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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 $U=\{z:$ $\vert z\vert <1\}$, 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 $$\vert f''(z)/f'(z)\vert \le 2\quad or\quad \vert zf''(z)/f'(z)+1\vert <2,\quad (z\in U).$$ Then f is starlike in U.
Reviewer: J.Waniurski

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