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New support points of $ \mathcal{S}$ and extreme points of $ \mathcal{HS}$


Author: Kent Pearce
Journal: Proc. Amer. Math. Soc. 81 (1981), 425-428
MSC: Primary 30C45
DOI: https://doi.org/10.1090/S0002-9939-1981-0597655-6
MathSciNet review: 597655
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Abstract: Let $ \mathcal{S}$ be the usual class of univalent analytic functions $ f$ on $ \{ z\left\vert {\vert z\vert} \right. < 1\}$ normalized by $ f(z) = z + {a_2}{z^2} + \cdots $. We prove that the functions

$\displaystyle {f_{x,y}}(z) = \frac{{z - \tfrac{1} {2}(x + y){z^2}}} {{{{(1 - yz)}^2}}},\quad \left\vert x \right\vert = \left\vert y \right\vert = 1,x \ne y,$

which are support points of $ \mathcal{C}$, the subclass of $ \mathcal{S}$ of close-to-convex functions, and extreme points of $ \mathcal{H}\mathcal{C}$, are support points of $ \mathcal{S}$ and extreme points of $ \mathcal{H}\mathcal{S}$ whenever $ 0 < \left\vert {\arg ( - x/y)} \right\vert \leqslant \pi /4$. We observe that the known bound of $ \pi /4$ for the acute angle between the omitted arc of a support point of $ \mathcal{S}$ and the radius vector is achieved by the functions $ {f_{x,y}}$ with $ \left\vert {\arg ( - x/y)} \right\vert = \pi /4$.

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

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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1981-0597655-6
Keywords: Support points, extreme points, univalent functions, close-to-convex functions
Article copyright: © Copyright 1981 American Mathematical Society

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