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On some classes of first-order differential subordinations. (English) Zbl 0575.30019
The paper contains several results of the following type: Theorem. Let h be analytic in $$U=\{z:$$ $$| z| <1\}$$, let $$\phi$$ be analytic in a domain D containing h(U) and suppose a) Re $$\phi$$ (h(z))$$>0$$, $$z\in U$$ and either b) h(z) is convex or b) $$H(z)=zh'(z)\phi (h(z))$$ is starlike w.r.t. the origin. If p is analytic in U with $$p(0)=h(0)$$, p(U)$$\subset D$$, and $p(z)+zp'(z)\phi (p(z))\prec h(z),$ then p(z)$$\prec h(z)$$. An application to univalent functions is given: Let f be analytic in U, $$f(0)=0$$ and either $| f''(z)/f'(z)| \leq 2\quad or\quad | zf''(z)/f'(z)+1| <2,\quad (z\in U).$ Then f is starlike in U.
Reviewer: J.Waniurski

##### MSC:
 30C45 Special classes of univalent and multivalent functions of one complex variable (starlike, convex, bounded rotation, etc.) 30C80 Maximum principle, Schwarz’s lemma, Lindelöf principle, analogues and generalizations; subordination
##### Keywords:
differential subordination; convex; starlike
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