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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 

 

Some sharp neighborhoods of univalent functions


Author: Johnny E. Brown
Journal: Trans. Amer. Math. Soc. 287 (1985), 475-482
MSC: Primary 30C45
MathSciNet review: 768720
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Abstract: For $ \delta \geqslant 0$ and $ f(z) = z + {a_2}{z^2} + \cdots $ analytic in $ \vert z\vert < 1$ let the $ \delta $-neighborhood of $ f$, $ {N_\delta }(f)$, consist of those analytic functions $ g(z) = z + {b_z}{z^2} + \cdots $ with $ \sum\nolimits_{k = 2}^\infty {k\vert{a_k} - {b_k}\vert \leqslant \delta } $. We determine sufficient conditions guaranteeing which neighborhoods of certain classes of convex functions belong to certain classes of starlike functions. We extend some recent results of St. Ruscheweyh and R. Fournier and, at the same time, provide much simpler proofs. We also prove precisely how boundaries affect the value of $ \delta $ for some general classes of functions.


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DOI: https://doi.org/10.1090/S0002-9947-1985-0768720-3
Keywords: Univalent functions, Hadamard product, subordination, starlike functions, convex functions
Article copyright: © Copyright 1985 American Mathematical Society