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Starlike functions with a fixed coefficient. (English) Zbl 0653.30004

The author investigates the radius of convexity for a certain class of functions: \[ S^*_{k,b}(A,B)=\{f(z)=z+a_{k+1}z^{k+1}+a_{2k+1}z^{2k+1}+...,\quad zf'(z)/f(z)\in P_{k,b}(A,B)\}, \] where \[ P_{k,b}(A,B)=\{p(z)=1+b(A-B)z^ k+p_{2k}z^{2k}+...,\quad p(z)\prec (1+Az^ k)/(1+Bz^ k)\}, \] k\(=1,2,3,...\), \(-1\leq B<A\leq 1\), \(0\leq b\leq 1\). One of the tools is a theorem of Pfaltzgraff and Pinchuk (Lemma 1 in the paper).
Reviewer: D.Aharonov

MSC:

30C45 Special classes of univalent and multivalent functions of one complex variable (starlike, convex, bounded rotation, etc.)
30C75 Extremal problems for conformal and quasiconformal mappings, other methods
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