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

New subclasses of the class of close-to-convex functions


Author: Pran Nath Chichra
Journal: Proc. Amer. Math. Soc. 62 (1977), 37-43
MSC: Primary 30A32
MathSciNet review: 0425097
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Abstract: In this paper we introduce new subclasses of the class of close-to-convex functions. We call a regular function $ f(z)$ an alpha-close-to-convex function if $ (f(z)f'(z)/z) \ne 0$ for z in E and if for some nonnegative real number $ \alpha $ there exists a starlike function $ \phi (z) = z + \cdots $ such that

$\displaystyle \operatorname{Re} \;\left[ {(1 - \alpha )\frac{{zf'(z)}}{{\phi (z)}} + \alpha \frac{{(zf'(z))'}}{{\phi '(z)}}} \right] > 0$

for z in E.

We have proved that all alpha-close-to-convex functions are close-to-convex and have obtained a few coefficient inequalities for $ \alpha $-close-to-convex functions and an integral formula for constructing these functions.

Let $ {\mathfrak{F}_\alpha }$ be the class of regular and normalised functions $ f(z)$ which satisfy $ \operatorname{Re} \;(f'(z) + \alpha zf''(z)) > 0$ for z in E. $ f(z) \in {\mathfrak{F}_\alpha }$ gives $ \operatorname{Re} f'(z) > 0$ for z in E provided $ \operatorname{Re} \alpha \geqslant 0$. A sharp radius of univalence of the class of functions $ f(z)$ for which $ zf'(z) \in {\mathfrak{F}_\alpha }$ has also been obtained.


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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9939-1977-0425097-1
PII: S 0002-9939(1977)0425097-1
Keywords: Alpha-starlike functions, convex functions, starlike functions, close-to-convex functions, radius of univalence
Article copyright: © Copyright 1977 American Mathematical Society