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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

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

The 2024 MCQ for Transactions of the American Mathematical Society is 1.48 .

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On a problem of Gronwall for Bazilevič functions
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by John L. Lewis
Trans. Amer. Math. Soc. 195 (1974), 231-242
DOI: https://doi.org/10.1090/S0002-9947-1974-0342687-7

Abstract:

Let $B(\alpha ,\beta ),\alpha$ positive, $\beta$ real, denote the class of normalized univalent Bazilevič functions in $K = \{ z:|z| < 1\}$ 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 $|f({z_0})|$. Also, a sharp asymptotic bound is obtained for ${(1 - r)^2}{\max _{|z| = r}}|f(z)|$. Finally, a sharp asymptotic bound is found for ${(1 - r)^2}{\max _{|z| = r}}|f(z)|$ when f is in B with second coefficient a.
References
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Bibliographic Information
  • © Copyright 1974 American Mathematical Society
  • 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