Jianfei WANG (王建飛)

School of Mathematical Sciences，Huaqiao Universitg，Quanzhou 362021，China E-mail: jfwang@hqu. edu.cn

Xiaofei ZHANG(張曉飛)+

School of Mathematics and Statistics,Pingdingshan University，Pingdingshan 4671000,China E-mail : zhxfei@mail.ustc.edu.cn

(iv) if f ∈S(U), then Φn,12(f) can be imbedded in a Loewner chain on Bn.

The convexity property (i) was obtained by Roper and Suffridge [1]. Graham and Kohr [2]provided a simplified proof in property (i) and also proved properties (ii) and (iii). In 2018,Wang and Liu [3] introduced a new idea to prove (i) and (ii). The property (iv) was proved by Graham et al. in [4]. By using the Roper-Suffridge extension operator, a lot of convex mappings and starlike mappings on Bncan be easily constructed; this is an important reason that people are interested in the Roper-Suffridge operator.

In general, let β ∈[0,1], γ ∈[0,1/2] and β +γ ≤1. Graham et al. [4] generalized the above Roper-Suffridge operator as

For more general domains in a complex n dimensional space,Liu[6]proved that starlikeness of order α ∈(0,1) is preserved by the Roper-Suffridge extension operator

The main purpose of this article is to introduce a subordination principle for dealing with the Roper-Suffridge extension operator and the Pfaltzgraff-Suffridge extension operator in complex Banach spaces; this seems to be a new and simple idea. Specially, we will first prove that both of the two extension operators preserve a subordination relation. As applications, we obtain that the Roper-Suffridge extension operator

2 Preliminaries

In this section some definitions will be given.

Starlike Mappings We say that a domain Ω ?X is starlike (with respect to 0 ∈Ω) if λw ∈Ω whenever w ∈Ω and λ ∈[0,1]. If f :Ω →X is a biholomorphic mapping, we say that f is a starlike mapping if f(Ω) is a starlike domain in X.

Subordination Suppose that X is a complex Banach space and that Ω ?X is a domain containing the origin. Let f : Ω →X and g : Ω →X be two holomorphic mappings. If there is a Schwarz mapping φ:Ω →Ω such that φ(0)=0 and f =g ?φ, then f is subordinate to g and is denoted by f ?g on Ω.

Loewner Chain Suppose that X is a complex Banach space and that Ω ?X is a domain containing the origin. A mapping f : Ω × [0,∞) →X is called a biholomorphic subordination chain if f(·,t) is biholomorphic on Ω, f(0,t) = 0 for t ≥0, and f(·,s) ?f(·,t)when 0 ≤s ≤t ＜∞. If Df(0,t)=etIXfor t ≥0, we say that f(z,t) is a Loewner chain.

Definition 2.1 ([7]) Suppose that X is a complex Banach space with the unit ball BX={z ∈X : ||z|| ＜1}. Let f : BX→X be a normalized locally biholomorphic mapping on BX,and let λ ∈C with Reλ ≤0. If

then f is called an almost starlike mapping of complex order λ.

3 Subordination Theorem and Applications

3.1 Five lemmas

In order to give the subordination theorem, we need five lemmas. The first lemma is the well-known Schwarz-Pick Lemma; see [23].

Lemma 3.1 If f :U →U is holomorphic, then

is a holomorphic mapping from Ωn,2,rinto Ωn,2,r.

Hence, it follows that F(z)∈Ωn,2,r. Accordingly, F ∈H(Ωn,2,r,Ωn,2,r). □

From the results of Zhao [7] and Zhang [24], we know that the following lemma holds:

Lemma 3.5 Let f be a normalized locally biholomorphic mapping on BX, and let λ ∈C with Reλ ≤0. Then f is an almost starlike mapping of complex order λ on BXif and only if

is a Loewner chain.

3.2 Subordination principle

so we get Ψn,r(f)=Ψn,r(g)?Ψn,r(v).

According to Lemma 3.4, this yields that Ψn,r(v) ∈H(Ωn,2,r,Ωn,2,r); namely, Ψn,r(f) ?Ψn,r(g) on Ωn,2,r. □

3.3 Applications of the subordination principle

From Lemma 3.5, we get that Φβ,γ(f)(z) is an almost starlike mapping of complex order λ on Ωp,r. □

4 Lower Bound for the Distortion Theorem of a Subclass of Almost Starlike Mappings of Complex Order λ

In one complex variable, there is a well known distortion theorem for normalized biholomorphic functions f on the unit disk U,

In the past thirty years,various distortion theorems have been established for convex mappings. However, there are few results for starlike mappings. In what follows, we give the lower bound of distortion theorem for almost starlike mappings of complex order λ on the domain Ωp,r; this generalizes some results in [28].