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

Keywords:

starlike
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