Solutions for the generalized Loewner differential equation in several complex variables

摘要

We generalize a one-variable result of J. Becker to several complex variables. We determine the form of arbitrary solutions of the Loewner differential equation that is satisfied by univalent subordination chains of the form where has the property k ₊(A) < 2m(A). Here and . (The notion of parametric representation has a useful generalization under these conditions, so that one has a canonical solution of the Loewner differential equation.) In particular, we determine the form of the univalent solutions. The results are applied to subordination chains generated by spirallike mappings on the unit ball in . Finally, we determine the form of the solutions in the presence of certain coefficient bounds.