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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)


Starlikeness and convexity from integral means of the derivative

Author: Shinji Yamashita
Journal: Proc. Amer. Math. Soc. 103 (1988), 1094-1098
MSC: Primary 30C45
MathSciNet review: 954989
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Abstract: If $ f$ is analytic in $ \vert z\vert < 1$ and normalized: $ f(0) = f'(0) - 1 = 0$, then $ f$ is univalent and starlike in $ \vert z\vert < 1(f)$, where

$\displaystyle I(f) = \sup \,r{\left\{ {{{(2\pi )}^{ - 1}}\int_0^{2\pi } {\vert f'(r{e^{it}}){\vert^2}dt} } \right\}^{ - 1/2}},\quad 0 \leq r \leq 1.$

Furthermore, there exists a normalized $ f$ such that $ I(f) < 1$ and that $ f'$ vanishes at a point on $ \vert z\vert = I(f)$.

If $ f$ is analytic and normalized in $ \vert z\vert < 1$, then $ f$ is univalent and convex in $ \vert z\vert < I(f)/2$.

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PII: S 0002-9939(1988)0954989-7
Article copyright: © Copyright 1988 American Mathematical Society

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