Research Problems in Function Theory — Problem 5.51

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.

Let $\mathbb D=\{z\in\mathbb C:|z|<1\}$ and $\mathbb T=\partial\mathbb D$. A Blaschke product is a function \[B(z)=\eta z^m\prod_j\frac{|a_j|}{a_j}\frac{a_j-z}{1-\overline{a_j}z},\qquad |\eta|=1,\quad m\in\{0,1,\ldots\},\quad 0<|a_j|<1,\quad\sum_j(1-|a_j|)<\infty,\] where the product can be finite or infinite and repeated zeros are counted with multiplicity. An analytic function $g$ is Bloch if \[\sup_{z\in\mathbb D}(1-|z|^2)|g^{\prime}(z)|<\infty.\]

Statement recorded in the supplied source.

Give an explicit construction of a Blaschke product $B$ with $B(0)=0$ such that $g=(1+B)/(1-B)$ is Bloch. Equivalently, the construction must satisfy \[\sup_{z\in\mathbb D}\frac{(1-|z|^2)|B^{\prime}(z)|}{|1-B(z)|^2}<\infty.\] [S2]

Short English statement

Construct explicitly a Blaschke product vanishing at zero whose Cayley transform is a Bloch function.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 2305051 (AMR-022-5051). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Walter K. Hayman and Edward F. Lingham, Research Problems in Function Theory (2018), arXiv:1809.07200. Problem 5.51, complete original question, all following updates, and the relevant chapter definitions. https://arxiv.org/abs/1809.07200.