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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.

##### MSC:
 30A10 Inequalities in the complex plane 30C99 Geometric function theory
##### Keywords:
integral operators
Full Text:
##### References:
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