Research Problems in Function Theory — Problem 5.58

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 $\mathbb T=\partial\mathbb D$. Let $f$ be analytic, injective and nowhere zero on $\mathbb D$. Write $H^{\infty}(\mathbb D)=\{h\text{ analytic}:\sup_{\mathbb D}|h|<\infty\}$. An analytic $F$ with $\operatorname{Re}F>0$ admits the canonical logarithm on the right half-plane, so $F^{1/p}=\exp(p^{-1}\log F)$ is well defined. The known factorization for $0<p<1/2$ is $f=B_pF_p^{1/p}$ with $B_p,1/B_p\in H^{\infty}$ and $\operatorname{Re}F_p>0$.

Statement recorded in the supplied source.

Does the same factorization exist at $p=1/2$; that is, for every such $f$, are there $B,1/B\in H^{\infty}(\mathbb D)$ and analytic $F$ with $\operatorname{Re}F>0$ such that $f=BF^2$? The square of the right half-plane maps onto a slit plane, giving the source’s equivalent bounded-factor/slit-plane formulation. The update distinguishes this question from a stronger conjecture that has a counterexample. [S2]

Short English statement

Does every zero-free univalent disk function admit the endpoint factorization into an invertible bounded analytic factor and the square of a positive-real-part function?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 2305058 (AMR-022-5058). 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 5.58, complete original question, all following updates, and the relevant chapter definitions. https://arxiv.org/abs/1809.07200.