The Subordination Principle and Its Application to the Generalized Roper-Suffridge Extension Operator

This note is devoted to applying the principle of subordination in order to explore the Roper-Suffridge extension operator and the Pfaltzgraff-Suffridge extension operator with special analytic properties. First, we prove that both the Roper-Suffridge extension operator and the Pfaltzgraff-Suffridge extension operator preserve subordination. As applications, we obtain that if \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\beta \in [0,1],\gamma \in [0,{1 \over r}]$$\end{document} and β+γ ≤ 1, then the Roper-Suffridge extension operator \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\Phi _{\beta,\gamma }}(f)(z) = \left( {f({z_1}),{{\left( {{{f({z_1})} \over {{z_1}}}} \right)}^\beta }{{({f^\prime }({z_1}))}^\gamma }w} \right),\,\,z \in {\Omega _{p,r}}$$\end{document}