reports Bakhtin's affirmative results for certain coefficient-functionals classes. [S2]

Short English statement

Must a common maximizer of two positively nonproportional linear objectives be a rotated Koebe map?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 2306087 (AMR-022-6087). 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 6.87 and its complete update; Chapter 6 initial notation. https://arxiv.org/abs/1809.07200.