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Transactions of the American Mathematical Society

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

 

 

On coefficient means of certain subclasses of univalent functions


Authors: F. Holland and J. B. Twomey
Journal: Trans. Amer. Math. Soc. 185 (1973), 151-163
MSC: Primary 30A34
DOI: https://doi.org/10.1090/S0002-9947-1973-0328054-X
MathSciNet review: 0328054
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Abstract: Let $ \mathcal{R}$ denote the class of regular functions whose derivatives have positive real part in the unit disc $ \gamma $ and let $ \mathcal{S}$ denote the class of functions starlike in $ \gamma $. In this paper we investigate the rates of growth of the means $ {s_n}(\lambda ) = {n^{ - 1}}\Sigma _1^n\vert{a_k}{\vert^\lambda }(0 < \lambda \leq 1)$ and $ {t_n}(\lambda ) = {n^{ - 1}}\Sigma _1^n{k^\lambda }\vert{a_k}{\vert^\lambda }\;(\lambda > 0)$ as $ n \to + \infty $ for bounded $ f(z) = \Sigma _1^\infty {a_k}{z^k} \in \mathcal{R} \cup \mathcal{S}$. It is proved, for example, that the estimate $ {t_n}(\lambda ) = o(1){(\log n)^{ - \alpha (\lambda )}}(n \to + \infty )$, where $ \alpha (\lambda ) = \lambda /2$ for $ 0 < \lambda < 2$ and $ \alpha (\lambda ) = 1$ for $ \lambda \geq 2$, holds for such functions f, and that it is best possible for each fixed $ \lambda > 0$ within the class $ \mathcal{R}$ and for each fixed $ \lambda \geq 2$ within the class $ \mathcal{S}$.

It is also shown that the inequality $ {s_n}(1) = o(1){n^{ - 1}}{(\log n)^{1/2}}$, which holds for all bounded univalent functions, cannot be improved for bounded $ f \in \mathcal{R}$. The behavior of $ {t_n}(\lambda )$ as $ n \to + \infty $ when $ {a_k} \geq 0(k \geq 1)$ and $ \lambda \geq 1$ is also examined.


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DOI: https://doi.org/10.1090/S0002-9947-1973-0328054-X
Keywords: Coefficient means, bounded univalent functions, starlike functions, derivatives with positive real part
Article copyright: © Copyright 1973 American Mathematical Society