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A class of univalent functions. (English) Zbl 0908.30009
Let $A$ denote the class of normalized analytic functions in the open unit disc $U$ of the complex plane. Let $S^*(\beta)$ denote functions in $A$ which are starlike of order $\beta$ and $S^*= S^*(0)$. In this paper, the author investigates conditions on $f\in A$ so that $f\in S^*(\beta)$. For example if $f\in A$ satisfies $$\Biggl| f'(z)\Biggl({z\over f(z)}\Biggr)^{1+ \mu}- 1\Biggr|< \lambda$$ with $0< \mu< 1$ and $0< \lambda\le(1- \mu)/\sqrt{(1- \mu)^2+ \mu^2}$ then it is shown that $f\in S^*$. The author also investigates conditions on $f$ which ensures a certain integral operator acting on $f$ belongs to $S^*(\beta)$. The results are too complicated to be reproduced here.

30C45Special classes of univalent and multivalent functions