A study on the univalent mappings in several complex variables

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1. We considered a theory of partial differential subordinations on the unit ball in infinite dimensional complex Banach spaces and gave some applications.2. We investigated the growth of normalized spirallike mappings on the Euclidean unit ball in C^n.3. We gave a characterization of normalized spirallike mappings on bounded balanced pseudo-convex domains using subordination chains and showed the growth theorem. Moreover, we obtained a quasiconformal extension of a quasiconformal strongly spirallike mapping.4. We studied linear invariant families on the unit polydisc.5. We showed a sharp growth theorem for normalized starlike mappings of order α on the unit ball in complex Banach spaces.6. We gave an analytic sufficient condition for locally diffeomorphism on the unit ball with respect to an arbitrary norm on C^n to be univalent.7. We gave an analytic characterization for locally biholomorphic mappings on the unit ball in infinite dimensional complex Banach spaces to be biholomorphic. We also give an analytic characterization for locally biholomorphic mappings on the unit ball in complex Hilbert spaces to be a convex mapping.