Research Problems in Function Theory — Problem 5.53

Definitions and mathematical statement

A Blaschke sequence $(a_n)\subset\mathbb D$, counted with multiplicity, satisfies $\sum_n(1-|a_n|)<\infty$. Its Blaschke product is \[ B_a(z)=z^m\prod_{a_n\ne0}\frac{|a_n|}{a_n}\frac{a_n-z}{1-\overline{a_n}z}, \] up to a unimodular constant, where $m$ counts zeros at zero. For two such sequences define the meromorphic quotient $R=B_a/B_b$, cancelling common zeros with their multiplicities. Characterize exactly when $R$ is normal in the disc, that is, \[ \sup_{z\in\mathbb D}(1-|z|^2)R^{\#}(z)<\infty, \qquad R^{\#}=\frac{|R'|}{1+|R|^2}, \] with the spherical derivative extended through poles by using $1/R$. A distinct sequence is interpolating if arbitrary bounded complex values on it can be interpolated by a bounded holomorphic function on $\mathbb D$. The source gives a criterion when the two zero sequences are interpolating: after cancellation, normality corresponds to their union being interpolating. The problem asks for a criterion covering general Blaschke sequences, including clustering and multiplicities. Separate convergence of the two Blaschke products only guarantees bounded characteristic of the quotient and does not control the displayed spherical derivative.

Short English statement

Give a necessary and sufficient condition on two general Blaschke zero sequences for their quotient to be a normal meromorphic function in the disc.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 2305053 (AMR-022-5053), 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 5.53. Original source indexed by AMR-022-5053. https://arxiv.org/abs/1809.07200.