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Closure properties of families of Cauchy-Stieltjes transforms

Authors: R. A. Hibschweiler and T. H. MacGregor
Journal: Proc. Amer. Math. Soc. 105 (1989), 615-621
MSC: Primary 30E20
MathSciNet review: 938912
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Abstract: For $ \alpha > 0$ let $ {\mathcal{F}_\alpha }$ denote the class of functions defined for $ \left\vert z \right\vert < 1$ by integrating $ 1/{(1 - xz)^\alpha }$ against a complex measure on $ \left\vert x \right\vert = 1$. The main results in this paper assert that $ {\mathcal{F}_\alpha }$ is closed under multiplication by a function holomorphic for $ \left\vert z \right\vert \leq 1$ and under composition with a function $ \varphi $ holomorphic and satisfying $ \left\vert {\varphi (z)} \right\vert < 1$ for $ \left\vert z \right\vert < 1$ when $ \alpha \geq 1$. The last result is shown to be false when $ 0 < \alpha < 1$.

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Keywords: Holomorphic function, Cauchy-Stieltjes transforms, complex measures
Article copyright: © Copyright 1989 American Mathematical Society

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