## 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 $$0< \lambda\leq(1- \mu)/\sqrt{(1- \mu)^2+ \mu^2}$$ 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.

### MSC:

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

starlike
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