An application of certain integral operators. (English) Zbl 0963.30001

Aouf, M. K.; Hossen, H. M.; Lashin, A. Y.

J. Math. Anal. Appl. 248, No. 2, 475-481 (2000).

Let \(A\) denote the class of functions of the form \[ f(z)= z+ \sum^\infty_{n=2} a_nz^n \] which are analytic in the open disk \(U= \{z\in \mathbb{C};|z|< 1\}\). Consider the following integral operators \(P^\alpha\) and \(Q^\alpha_\beta\) defined by the formulas \[ P^\alpha f(z)= {2^\alpha\over z\Gamma(\alpha)} \int^z_0 \Biggl(\log{1\over p}\Biggr)^{\alpha- 1} f(t) dt \] and \[ Q^\alpha_\beta f(z)= {\alpha+\beta\choose \beta}{\alpha\over z^\beta} \int^z_0 \Biggl(1-{t\over z}\Biggr)^{\alpha- 1} t^{\beta- 1}f(t) dt, \] where \(\alpha> 0\), \(\beta>-1\) and \(\Gamma\) is the familiar Euler’s function. In the paper the authors prove, under suitable assumptions, that \(|P^\alpha f(z)|< 1\) and \(|Q^\alpha_\beta f(z)|< 1\) for all \(z\in U\), and all \(f\in A\).
In the opinion of the reviewer the title of the work is not adequate to the contents of this paper.

Reviewer: Stanislaw Walczak (Łódź)

MSC:

30A10	Inequalities in the complex plane
30C99	Geometric function theory

Keywords:

integral operators

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