J. Aust. Math. Soc.  80 (2006), 367-373
Omitted rays and wedges of fractional Cauchy transforms

R. A. Hibschweiler
  University of New Hampshire
  Department of Mathematics and Statistics
  Durham, NH 03824
T. H. Macgregor
  Bowdoin College
  Department of Mathematics
  Brunswick, ME 04011

For $\alpha>0$ let $\mathcal{F}_{\alpha}$ denote the set of functions which can be expressed
\[ f(z)= \int_{|\zeta|=1} 
 \frac1{(1-\overline{\zeta}z)^{\alpha}}\, d\mu(\zeta) 
 \quad\text{for }\ |z|<1, \]
where $\mu$ is a complex-valued Borel measure on the unit circle. We show that if $f$ is an analytic function in $\Delta=\{z\in\mathbb{C}: |z|<1\}$ and there are two nonparallel rays in $\mathbb{C}\backslash f(\Delta)$ which do not meet, then $f\in \mathcal{F}_{\alpha}$ where $\alpha\pi$ denotes the largest of the two angles determined by the rays. Also if the range of a function analytic in $\Delta$ is contained in an angular wedge of opening $\alpha\pi$ and $1<\alpha<2$, then $f\in \mathcal{F}_{\alpha}$.
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