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Zbl 0898.30021
Bulboacă, Teodor
Integral operators that preserve the subordination.
(English)
[J] Bull. Korean Math. Soc. 34, No.4, 627-636 (1997). ISSN 1015-8634

Let $H(U)$ denote the space of all analytic functions in the open unit disc $U$ of the complex plane. For $f\in H(U)$, $\beta,\gamma\in \bbfC$ define $$A_{\beta,\gamma}(f)(z)= \Biggl[{\beta+ \gamma\over z^\gamma} \int^z_0 f^\beta(t) t^{\gamma- 1}dt\Biggr]^{1/\beta}.$$ Let $f\prec g$ mean $f$ is subordinate to $g$ in $U$. The author investigates conditions on $g$, $\beta$, $\gamma$ so that $$z(f(z)/z)^\beta\prec z(g(z)/z)^\beta\Rightarrow z\Biggl({A_{\beta, \gamma}(f)(z)\over z}\Biggr)^\beta\prec z\Biggl({A_{\beta, \gamma}(g)(z)\over z}\Biggr)^\beta.$$ Similar results were earlier obtained by S. S. Miller, P. T. Mocanu and M. O. Reade.
[V.Karunakaran (Mandurai)]
MSC 2000:
*30C80 Maximum principle, etc. (one complex variable)
30C25 Covering theorems in conformal mapping theory
30C45 Special classes of univalent and multivalent functions

Cited in: Zbl 1182.30022 Zbl 1026.30021

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