Cyclic Nevanlinna class functions in Bergman spaces

Abstract:

Let $f$ be a function which is in both the Bergman space ${A^p}$ $(p \geq 1)$ and the Nevanlinna class $N$. We show that if $f$ is expressed as the quotient of ${H^\infty }$ functions, then the inner part of its denominator is cyclic. As a corollary, we obtain that $f$ is cyclic if and only if the inner part of its numerator is cyclic. These results extend those of Berman, Brown, and Cohn [2]. Using more difficult methods, they have obtained them for the case $f \in {A^2} \cap N$. Finally, we show that the condition $|f(z)| \geq \delta {(1 - |z|)^c}$ ($z \in D$; $\delta ,c$ positive constants) is sufficient for cyclicity for $f \in {A^p} \cap N$, which answers a question of Aharonov, Shapiro, and Shields [1].