On the composition of Frostman Blaschke products (Q2866765)

A Blaschke product is called a uniform Frostman Blaschke product if the set of its zeros \(\{z_n\}\) has the property NEWLINE\[NEWLINE \sup_{|z|=1}\sum_{n=1}^\infty \frac{1-|z_n|}{|z-z_n|}<\infty. NEWLINE\]NEWLINE The authors construct an infinite uniform Frostman Blaschke product \(B\) such that \(B\circ B\) is also a uniform Frostman Blaschke product. This result answers a question posed by \textit{P. Gorkin} et al. [Result. Math. 25, No. 3--4, 252--269 (1994; Zbl 0799.30023)]. The authors also show that the set of uniform Frostman Blaschke products is open in the set of inner functions with the uniform norm. The paper contains an introduction with a comprehensive discussion of these and several related results. Finally the authors pose a conjecture: The class of prime Blaschke products is uniformly dense in the set of all Blaschke products. A Blaschke product \(B\) is prime whenever \(B=U\circ V\) then either of \(U\) or \(V\) is a Möbius transformation.