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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.)
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