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Simple sufficient conditions for univalence. (English) Zbl 0992.30005

As usual, let \(A\) denote the class of functions \(f\) analytic in the unit disc \(U=\{z:|z|< 1 \}\) with \(f(0)=f'(0)-1=0\).
The author proves: a) if \(f \in A\) and \(f^{\prime\prime }(z) \leq 1\), \(z \in U\), then \(f\) is starlike; b) if \(f \in A\) and \(f^{\prime\prime }(z) \leq {{1}\over {2}}\), \(z \in U\), then \(f\) is convex.

MSC:

30C55 General theory of univalent and multivalent functions of one complex variable
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