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Zbl 1019.30023
Bulboacă, Teodor
A class of superordination-preserving integral operators. (English)
[J] Indag. Math., New Ser. 13, No.3, 301-311 (2002). ISSN 0019-3577

Let $H(U)$ denote the class of analytic functions in the unit disk $U$ and let the integral operator $A_{\beta,\gamma} (f)(z):K\to H(U)$, $K \subset H(U)$ be defined by $$A_{\beta,\gamma} (f)(z)= \bigl(\beta+ \gamma)/ z^\gamma \int^z_0 f^\beta(t) t^{\gamma-1}dt\bigr]^{1/ \beta},\quad \beta,\gamma \in\bbfC.$$ If $f,F\in H(U)$ and $F$ is univalent in $U$ we say that $f$ is subordinate to $F$ or $F$ is superordinate to $f$, written $f(z) \prec F(z)$, if $f(0)= F(0)$ and $f(U)\subseteq F(U)$. In a recent paper S. S. Miller and P. T. Mocanu have determined conditions on $\varphi$ such that $$h(z) \prec\varphi \bigl(p(z),zp'(z), z^2p''(z); z\bigr) \text{ implies }q(z)\prec p(z),$$ for all functions $p$ that satisfy the above superordination. In this paper the author determines sufficient conditions on $g,\beta$ and $\gamma$ such that the following differential superordination holds: $$z\bigl[g(z)/z^\beta\prec z\bigl[ f(z)/z \bigr]^\beta \text{ implies }z\bigl[A_{\beta,\gamma} (g)(z)/z \bigr ]^\beta \prec z\bigl[A_{\beta, \gamma}(f)(z)/z \bigr]^\beta.$$ The function $z [A_{\beta,\gamma} (g)(z)/z\bigr]^\beta$ is the largest function so that the right-hand side holds, for all functions $f$ satisfying the left-hand side differential super-ordination. The particular case $g(z)=ze^{\lambda z}$ is considered.
[O.Fekete (Freiburg)]

MSC 2000:: *30C80 Maximum principle, etc. (one complex variable)
30C45 Special classes of univalent and multivalent functions

Keywords: differential superordination; integral operator

Cited in: Zbl 1039.30011

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