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Zbl 1084.30009
Acu, Mugur; Owa, Shigeyoshi
On some subclasses of univalent functions. (English)
[J] JIPAM, J. Inequal. Pure Appl. Math. 6, No. 3, Paper No. 70, 6 p., electronic only (2005). ISSN 1443-5756/e

In 1999, S. Kanas and F. Rønning introduced the classes of functions starlike and convex, which are normalized with $f(w)=f^{\prime}(w)-1=0$ and $w$ is a fixed point in $U$. In this paper the authors continue the investigation of the univalent functions normalized with $f(w)=f^{\prime}(w)-1=0$, where $w$ is a fixed point in $U$. They consider the integral operator $L_{a}: A(w)\rightarrow A(w)$ defined by $$ f(z)=L_{a}F(z)=\frac{1+a}{(z-w)^{a}}\cdot\int\limits^{z}_{w}F(t)\cdot (t-w)^{a-1}dt \ , \ \ \ a\in{\bold R} \ , \ \ a\geq 0,$$ and they give a transformation theorem of the starlike functions normalized with $f(w)=f^{\prime}(w)-1=0$, where $w$ is a fixed point in $U$. Also the authors define the classes of close-to-convex and $\alpha$-convex functions normalized in the same way and give bounds for the coefficients of these functions and some inclusion results.
[Dorin Blezu (Sibiu)]

MSC 2000:: *30C45 Special classes of univalent and multivalent functions

Keywords: Close-to-convex functions; $\alpha$-convex functions; Briot-Bouquet differential subordination

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