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 operatorΦβγ(f)(z)=(f(z1),(f(z1)/z1)β(f'(z1))γω),z ∈ Ωp,r preserves an almost starlike mapping of complex order λ on Qp,r={z=(z1,w)∈ C x X:|z1|p+||w||x＜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=2to1≤p≤2.

Jianfei WANG;Xiaofei ZHANG

School of Mathematical Sciences,Huaqiao University,Quanzhou 362021,China;School of Mathematics and Statistics,Pingdingshan University,Pingdingshan 467000,China

数学物理学报（英文版）

[1]Jianfei WANG;Xiaofei ZHANG-.THE SUBORDINATION PRINCIPLE AND ITS APPLICATION TO THE GENERALIZED ROPER-SUFFRIDGE EXTENSION OPERATOR)[J].数学物理学报（英文版）,2022(02):611-622

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