Research Problems in Function Theory — Problem 5.68

Definitions and mathematical statement

A holomorphic function $f$ on the unit disc $\mathbb D$ is a Bloch function if $\beta(f)=\sup_{z\in\mathbb D}(1-|z|^2)|f'(z)|<\infty$. Consider the additional normalization $f(0)=1$ and positivity $\operatorname{Re}f(z)>0$ in $\mathbb D$, and write $f(z)=1+\sum_{n\ge1}a_nz^n$. For $N\ge1$ put $S_f(N)=\sum_{n=1}^N|a_n|^2$. Determine the sharp possible rates of growth of $S_f(N)$ as $N\to\infty$ for this class. A quantitative universal formulation fixes $B>0$ and asks for optimal asymptotic bounds on \[ G_B(N)=\sup\{S_f(N):f(0)=1, \operatorname{Re}f>0,\ \beta(f)\le B\}. \] One should also distinguish bounds on this supremum from growth rates attained by a single fixed function along all $N$ or a subsequence. Normalizing the Bloch seminorm prevents an implicit constant from hiding arbitrary dependence on $f$. For comparison, positivity alone gives a uniform bound on each individual coefficient, but summing those separate bounds need not reflect the added Bloch condition. Parseval's identity rewrites the corresponding weighted coefficient sum as \[ 1+\sum_{n\ge1}|a_n|^2r^{2n} =\frac1{2\pi}\int_0^{2\pi}|f(re^{i\theta})|^2\,d\theta. \] The issue is the best joint consequence of positivity and Bloch growth, with matching examples rather than only a nonsharp estimate.

Short English statement

How quickly can the squared Taylor coefficients accumulate for a normalized Bloch function with positive real part?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 2305068 (AMR-022-5068), 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 5.68. Original source indexed by AMR-022-5068. https://arxiv.org/abs/1809.07200.