Transactions of the American Mathematical Society
Not many convex mappings on the unit ball in Cn for n > 1 are known. We introduce two families of mappings, which we believe are actually identical, that both contain the convex mappings. These families which we have named the "Quasi-Convex Mappings, Types A and B" seem to be natural generalizations of the convex mappings in the plane. It is much easier to check whether a function is in one of these classes than to check for convexity. We show that the upper and lower bounds on the growth rate of such mappings is the same as for the convex mappings.
Roper, Kevin A. and Suffridge, T. J., "Convexity Properties of Holomorphic Mappings in Cn" (1999). Science and Mathematics Faculty Publications. 202.