Characterizations of symmetrized polydisc

Let Γn, n ≥ 2, denote the symmetrized polydisc in ℂn, and Γ1 be the closed unit disc in ℂ. We provide some characterizations of elements in Γn. In particular, an element (s1,..., sn−1, p) ∈ ℂn is in Γn if and only if sj=βj+βn−j¯p\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${s_j} = {\beta _j} + \overline {{\beta _{n - j}}}p$$\end{document}, j = 1,..., n − 1, for some (β1,..., βn−1) ∈ Γn−1, and |p| ≤ 1.