Research Problems in Function Theory — Problem 6.50

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 $\Sigma$ be the source's class of holomorphic injective exterior functions $f(z)=z+\sum_{n\ge1}b_nz^{-n}$ on $|z|>1$.For $0\le\alpha\le1$, define the Laurent germ near $\infty$ by \[F(z)=z\exp\left((1-\alpha)\log(f(z)/z)+\alpha\log(g(z)/z)\right)=z+\sum_{n\ge0}A_nz^{-n},\] with the logarithms tending to $0$ at $\infty$. This fixes all coefficients even if fractional powers cannot be continued across exterior zeros.

Statement recorded in the supplied source.

For every $f,g\in\Sigma$ and $0\le\alpha\le1$, is $\sum_{n\ge1}n|A_n|^2\le1$? [S2]

Short English statement

Does the coefficient area inequality survive a weighted geometric product of two exterior univalent maps?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 2306050 (AMR-022-6050). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 W. K. Hayman and E. F. Lingham, Research Problems in Function Theory (2018), arXiv:1809.07200, Chapter 6, Problem 6.50, complete update and preceding notation. https://arxiv.org/abs/1809.07200.