Research Problems in Function Theory — Problem 5.43

Definitions and mathematical statement

Fix $0<r<1$ and let $A_r=\{z\in\mathbb C:r<|z|<1\}$. Let $\mathcal H_r$ be the class of analytic functions $f:A_r\to\mathbb C$ with $|f(z)|\leq1$ throughout the annulus. Every such function has a Laurent expansion $f(z)=\sum_{k\in\mathbb Z}c_kz^k$, converging on $A_r$, where \[ c_k=\frac1{2\pi i}\int_{|z|=\rho} \frac{f(z)}{z^{k+1}}\,dz \quad(r<\rho<1). \] The integral is independent of the chosen intermediate radius.

For arbitrary nonnegative integers $m,n$, determine the coefficient body \[ \mathcal C_{m,n}(r)= \{(c_{-m},\ldots,c_n):f\in\mathcal H_r\} \subset\mathbb C^{m+n+1}. \] Thus one seeks necessary and sufficient joint restrictions on these finitely many coefficients, not only separate bounds for each coefficient. The remaining Laurent coefficients are free to vary in a realization.

Equivalently, determine the sharp support value $\sup_{f\in\mathcal H_r}\operatorname{Re} \sum_{k=-m}^{n}\lambda_kc_k$ for every complex coefficient vector $(\lambda_{-m},\ldots,\lambda_n)$, and characterize all functions attaining it. These support values specify the compact convex coefficient body completely. The problem includes its dependence on the annulus modulus $r$, all finite coefficient ranges, and the corresponding extremal functions. Functions need not extend analytically across either boundary circle; boundedness on the open annulus is the only boundary restriction.

Short English statement

Describe all finite collections of Laurent coefficients possible for bounded analytic functions on an annulus, and their extremizers.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 2305043 (AMR-022-5043), AMR Open Problem Lists. Catalogue identity and source transcription; CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Hayman - Research Problems in Function Theory (2018), Problem 5.43. Original source indexed by AMR-022-5043. https://arxiv.org/abs/1809.07200.