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Transactions of the American Mathematical Society

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On a problem of Gronwall for Bazilevič functions


Author: John L. Lewis
Journal: Trans. Amer. Math. Soc. 195 (1974), 231-242
MSC: Primary 30A32
DOI: https://doi.org/10.1090/S0002-9947-1974-0342687-7
MathSciNet review: 0342687
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Abstract: Let $ B(\alpha ,\beta ),\alpha $ positive, $ \beta $ real, denote the class of normalized univalent Bazilevič functions in $ K = \{ z:\vert z\vert < 1\} $ of type $ \alpha ,\beta $. Let $ B = { \cup _{\alpha ,\beta }}B(\alpha ,\beta )$. Let $ \alpha ,0 \leq \alpha \leq 2$, and $ \alpha ,0 < \alpha < \infty $, be fixed and suppose that $ f(z) = z + a{z^2} + \cdots $ is in $ B(\alpha ,0)$. In this paper for given $ {z_0} \in K$, the author finds a sharp upper bound for $ \vert f({z_0})\vert$. Also, a sharp asymptotic bound is obtained for $ {(1 - r)^2}{\max _{\vert z\vert = r}}\vert f(z)\vert$. Finally, a sharp asymptotic bound is found for $ {(1 - r)^2}{\max _{\vert z\vert = r}}\vert f(z)\vert$ when f is in B with second coefficient a.


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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1974-0342687-7
Keywords: Univalent functions, Bazilevič functions, extremal functions, asymptotic growth
Article copyright: © Copyright 1974 American Mathematical Society

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