## 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

### Keywords:

convex and starlike functions; subordinate
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