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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.)