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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



Subordination by univalent functions

Authors: Sunder Singh and Ram Singh
Journal: Proc. Amer. Math. Soc. 82 (1981), 39-47
MSC: Primary 30C55
MathSciNet review: 603598
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Abstract: Let $ K$ be the class of functions $ f(z) = z + {a_2}{z^2} + \cdots $, which are regular and univalently convex in $ \left\vert z \right\vert < 1$. In this paper we establish certain subordination relations between an arbitrary member $ f$ of $ K$, its partial sums and the functions $ (\lambda /z)\int_0^z {f(t)dt} $ and $ \mu \int_0^z {{t^{ - 1}}f(t)dt} $. The well-known result that $ z/2$ is subordinate to $ f(z)$ in $ \left\vert z \right\vert < 1$ for every $ f$ belonging to $ K$ follows as a particular case from our results. We also improve certain results of Robinson regarding subordination by univalent functions. A sufficient condition for a univalent function to be convex of order $ \alpha $ is also given.

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Keywords: Regular functions, univalent, starlike and convex functions, subordination, subordinating factor sequences, Hadamard product
Article copyright: © Copyright 1981 American Mathematical Society