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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Locally univalent functions with locally univalent derivatives


Author: Douglas Michael Campbell
Journal: Trans. Amer. Math. Soc. 162 (1971), 395-409
MSC: Primary 30.42
MathSciNet review: 0286992
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Abstract: S. M. Shah and S. Y. Trimble have discovered that the behavior of an analytic function $ f(z)$ is strongly influenced by the radii of univalence of its derivatives $ {f^{(n)}}(z)\;(n = 0,1,2, \ldots )$. In this paper many of Shah and Trimble's results are extended to large classes of locally univalent functions with locally univalent derivatives. The work depends on the concept of the $ {\mathcal{U}_\beta }$-radius of a locally univalent function that is introduced and developed in this paper. Ch. Pommerenke's definition of a linear invariant family of locally univalent functions and the techniques of that theory are employed in this paper. It is proved that the universal linear invariant families $ {\mathcal{U}_\alpha }$ are rotationally invariant. For fixed $ f(z)$ in $ {\mathcal{U}_\alpha }$, it is shown that the function $ r \to {\text{order}}\;[f(rz)/r]\;(0 < r \leqq 1)\;$ is a continuous increasing function of r.


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DOI: http://dx.doi.org/10.1090/S0002-9947-1971-0286992-9
PII: S 0002-9947(1971)0286992-9
Keywords: Analytic function, univalent function, entire function, linear invariant family of locally univalent functions, order of a family, admissible property, the $ {\mathcal{U}_\beta }$-radius, rotational invariance, contractional invariance
Article copyright: © Copyright 1971 American Mathematical Society