Research Problems in Function Theory — Problem 5.69

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 $f(z)=1+\sum_{n\ge1}a_nz^n$. Assume in addition that every Taylor coefficient $a_n$ is a nonnegative real number. Is it necessarily true that \[ \sum_{n\ge1}a_n^2<\infty? \] Equivalently, must every function in this smaller class belong to the Hardy space $H^2$, defined by $\sup_{0<r<1}(2\pi)^{-1}\int_0^{2\pi}|f(re^{i\theta})|^2\,d\theta<\infty$? The equivalence follows from Parseval's identity. The Bloch condition by itself and positivity of the real part by itself are not replaced by coefficient positivity; all three assumptions are imposed simultaneously. The source also asks a related question without the nonnegative-coefficient assumption: does every normalized positive-real-part Bloch function satisfy $\sum_n|a_n|^4<\infty$, or at least $\sum_n|a_n|^p<\infty$ for one universal finite $p>4$? In that last formulation $p$ must work for the whole class, while the value of the sum may depend on the function. A bound on every finite partial sum that still diverges with its endpoint would not establish the required summability.

Short English statement

Are nonnegative Taylor coefficients of positive-real-part Bloch functions square summable, and is some universal higher-power summability available without nonnegativity?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 2305069 (AMR-022-5069), 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 5.69. Original source indexed by AMR-022-5069. https://arxiv.org/abs/1809.07200.