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

 

Analytic functions close to mappings convex in one direction


Authors: Walter Hengartner and Glenn Schober
Journal: Proc. Amer. Math. Soc. 28 (1971), 519-524
MSC: Primary 30.42
MathSciNet review: 0277704
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Abstract: By analogy to the class of close-to-convex functions we define a class of analytic functions which are close to a family $ \Sigma $ of mappings onto domains convex in one direction. In contrast to the close-to-convex class the close-to-$ \Sigma $ functions are not necessarily univalent. However, we determine the radius of convexity for $ \Sigma $, and this gives a lower bound for the radius of univalence of close-to-$ \Sigma $ functions. We next derive the coefficient estimate $ \vert{A_n}\vert \leqq n\vert{A_1}\vert$ for close-to-$ \Sigma $ functions and conclude with an elementary distortion theorem.


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DOI: http://dx.doi.org/10.1090/S0002-9939-1971-0277704-9
PII: S 0002-9939(1971)0277704-9
Keywords: Close-to-convex functions, convexity in one direction, radius of convexity, coefficient estimates, distortion theorems
Article copyright: © Copyright 1971 American Mathematical Society