The coefficient bounds and the
Growth and Distortion Theorems for convex functions in one complex variable are
generalized to several variables. The holomorphic mappings studied are defined in the
unit ball or some other domain of one of the first three classical types. Each mapping
takes its domain onto a convex set in a one-to-one fashion. The coordinate
functions of each mapping have multivariable power series about the origin.
The best possible upper bounds are found for certain combinations of the
coefficients of these power series. In case the domain is the unit disk in the
plane, these bounds reduce to the classical coefficient estimates for convex
functions. As an application, these coefficient bounds are used to obtain
the best possible upper and lower bounds on the growth of the magnitude
of each mapping in terms of the magnitude of the independent variable.
Also, estimates on the magnitudes of various derivatives of each mapping are
found.
