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

Neighborhoods of univalent functions


Author: Stephan Ruscheweyh
Journal: Proc. Amer. Math. Soc. 81 (1981), 521-527
MSC: Primary 30C45; Secondary 30C75
MathSciNet review: 601721
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Abstract: For an analytic function $ f(z) = z + \Sigma _{k = 2}^\infty {a_k}{z^k}$ in the unit disc $ E$ conditions are established such that all functions $ g(z) = z + \Sigma _{k = 2}^\infty {b_k}{z^k} \in {N_\delta }(f)$, i.e. $ \Sigma _{k = 2}^\infty k\left\vert {{a_k} - {b_k}} \right\vert \leqslant \delta $, are in some class of univalent functions in $ E$. For instance, we prove that every $ g \in {N_{1/4}}(f)$ is starlike univalent in $ E$ if $ f$ is convex univalent.


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DOI: http://dx.doi.org/10.1090/S0002-9939-1981-0601721-6
PII: S 0002-9939(1981)0601721-6
Keywords: Univalent functions, special classes of univalent functions, Hadamard product
Article copyright: © Copyright 1981 American Mathematical Society