Regular functions of restricted growth and their zeros in tangential regions

Abstract:

For a given function $k$, positive, continuous, nondecreasing and unbounded on $[0,1)$, let ${A^{(k)}}$ denote the class of functions regular in the unit disc for which log $|f(z)| < k(|z|)$ when $|z| < 1$. Hayman and Korenblum have shown that a necessary and sufficient condition for the sets of positive zeros of all functions in ${A^{(k)}}$ to be Blaschke is that \[ \int _0^1 {\sqrt {(k(t)/(1 - t)) dt} } \] is finite. It is shown that the imposition of a further regularity condition on the growth of $k$ ensures that in some tangential region the zero set of each function in ${A^{(k)}}$ is also Blaschke.