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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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[1] Bulboacǎ, T., Integral operators that preserve the subordination, Bull. Korean Math. Soc., 34, no. 4, 627-636 (1997) · Zbl 0898.30021
[2] Bulboacǎ, T. — On a class of integral operators that preserve the subordination. Pure Math. and Appl. (To appear).; Bulboacǎ, T. — On a class of integral operators that preserve the subordination. Pure Math. and Appl. (To appear).
[3] Kaplan, W., Close to convex schlicht functions, Michig. Math. J., 2, 1, 169-185 (1952) · Zbl 0048.31101
[4] Miller, S. S.; Mocanu, P. T., Univalent solutions of Briot-Bouquet differential equations, J. Differential Equations, 67, 199-211 (1987)
[5] Miller, S. S.; Mocanu, P. T., Integral operators on certain classes of analytic functions, (Univalent Functions, Fractional Calculus and their Applications (1989), HalsteadPress, J. Wiley & Sons: HalsteadPress, J. Wiley & Sons New York), 153-166
[6] Miller, S. S.; Mocanu, P. T., Differential Subordinations, (Theory and Applications (1999), Marcel Dekker: Marcel Dekker New York) · Zbl 0954.34003
[7] Miller, S.S. and P.T. Mocanu — Subordinants of differential superordinations. (To appear).; Miller, S.S. and P.T. Mocanu — Subordinants of differential superordinations. (To appear).
[8] Mocanu, P. T.; Ripeanu, D.; Serb, I., The order of starlikeness of certain integral operators, Mathematica (Cluj), 23, 46, 225-230 (1981), no. 2 · Zbl 0502.30007
[9] Pommerenke, Ch, Univalent Functions (1975), Vanderhoeck and Ruprecht: Vanderhoeck and Ruprecht Göttingen
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