Publications Meetings The Profession Membership Programs Math Samplings Policy & Advocacy In the News About the AMS

   
Mobile Device Pairing
Green Open Access
Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

On the coefficients of meromorphic univalent functions


Author: D. K. Thomas
Journal: Proc. Amer. Math. Soc. 47 (1975), 161-166
MathSciNet review: 0357765
Full-text PDF Free Access

Abstract | References | Additional Information

Abstract: Let $ f \in \Sigma $, the class of all analytic univalent functions defined in $ \gamma = \{ z:\vert z\vert > 1\} $. For $ f,g \in \Sigma $ define $ h$ in $ \gamma $ by $ h(z) = f{(z)^{1 - \alpha }}g{(z)^\alpha },0 < \alpha < 1$. If $ h(z) = z + \Sigma _{n = 0}^\infty {c_n}{z^{ - n}}$, it is shown that $ \Sigma _{n = 1}^\infty n\vert{c_n}{\vert^2} < \infty $. This result is used to show that if $ {B_\alpha }$ denotes the class of all meromorphic Bazilevič functions of type $ \alpha $ and $ f \in {B_\alpha }$ with $ f(z) = z + \Sigma _{n = 0}^\infty {a_n}{z^{ - n}}$, then $ n{a_n} = O(1)$ as $ n \to \infty $, the result being best possible.


References [Enhancements On Off] (What's this?)


Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9939-1975-0357765-2
PII: S 0002-9939(1975)0357765-2
Article copyright: © Copyright 1975 American Mathematical Society