Bounded support points for mappings with g-parametric representation in C2

In this paper we consider support points for the family S-g(0)(B-2) of mappings with g-parametric representation on the Euclidean unit ball B-2 in C-2, where g is a univalent function on the unit disc U in C, which satisfies certain natural assumptions. We shall use the shearing process recently introduced by Bracci, to prove the existence of bounded support points for the family S-g(0)(B-2). This result is in contrast to the one dimensional case, where all support points of the family S are unbounded. We also study the case of time-log M reachable families (R) over tilde (log) M(id(B2),M-g) generated by the Caratheodory family M-g, and obtain certain results and applications, which show a basic difference between the theory in the case of one complex variable and that in higher dimensions. Of particular interest is the case where g is a convex (univalent) function on U. Finally, various consequences and certain conjectures are also considered. (C) 2017 Elsevier Inc. All rights reserved.