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A class of superordination-preserving integral operators. (English) Zbl 1019.30023

Let $H\left(U\right)$ denote the class of analytic functions in the unit disk $U$ and let the integral operator ${A}_{\beta ,\gamma }\left(f\right)\left(z\right):K\to H\left(U\right)$, $K\subset H\left(U\right)$ be defined by

${A}_{\beta ,\gamma }\left(f\right)\left(z\right)={\left(\beta +\gamma \right)/{z}^{\gamma }{\int }_{0}^{z}{f}^{\beta }\left(t\right){t}^{\gamma -1}dt\right]}^{1/\beta },\phantom{\rule{1.em}{0ex}}\beta ,\gamma \in ℂ·$

If $f,F\in H\left(U\right)$ and $F$ is univalent in $U$ we say that $f$ is subordinate to $F$ or $F$ is superordinate to $f$, written $f\left(z\right)\prec F\left(z\right)$, if $f\left(0\right)=F\left(0\right)$ and $f\left(U\right)\subseteq F\left(U\right)$. In a recent paper S. S. Miller and P. T. Mocanu have determined conditions on $\phi$ such that

$h\left(z\right)\prec \phi \left(p\left(z\right),z{p}^{\text{'}}\left(z\right),{z}^{2}{p}^{\text{'}\text{'}}\left(z\right);z\right)\phantom{\rule{4.pt}{0ex}}\text{implies}\phantom{\rule{4.pt}{0ex}}q\left(z\right)\prec p\left(z\right),$

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:

$z\left[g\left(z\right)/{z}^{\beta }\prec z{\left[f\left(z\right)/z\right]}^{\beta }\phantom{\rule{4.pt}{0ex}}\text{implies}\phantom{\rule{4.pt}{0ex}}z{\left[{A}_{\beta ,\gamma }\left(g\right)\left(z\right)/z\right]}^{\beta }\prec z{\left[{A}_{\beta ,\gamma }\left(f\right)\left(z\right)/z\right]}^{\beta }·$

The function $z\left[{A}_{\beta ,\gamma }\left(g\right)\left(z\right)/z{\right]}^{\beta }$ 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 $g\left(z\right)=z{e}^{\lambda z}$ is considered.

##### MSC:
 30C80 Maximum principle; Schwarz’s lemma, Lindelöf principle, etc. (one complex variable) 30C45 Special classes of univalent and multivalent functions
##### Keywords:
differential superordination; integral operator