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Transactions of the American Mathematical Society

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On some starlike and convex functions


Author: G. M. Shah
Journal: Trans. Amer. Math. Soc. 154 (1971), 83-91
MSC: Primary 30.42
DOI: https://doi.org/10.1090/S0002-9947-1971-0269826-8
MathSciNet review: 0269826
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Abstract: In this paper we study functions of the form $ \smallint _0^z(g(t)/\Pi _{k = 1}^n{(1 - t{z_k})^{{\alpha _k}}})$ for $ \vert z\vert < 1$ and show under what conditions such a function is convex, convex in one direction and hence univalent in $ \vert z\vert < 1$. We also study the functions $ g(z)$ where $ g(0) = 1,g(z) \ne 0$ and $ \operatorname{Re} \;[zg'(z)/g(z)] \geqq - \alpha ,0 \leqq \alpha < 1$, for $ \vert z\vert < 1$.


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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1971-0269826-8
Keywords: Univalent, starlike, convex, Schwarz-Christoffel transformation, Herglotz's representation, extremal functions, contour
Article copyright: © Copyright 1971 American Mathematical Society

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