On some convex combinations of biholomorphic mappings in several complex variables

Abstract

In this paper, our interest is devoted to study the convex combinations of the form (1 ? ?) f + ?1, where ? ? (0, 1), of biholomorphic mappings on the Euclidean unit ball Bn in the case of several complex variables. Starting from a result proved by S. Trimble [26] and then extended by P.N. Chichra and R. Singh [3, Theorem 2] which says that if f is starlike such that Re[f ?(z)] > 0, then (1??)z+?f (z) is also starlike, we are interested to extend this result to higher dimensions. In the first part of the paper, we construct starlike convex combinations using the identity mapping on Bn and some particular starlike mappings on Bn. In the second part of the paper, we define the class L* ?(Bn) and prove results involving convex combinations of normalized locally biholomorphic mappings and Loewner chains. Finally, we propose a conjecture that generalize the result proved by Chichra and Singh.