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A class of superordination-preserving integral operators. (English) Zbl 1019.30023
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 $A_{\beta,\gamma} (f)(z)= \bigl(\beta+ \gamma)/ z^\gamma \int^z_0 f^\beta(t) t^{\gamma-1}dt\bigr]^{1/ \beta},\quad \beta,\gamma \in\mathbb{C}.$ 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.

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