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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.)
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