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

Regular functions $ f(z)$ for which $ z f'(z)$ is $ \alpha$-spiral


Authors: Richard J. Libera and Michael R. Ziegler
Journal: Trans. Amer. Math. Soc. 166 (1972), 361-370
MSC: Primary 30A32
MathSciNet review: 0291433
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Abstract: A function $ f(z) = z + \Sigma _{n = 2}^\infty {a_n}{z^n}$ regular in the open unit disk $ \Delta = \{ z:\vert z\vert < 1\} $ is a (univalent) $ \alpha $-spiral function for real $ \alpha ,\vert\alpha \vert < \pi /2$, if $ \operatorname{Re} \{ {e^{i\alpha }}zf'(z)/f(z)\} > 0$ for z in $ \Delta $; in this case we write $ f(z) \in {\mathcal{F}_\alpha }$. A fundamental result of this paper shows that the transformation

$\displaystyle {f_ \ast }(z) = \frac{{azf((z + a)/(1 + \bar az))}}{{f(a)(z + a){{(1 + \bar az)}^{{e^{ - 2i\alpha }}}}}}$

defines a function in $ {\mathcal{F}_\alpha }$ whenever $ f(z)$ is in $ {\mathcal{F}_\alpha }$ and a is in $ \Delta $.

If $ g(z)$ is regular in $ \Delta ,g(0) = 0$ and $ g'(0) = 1$, then $ g(z)$ is in $ {\mathcal{G}_\alpha }$ if and only if $ zg'(z)$ is in $ {\mathcal{F}_\alpha }$. The main result of the paper is the derivation of the sharp radius of close-to-convexity for each class $ {\mathcal{G}_\alpha }$; it is given as the solution of an equation in r which is dependent only on $ \alpha $. (Approximate solutions of this equation were made by computer and these suggest that the radius of close-to-convexity of the class $ \mathcal{G} = { \cup _\alpha }{\mathcal{G}_\alpha }$ is approximately $ .99097^{+}$.) Additional results are also obtained such as the radius of convexity of $ {\mathcal{G}_\alpha }$, a range of $ \alpha $ for which $ g(z)$ in $ {\mathcal{G}_\alpha }$ is always univalent is given, etc. These conclusions all depend heavily on the transformation cited above and its analogue for $ {\mathcal{G}_\alpha }$.


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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9947-1972-0291433-2
PII: S 0002-9947(1972)0291433-2
Keywords: Spiral functions, univalence preserving transformations, radius of close-to-convexity
Article copyright: © Copyright 1972 American Mathematical Society