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A method for investigating geometric properties of support points and applications


Author: Johnny E. Brown
Journal: Trans. Amer. Math. Soc. 287 (1985), 285-291
MSC: Primary 30C55; Secondary 30C50
DOI: https://doi.org/10.1090/S0002-9947-1985-0766220-8
MathSciNet review: 766220
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Abstract: A normalized univalent function $ f$ is a support point of $ S$ if there exists a continuous linear functional $ L$ (which is nonconstant on $ S$) for which $ f$ maximizes $ \operatorname{Re} L(g),g \in S$. For such functions it is known that $ \Gamma = {\text{C}} - f(U)$ is a single analytic arc that is part of a trajectory of a certain quadratic differential $ Q(w)\;d{w^2}$. A method is developed which is used to study geometric properties of support points. This method depends on consideration of $ \operatorname{Im} \{ {w^2}Q(w)\} $ rather than the usual $ \operatorname{Re} \{ {w^2}Q(w)\} $. Qualitative, as well as quantitative, applications are obtained. Results related to the Bieberbach conjecture when the extremal functions have initial real coefficients are also obtained.


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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1985-0766220-8
Article copyright: © Copyright 1985 American Mathematical Society

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