Convex Mappings Associated with the Roper-Suffridge Extension Operator

Let λG(z)∣dz∣ be the hyperbolic metric on a simply connected proper domain G ⊂ ℂ containing the origin, and let ∥ · ∥j be the Banach norms of ℂnj\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb{C}^{{n_j}}}$$\end{document} for j = 1, 2, ⋯, k. This note is to prove that if f is a normalized biholomorphic convex function on G, then