Research Problems in Function Theory — Problem 5.28

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 $0$.Let $f$ be a nonconstant function holomorphic on $\mathbb D$ and continuous on $\overline{\mathbb D}$. For $0<\delta<2$, define \[\omega(f,\delta)=\sup_{\substack{z,w\in\mathbb D\\|z-w|\le\delta}}|f(z)-f(w)|,\qquad \widetilde\omega(f,\delta)=\sup_{\substack{z,w\in\mathbb T\\|z-w|\le\delta}}|f(z)-f(w)|. \] Nonconstancy makes the denominator positive; constant functions have zero modulus in both definitions.

Statement recorded in the supplied source.

Must every such $f$ satisfy $\lim_{\delta\downarrow0}\omega(f,\delta)/\widetilde\omega(f,\delta)=1$? [S2]

Short English statement

Historical question: are the small-scale interior and boundary moduli of continuity asymptotically equal for every disc-algebra function? The source update records a negative answer.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 2305028 (AMR-022-5028). 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 5.28, its update and the preceding shared notation. https://arxiv.org/abs/1809.07200.