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
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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