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 β∈ [0,1],γ∈[0,1/r] and β+γ≤1,then the Roper-Suffridge extension operatorΦβ,γ(f)(z)=(f(z1),(f(z1)/z1)~β(f’(z1))γw),z∈Ωp,rpreserves an almost starlike mapping of complex order λ on Ωp,r={z=(z1,w) ∈ C × X:|z1|~p+‖w‖~rX<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.