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

Abstract

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 $\beta \in [0,1],\gamma \in [0,{1 \over r}]$ and β+γ ≤ 1, then the Roper-Suffridge extension operator

$${\Phi _{\beta,\gamma }}(f)(z) = \left( {f({z_1}),{{\left( {{{f({z_1})} \over {{z_1}}}} \right)}^\beta }{{({f^\prime }({z_1}))}^\gamma }w} \right),\,\,z \in {\Omega _{p,r}}$$

preserves an almost starlike mapping of complex order λ on ${\Omega _{p,r}} = \{ z = ({z_1},w) \in \mathbb{C} \times X:{\left| {{z_1}} \right|^p} + \left\| w \right\|_X^r < 1\}$, where 1 ≤ p ≤ 2, r ≥ 1 and X is a complex Banach space. Second, by applying the principle of subordination, we will prove that the Pfaltzgraff-Suffridge extension operator preserves an almost starlike mapping of complex order λ. Finally, we will obtain the lower bound of distortion theorems associated with the Roper-Suffridge extension operator. This subordination principle seems to be a new idea for dealing with the Loewner chain associated with the Roper-Suffridge extension operator, and enables us to generalize many known results from p = 2 to 1 ≤ p ≤ 2.