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

 

Regular functions $ f(z)$ for which $ zf\sp{'} (z)$ is $ \alpha $-spiral-like


Author: Pran Nath Chichra
Journal: Proc. Amer. Math. Soc. 49 (1975), 151-160
MSC: Primary 30A32
MathSciNet review: 0361033
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Abstract: Let $ \mathfrak{F}_\alpha ^\lambda $ be the class of functions $ f(z) = z + {a_2}{z^2} + \cdots $ which are regular in $ E = \{ z/\vert z\vert < 1\} $ and satisfy

$\displaystyle \operatorname{Re} \{ {e^{i\alpha }}(1 + zf''(z)/f'(z))\} > \lambda \cos \alpha $

for some $ \alpha ,\vert\alpha \vert < \pi /2$, and for some $ \lambda ,0 \leq \lambda < 1$. The author finds a range on $ \alpha $ for which $ f(z)$ in $ \mathfrak{F}_\alpha ^\lambda $ is univalent in $ E$. In particular, the author improves upon the range on a for which $ f(z) \in \mathfrak{F}_\alpha ^0$ is known to be univalent in $ E$. Also a corresponding result is obtained for those functions $ f(z)$ in $ \mathfrak{F}_\alpha ^\lambda $ for which $ f''(0) = 0$.

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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9939-1975-0361033-2
PII: S 0002-9939(1975)0361033-2
Keywords: Regular function, starlike function, spiral-like function, univalent function, radius of convexity
Article copyright: © Copyright 1975 American Mathematical Society