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Zbl 0575.30019
Miller, Sanford S.; Mocanu, Petru T.
On some classes of first-order differential subordinations.
(English)
[J] Mich. Math. J. 32, 185-195 (1985). ISSN 0026-2285

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.
[J.Waniurski]
MSC 2000:
*30C45 Special classes of univalent and multivalent functions
30C80 Maximum principle, etc. (one complex variable)

Keywords: differential subordination; convex; starlike

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