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Proceedings of the American Mathematical Society

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

 

 

Odd starlike functions


Authors: Ram Singh and Sangita Puri
Journal: Proc. Amer. Math. Soc. 94 (1985), 77-80
MSC: Primary 30C45
MathSciNet review: 781060
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Abstract: In the present paper, among other things, we prove that if $ f(f(0) = 0,f'(0) = 1)$ is regular and odd starlike in $ \left\vert z \right\vert < 1$, then $ {\text{Re }}f(z)/{s_n}(z,f) > 1/2,\;\left\vert z \right\vert < 1$, where $ {s_n}(z,f)$ denotes the $ n$th partial sum of $ f,\;n = 1,2,3, \ldots ,$, thus generalising the known result: $ \operatorname{Re} {\text{ }}f(z)/z > 1/2,\;\left\vert z \right\vert < 1$. As an application, we show that each partial sum of an odd convex function is close-to-convex in $ \left\vert z \right\vert < 1$.


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DOI: http://dx.doi.org/10.1090/S0002-9939-1985-0781060-7
Keywords: Univalent, starlike, convex and close-to-convex functions, convolution/Hadamard product
Article copyright: © Copyright 1985 American Mathematical Society