A Loewner approach to a coefficient inequality for bounded univalent functions
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- by Duane W. De Temple and James A. Jenkins
- Proc. Amer. Math. Soc. 65 (1977), 125-126
- DOI: https://doi.org/10.1090/S0002-9939-1977-0444932-4
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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
- D. W. DeTemple, Generalizations of the Grunsky-Nehart inequalities, Arch. Rational Mech. Anal. 44 (1971), 93-120.
- W. K. Hayman, Multivalent functions, Cambridge Tracts in Mathematics and Mathematical Physics, No. 48, Cambridge University Press, Cambridge, 1958. MR 0108586
- James A. Jenkins, On certain coefficients of univalent functions, Analytic functions, Princeton Univ. Press, Princeton, N.J., 1960, pp. 159–194. MR 0117345
- George B. Leeman Jr., A new proof for an inequality of Jenkins, Proc. Amer. Math. Soc. 54 (1976), 114–116. MR 393457, DOI 10.1090/S0002-9939-1976-0393457-2
Bibliographic Information
- © Copyright 1977 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 65 (1977), 125-126
- MSC: Primary 30A34
- DOI: https://doi.org/10.1090/S0002-9939-1977-0444932-4
- MathSciNet review: 0444932