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Second order differential subordinations of holomorphic mappings on bounded convex balanced domains in \(\mathbb C^n\). (English) Zbl 1137.32009

Summary: In this paper, we obtain some second order differential subordinations of holomorphic mappings on a bounded convex balanced domain \(\Omega\) in \(\mathbb{C}^n\). These results imply some first order differential subordinations of holomorphic mappings on a bounded convex balanced domain \(\Omega\) in \(\mathbb{C}^n\). When \(\Omega\) is the unit disc in the complex plane \(\mathbb{C}\), these results are just the ones of Miller and Mocanu et al. about differential subordinations of analytic functions on the unit disc in the complex plane \(\mathbb{C}\).

MSC:

32H02 Holomorphic mappings, (holomorphic) embeddings and related questions in several complex variables
30C45 Special classes of univalent and multivalent functions of one complex variable (starlike, convex, bounded rotation, etc.)
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