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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

On a subclass of spiral-like functions


Author: E. M. Silvia
Journal: Proc. Amer. Math. Soc. 44 (1974), 411-420
MSC: Primary 30A32
MathSciNet review: 0342688
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Abstract: Let $ \alpha \geqq 0,0 \leqq \beta < 1,\vert\lambda \vert < \pi /2$ and suppose that $ f(z) = z + \sum\nolimits_{n = 2}^\infty {{a_n}{z^n}} $ is holomorphic in $ U = \{ z:\vert z\vert < 1\} $. If

$\displaystyle \operatorname{Re} \left[ {{e^{i\lambda }}\frac{{zf'(z)}}{{f(z)}} ... ...{{f'(z)}} + 1 - \frac{{zf'(z)}}{{f(z)}}} \right)} \right] > \beta \cos \lambda $

for $ z \in U$, then $ f(z)$ is said to be $ \alpha$ - $ \lambda $-spiral-like of order $ \beta $ and we write $ f(z) \in S_\alpha ^\lambda (\beta )$. The author shows that for each $ \alpha \geqq 0, \alpha$ - $ \lambda$ -spiral-like functions of order $ \beta $ are $ \lambda $-spiral-like of order $ \beta $. The following representation theorem is obtained: The function $ f(z) \in S_\alpha ^\lambda (\beta )(\alpha > 0,0 \leqq \beta < 1,\vert\lambda \vert < \pi /2)$, if and only if there exists a function $ F(\zeta )\lambda $-spiral-like of order $ \beta $ such that

$\displaystyle F(z) = {\left[ {({e^{i\lambda }}/\alpha )\int_0^z {F{{(\zeta )}^{... ...bda }}/\alpha }}{\zeta ^{ - 1}}d\zeta } } \right]^{\alpha {e^{ - i\lambda }}}}.$

A distortion theorem for $ \log \vert f(z)/z\vert$ and a rotation theorem for $ \arg f(z)/z$ are also proved for functions $ f(z) \in S_0^\lambda (\beta )$.

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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9939-1974-0342688-4
PII: S 0002-9939(1974)0342688-4
Keywords: Univalent, $ \alpha $-starlike, $ \lambda $-spiral-like
Article copyright: © Copyright 1974 American Mathematical Society