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 $(1-\lambda)f+\lambda g$, where $\lambda\in(0,1)$, of biholomorphic mappings on the Euclidean unit ball $\mathbb{B}^n$ in the case of several complex variables. Starting from a result proved by S. Trimble \cite{trimble} and then extended by P.N. Chichra and R. Singh\cite[Theorem 2]{chichra-singh} which says that if $f$ is starlike such that $\text{Re}[f'(z)]>0$, then $(1-\lambda)z+\lambda f(z)$ 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 $\mathbb{B}^n$ and some particular starlike mappings on $\mathbb{B}^n$. In the second part of the paper, we define the class $\mathcal{L}_\lambda^*(\mathbb{B}^n)$ 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.