An application of certain integral operators (Q1583954)

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.