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Some classes of univalent functions with negative coefficients. (English) Zbl 0671.30013

Let A denote the class of functions f of the form \(f(z)=z+\sum^{\infty}_{k=2}a_ kz^ k\) analytic in the unit disk \(\{z=z:\) \(| z| <1\}.\)
Let \(D^ nf(z)=z/(1-z)^{n+1}*f(z)\) (* denotes the Hadamard product), \(n=0,1,2,... \). Denote by \(K_ n\) the set of all functions \(f\in A\) satisfying \(Re(D^{n+1}f(z)/D^ nf(z))>1/2\), \(z\in E\). The class \(K_ n\) was introduced and studied by Ruscheweyh. Let T denote the subclass of A consisting of functions f of the form \(f(z)=z-\sum^{\infty}_{2}a_ kz^ k\) \((a_ k\geq 0)\) which are univalent in E and \(Q_ n[\alpha]\), \(0\leq \alpha <1\) denote the subclass of T whose members satisfy \(Re(D^ nf(z))'>\alpha\) for \(z\in E\). In this paper some properties of \(Q_ n[\alpha]\) are studied.
Reviewer: K.S.Padmanabhan

MSC:

30C45 Special classes of univalent and multivalent functions of one complex variable (starlike, convex, bounded rotation, etc.)
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