Growth and covering theorems associated with the Roper-Suffridge extension operator

Graham, Ian

doi:10.1090/S0002-9939-99-04963-1

Growth and covering theorems associated with the Roper-Suffridge extension operator
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by Ian Graham

Proc. Amer. Math. Soc. 127 (1999), 3215-3220

DOI: https://doi.org/10.1090/S0002-9939-99-04963-1

Published electronically: April 27, 1999

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Abstract:

The Roper-Suffridge extension operator, originally introduced in the context of convex functions, provides a way of extending a (locally) univalent function $f\in \operatorname {Hol}(\mathbb {D},\mathbb {C})$ to a (locally) univalent map $F\in \operatorname {Hol}(B_{n},\mathbb {C}^{n})$. If $f$ belongs to a class of univalent functions which satisfy a growth theorem and a distortion theorem, we show that $F$ satisfies a growth theorem and consequently a covering theorem. We also obtain covering theorems of Bloch type: If $f$ is convex, then the image of $F$ (which, as shown by Roper and Suffridge, is convex) contains a ball of radius $\pi /4$. If $f\in S$, the image of $F$ contains a ball of radius $1/2$.

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