Research Problems in Function Theory — Problem 5.15

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 $\mathbb T=\partial\mathbb D$. All Taylor coefficients below are taken at $0$. For a plane domain $D$, write $\ell_D(r)$ for the angular Lebesgue measure of $\{\theta:re^{i\theta}\in D\}$. Its circular symmetrization $D^*$ contains the whole circle of radius $r$ when $D$ does; contains no points on that circle when $D$ misses it; otherwise it contains the arc $\{re^{i\theta}:|\theta|<\ell_D(r)/2\}$. Include $0$ in $D^*$ exactly when $0\in D$. Let $a_0>0$ belong to $D$, let $f:\mathbb D\to D$ be holomorphic with $f(0)=a_0$, and suppose $D^*$ admits a universal covering map $f^*:\mathbb D\to D^*$. Normalize the covering map by $f^*(0)=a_0$ and $(f^*)'(0)>0$. Write $f(z)=a_0+\sum_{n\ge1}a_nz^n$ and $f^*(z)=a_0+\sum_{n\ge1}a_n^*z^n$.For $0<r<1$, put $m(r,h)=\inf_{|z|=r}|h(z)|$ and $M(r,h)=\sup_{|z|\le r}|h(z)|$. A domain is doubly connected if its complement in the Riemann sphere has two connected components.

Statement recorded in the supplied source.

Without assuming $D^*$ simply connected, do both \[|a_1|\le |a_1^*|\quad\text{and}\quad m(r,f^*)\le m(r,f)\le M(r,f)\le M(r,f^*)\qquad(0<r<1)\] hold for every $f$ as above? Ask in particular the same two questions when both $D$ and $D^*$ are doubly connected. [S2]

Short English statement

Does circular symmetrization still dominate the derivative and minimum and maximum modulus when the symmetrized domain is multiply connected, especially doubly connected?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 2305015 (AMR-022-5015). 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 5, Problem 5.15, its update and the preceding shared notation. https://arxiv.org/abs/1809.07200.