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

 

Convolution properties of a class of starlike functions


Authors: Ram Singh and Sukhjit Singh
Journal: Proc. Amer. Math. Soc. 106 (1989), 145-152
MSC: Primary 30C45
MathSciNet review: 994388
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Abstract: Let $ R$ denote the class of functions $ f(z) = z + {a_2}{z^2} + \cdots $ that are analytic in the unit disc $ E = \{ z:\left\vert z \right\vert < 1\} $ and satisfy the condition $ \operatorname{Re} (f'(z) + zf''(z)) > 0,z \in E$. It is known that $ R$ is a subclass of $ {S_t}$, the class of univalent starlike functions in $ E$. In the present paper, among other things, we prove (i) for every $ n \geq 1$, the $ n$th partial sum of $ f \in R,{s_n}(z,f)$, is univalent in $ E$, (ii) $ R$ is closed with respect to Hadamard convolution, and (iii) the Hadamard convolution of any two members of $ R$ is a convex function in $ E$.


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DOI: http://dx.doi.org/10.1090/S0002-9939-1989-0994388-6
PII: S 0002-9939(1989)0994388-6
Keywords: Univalent, starlike and convex functions, convex null sequences
Article copyright: © Copyright 1989 American Mathematical Society