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Classes of first-order differential superordinations. (English) Zbl 1010.30020
Let \(U\) denote the unit disc, let \(\varphi: C^2\to C\) be analytic in a domain and let \(p\) be an analytic function in \(U\) such that \(\varphi (p(z),zp'(z))\) is univalent in \(U\) and \(p\) satisfies \(h(z)\prec \varphi (p(z), zp'(z))\). ‘\(\prec\)’ denotes the usual differential subordination. In this paper the author considers the particular case \(\varphi(p(z), (z))=\alpha (p(z))+ \beta (p(z)) \gamma(zp'(z))\) and determines conditions on \(h,\alpha, \beta\) and \(\gamma\) so that the above subordination implies \(q(z)\prec p(z)\), where \(q\) is the largest function. The proof is based on a recent result of S. S. Miller and P. T. Mocanu on subordinants of differential superordinants and on Löwner chains. Some particular cases and examples are given.

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