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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



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
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 $|z| < 1$, a type of extremal problems is determined which can be solved already within the set $p(z) = (1 + \varepsilon z)/(1 - \varepsilon z),\;|\varepsilon | = 1$. One problem of this kind is to find the largest number $\rho (s,\;\mu )$ such that \[ \operatorname {Re} \{ p(z) + szp’(z)/(p(z) + \mu )\} > 0,\] $|z| \leqslant \rho (s,\;\mu )$, 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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Keywords: Functions with positive real part, Hadamard product, spiral-convexity
Article copyright: © Copyright 1976 American Mathematical Society