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A Loewner approach to a coefficient inequality for bounded univalent functions

Authors: Duane W. De Temple and James A. Jenkins
Journal: Proc. Amer. Math. Soc. 65 (1977), 125-126
MSC: Primary 30A34
MathSciNet review: 0444932
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Abstract: The Loewner theory is used to obtain the sharp upper bound for the functional $ \operatorname{Re} \{ {e^{2i\theta }}({a_3} - a_2^2) + 4\sigma {e^{i\theta }}{a_2}\} $ over the class of univalent functions $ f(z) = b(z + {a_2}{z^2} + {a_3}{z^3} + \ldots )$ which map the unit disc into itself; $ \theta \in {\mathbf{R}},\sigma \in [0,1]$ and $ b \in (0,1]$ are fixed parameters.

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

  • [1] D. W. DeTemple, Generalizations of the Grunsky-Nehart inequalities, Arch. Rational Mech. Anal. 44 (1971), 93-120.
  • [2] W. K. Hayman, Multivalent functions, Cambridge Univ. Press, London and New York, 1958. MR 0108586 (21:7302)
  • [3] J. A. Jenkins, On certain coefficients of univalent functions, Analytic Functions, Princeton Univ. Press, Princeton, N.J., 1960, pp. 159-194. MR 0117345 (22:8126a)
  • [4] G. B. Leeman, Jr., A new proof for an inequality of Jenkins, Proc. Amer. Math. Soc. 54 (1976), 114-116. MR 0393457 (52:14267)

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Keywords: Univalent functions, bounded univalent functions
Article copyright: © Copyright 1977 American Mathematical Society

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