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

 

Functions whose derivatives take values in a half-plane


Authors: Fernando Gray and Stephan Ruscheweyh
Journal: Proc. Amer. Math. Soc. 104 (1988), 215-218
MSC: Primary 30C45
MathSciNet review: 958069
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Abstract: We derive sharp upper and lower bounds for $ \left\vert {zf'(z)/f(z)} \right\vert$ where $ f \in R$, i.e. $ f$ analytic in $ {\mathbf{D}}$ with $ f(0) = 0,f'(0) = 1$ and $ {e^{i\alpha }}f'(z){\text{ > }}0$ in $ {\mathbf{D}}$ for a certain $ \alpha = \alpha (f) \in {\mathbf{R}}$. The extremal function is $ k(z) = - z - 2\log (1 - z)$. This result improves an earlier one of D. K. Thomas.


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DOI: http://dx.doi.org/10.1090/S0002-9939-1988-0958069-6
PII: S 0002-9939(1988)0958069-6
Article copyright: © Copyright 1988 American Mathematical Society