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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 $$A_{\beta,\gamma} (f)(z)= \bigl(\beta+ \gamma)/ z^\gamma \int^z_0 f^\beta(t) t^{\gamma-1}dt\bigr]^{1/ \beta},\quad \beta,\gamma \in\bbfC.$$ 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.

30C80Maximum principle; Schwarz’s lemma, Lindelöf principle, etc. (one complex variable)
30C45Special classes of univalent and multivalent functions
Full Text: DOI
[1] Bulboaca\check{}, T.: Integral operators that preserve the subordination. Bull. korean math. Soc. 34, No. no. 4, 627-636 (1997) · Zbl 0898.30021
[2] Bulboaca\check{}, 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, No. 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) · Zbl 0633.34005
[5] Miller, S. S.; Mocanu, P. T.: Integral operators on certain classes of analytic functions. Univalent functions, fractional calculus and their applications, 153-166 (1989)
[6] Miller, S. S.; Mocanu, P. T.: Differential subordinations. Theory and applications (1999) · Zbl 0439.30015
[7] 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, No. 46, 225-230 (1981) · Zbl 0502.30007
[9] Pommerenke, Ch: Univalent functions. (1975)