Closure properties of families of Cauchy-Stieltjes transforms

Abstract:

For $\alpha > 0$ let ${\mathcal {F}_\alpha }$ denote the class of functions defined for $\left | z \right | < 1$ by integrating $1/{(1 - xz)^\alpha }$ against a complex measure on $\left | x \right | = 1$. The main results in this paper assert that ${\mathcal {F}_\alpha }$ is closed under multiplication by a function holomorphic for $\left | z \right | \leq 1$ and under composition with a function $\varphi$ holomorphic and satisfying $\left | {\varphi (z)} \right | < 1$ for $\left | z \right | < 1$ when $\alpha \geq 1$. The last result is shown to be false when $0 < \alpha < 1$.