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On certain extremal problems for functions with positive real part


Authors: Stephan Ruscheweyh and Vikramaditya Singh
Journal: Proc. Amer. Math. Soc. 61 (1976), 329-334
MSC: Primary 30A32
DOI: https://doi.org/10.1090/S0002-9939-1976-0425102-1
MathSciNet review: 0425102
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Abstract: For the class $ P$ of analytic functions $ p(z),p(0) = 1$, with positive real part in $ \vert z\vert < 1$, a type of extremal problems is determined which can be solved already within the set $ p(z) = (1 + \varepsilon z)/(1 - \varepsilon z),\;\vert\varepsilon \vert = 1$. One problem of this kind is to find the largest number $ \rho (s,\;\mu )$ such that

$\displaystyle \operatorname{Re} \{ p(z) + szp'(z)/(p(z) + \mu )\} > 0,$

$ \vert z\vert \leqslant \rho (s,\;\mu )$, for all $ p \in P,\; - 1 \ne \mu \in {\mathbf{C}},\;s > 0$. Sharp upper bounds for two other functionals over $ P$ are also given.

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DOI: https://doi.org/10.1090/S0002-9939-1976-0425102-1
Keywords: Functions with positive real part, Hadamard product, spiral-convexity
Article copyright: © Copyright 1976 American Mathematical Society

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