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

In this paper we consider support points for the family Sg0(B2) of mappings with g-parametric representation on the Euclidean unit ball B2 in C2, 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 Sg0(B2). 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˜log⁡M(idB2,Mg) generated by the Carathéodory family Mg, 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.