Extensions of classical results in one complex variable to several complex variables
Carolyn R. Thomas · 1991
This dissertation extends certain results from geometric function theory of one complex variable to functions of several complex variables. Generalizations of bounds on the magnitudes of coefficients of convex mappings are found. From these inequalities, bounds are obtained on the norms of convex mappings from the ball in $C\sp n$ and from the classical domains, $R\sb I$, $R\sb{II}$, and $R\sb{III}$ and their partial derivatives. An interior variational method used by Marty in one variable is extended to several variables. Necessary conditions for extremal mappings are obtained in the form of values for certain combinations of coefficients. The technique is shown to apply to uniformly locally convex mappings from the ball, and it is seen that the union of these families is dense in the family of biholomorphic mappings.