On some classes of first-order differential subordinations. (English) Zbl 0575.30019

Miller, Sanford S.; Mocanu, Petru T.

Mich. Math. J. 32, 185-195 (1985).

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