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Integral operators of certain univalent functions. (English) Zbl 0594.30012
Let $$\Delta$$ denote the class of functions $$f(z)=z+\sum^{\infty}_{2}a_ nz^ n$$ which are regular in the unit disc $$\Delta =\{z:$$ $$| z| <1\}$$. A function f in A is said to belong to the class $$R_ n(\alpha)$$ for some $$0\leq \alpha <1$$ if $Re z\frac{(D^ nf)'}{D^ nf}>\alpha,\quad z\in \Delta,\quad where\quad D^ nf(z)=z(n^{n-1}f(z))^{(n)}/n!.$ One result in this paper states that if $$f\in R_ n(\alpha)$$, then $J(f)=\frac{\gamma +1}{z^{\gamma}}\int^{z}_{0}t^{\gamma -1}f(t)dt\in R_ n(\alpha),\quad where\quad Re \gamma \geq -\alpha,\quad \gamma \neq -1.$ Also the converse of this result is obtained, namely, if $$F\in R_ n(\alpha)$$, $$n\geq 0$$, $$0\leq \alpha <1,$$
$F(z)=\frac{\gamma +1}{z^{\gamma}}\int^{z}_{0}t^{\gamma - 1}f(t)dt,\quad Re \gamma \geq -\alpha \quad and\quad 0\leq \beta <1,$ then f is an element of $$R_ n(\beta)$$ for $$| z| <r_ 0$$ where $$r_ 0$$ is the smallest positive root of some quadratic equation. An argument theorem is also shown.
Reviewer: H.S.Al-Amiri

##### MSC:
 30C45 Special classes of univalent and multivalent functions of one complex variable (starlike, convex, bounded rotation, etc.)
##### Keywords:
starlike function; convex function; convolution
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