Research Problems in Function Theory — Problem 6.95

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 $f(z)=z+\sum_{n=2}^{\infty}a_nz^n$. Let $\mathcal H$ consist of holomorphic $h$ on $\mathbb D$ for which there is $f\in S$ whose inverse germ at $0$ continues holomorphically to $\mathbb D$ and satisfies $2h=f+f^{-1}$ there. The inverse is a compositional inverse continued from $0$, not a reciprocal. The definition forces $h(0)=0$ and $h'(0)=1$.

Statement recorded in the supplied source.

Give an intrinsic necessary-and-sufficient characterization of $\mathcal H$, expressed in terms of $h$ itself. In particular, is there any nonzero $a\in\mathbb C$ for which $h_a(z)=z+a\sin(2\pi z)$ belongs to $\mathcal H$? [S2]

Short English statement

Characterize averages of normalized univalent maps with their continued inverses, including membership of the specified sine perturbations.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 2306095 (AMR-022-6095). 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 6.95 and its complete update; Chapter 6 initial notation. https://arxiv.org/abs/1809.07200.