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

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.


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