Solution of a Loewner chain equation in several complex variables

We find a solution to the Loewner chain equation in the case when the infinitesimal generator satisfies h ( 0 , t ) = 0 , D h ( 0 , t ) = A for any A ∈ L ( C n , C n ) with m ( A ) > 0 . We also study the related classes of spirallike mappings, mappings with parametric representation and asymptotically spirallike mappings.