On Certain Classes of Biharmonic Mappings Defined by Convolution

Abstract

We introduce a class of complex-valued biharmonic mappings, denoted by 𝐵𝐻⁰(𝜙_𝑘;𝜎,𝑎,𝑏), together with its subclass 𝑇𝐵𝐻⁰(𝜙_𝑘;𝜎,𝑎,𝑏), and then generalize the discussions in Ali et al. (2010) to the setting of 𝐵𝐻⁰(𝜙_𝑘;𝜎,𝑎,𝑏) and 𝑇𝐵𝐻⁰(𝜙_𝑘;𝜎,𝑎,𝑏) in a unified way.

1. Introduction

A four times continuously differentiable complex-valued function 𝐹=𝑢+𝑖𝑣 in a domain 𝐷⊂ℂ is biharmonic if Δ𝐹, the Laplacian of 𝐹, is harmonic in 𝐷. Note that Δ𝐹 is harmonic in 𝐷 if 𝐹 satisfies the biharmonic equation Δ(Δ𝐹)=0 in 𝐷, where Δ represents the Laplacian operator 𝜕Δ=4²𝜕𝑧𝜕𝑧𝜕∶=²𝜕𝑥²+𝜕²𝜕𝑦². (1.1)

It is known that, when 𝐷 is simply connected, a mapping 𝐹 is biharmonic if and only if 𝐹 has the following representation: 𝐹(𝑧)=²_𝑘=1|𝑧|^2(𝑘−1)𝐺_𝑘(𝑧), (1.2) where 𝐺_𝑘 are complex-valued harmonic mappings in 𝐷 for 𝑘∈{1,2} (cf. [1–6]). Also it is known that 𝐺_𝑘 can be expressed as the form 𝐺_𝑘=ℎ_𝑘+𝑔_𝑘 (1.3) for 𝑘∈{1,2}, where all ℎ_𝑘 and 𝑔_𝑘 are analytic in 𝐷 (cf. [7, 8]).

Biharmonic mappings arise in a lot of physical situations, particularly, in fluid dynamics and elasticity problems, and have many important applications in engineering and biology (cf. [9–11]). However, the investigation of biharmonic mappings in the context of geometric function theory is a recent one (cf. [1–6]).

In this paper, we consider the biharmonic mappings in 𝔻={𝑧∈ℂ∶|𝑧|<1}. Let 𝐵𝐻⁰(𝔻) denote the set of all biharmonic mappings 𝐹 in 𝔻 with the following form: 𝐹(𝑧)=²_𝑘=1|𝑧|^2(𝑘−1)ℎ_𝑘(𝑧)+𝑔_𝑘(𝑧=_2𝑘=1)|𝑧|^{2(𝑘−1)∞}_𝑗=1𝑎_𝑘,𝑗𝑧^𝑗+^∞_𝑗=1𝑏_𝑘,𝑗𝑧^𝑗, (1.4) with 𝑎_1,1=1, 𝑎_2,1=0, 𝑏_1,1=0, and 𝑏_2,1=0.

In [12], Qiao and Wang proved that for each 𝐹∈𝐵𝐻⁰(𝔻), if the coefficients of 𝐹 satisfy the following inequality: ^2∞_𝑘=1𝑗=12(𝑘−1)+𝑗𝑎_𝑘,𝑗+𝑏_𝑘,𝑗≤2, (1.5) then 𝐹 is sense preserving, univalent, and starlike in 𝔻 (see [12, Theorems 3.1 and 3.2]).

Let 𝑆_𝐻 denote the set of all univalent harmonic mappings 𝑓 in 𝔻, where 𝑓(𝑧)=ℎ(𝑧)+𝑔(𝑧)=𝑧+^∞_𝑗=2𝑎_𝑗𝑧^𝑗+^∞_𝑗=1𝑏_𝑗𝑧^𝑗, (1.6) with |𝑏₁|<1. In particular, we use 𝑆⁰_𝐻 to denote the set of all mappings in 𝑆_𝐻 with 𝑏₁=0. Obviously, 𝑆⁰_𝐻⊂𝐵𝐻⁰(𝔻).

In 1984, Clunie and Sheil-Small [7] discussed the class 𝑆_𝐻 and its geometric subclasses. Since then, there have been many related papers on 𝑆_𝐻 and its subclasses (see [13, 14] and the references therein). In 1999, Jahangiri [15] studied the class 𝑆^∗_𝐻(𝛼) consisting of all mappings 𝑓=ℎ+𝑔 such that ℎ and 𝑔 are of the form ℎ(𝑧)=𝑧−^∞_𝑗=2𝑎_𝑗𝑧^𝑗,𝑔(𝑧)=^∞_𝑗=1𝑏_𝑗𝑧^𝑗 (1.7) and satisfy the condition 𝜕𝜕𝜃arg𝑓𝑟𝑒^𝑖𝜃=Re𝑧ℎ′−𝑧𝑔′ℎ+𝑔>𝛼 (1.8) in 𝔻, where 0≤𝛼<1.

For two analytic functions 𝑓₁ and 𝑓₂, if 𝑓₁(𝑧)=^∞_𝑗=1𝑎_𝑗𝑧^𝑗,𝑓₂(𝑧)=^∞_𝑗=1𝐴_𝑗𝑧^𝑗, (1.9) then the convolution of 𝑓₁ and 𝑓₂ is defined by 𝑓₁∗𝑓₂(𝑧)=𝑓₁(𝑧)∗𝑓₂(𝑧)=^∞_𝑗=1𝑎_𝑗𝐴_𝑗𝑧^𝑗. (1.10)

By using the convolution, in [16], Ali et al. introduced the class 𝑆⁰_𝐻(𝜙,𝜎,𝛼) of harmonic mappings in the form of (1.6) such that 𝑧Reℎ∗𝜙^′(𝑧)−𝜎𝑧𝑔∗𝜙^′(𝑧)ℎ∗𝜙(𝑧)+𝜎𝑔∗𝜙(𝑧)>𝛼 (1.11) and the class 𝑆𝑃⁰_𝐻(𝜙,𝜎,𝛼) such that Re1+𝑒^𝑖𝛾𝑧ℎ∗𝜙^′(𝑧)−𝜎𝑧𝑔∗𝜙^′(𝑧)ℎ∗𝜙(𝑧)+𝜎𝑔∗𝜙(𝑧)−𝑒^𝑖𝛾>𝛼, (1.12) where 𝜎∈ℝ and 𝛼∈[0,1) are constants, 𝛾∈ℝ and 𝜙(𝑧)=𝑧+∑^∞_𝑛=2𝜙_𝑛𝑧^𝑛 is analytic in 𝔻.

Now we consider a class of biharmonic mappings, denoted by 𝐵𝐻⁰(𝜙_𝑘;𝜎,𝑎,𝑏), as follows: 𝐹∈𝐵𝐻⁰(𝔻) with the form (1.4) is said to be in 𝐵𝐻⁰(𝜙_𝑘;𝜎,𝑎,𝑏) if and only if 𝑎Φ(𝑧)ReΨ(𝑧)−𝑏>0, (1.13) where Φ(𝑧)=𝑧²_𝑘=1|𝑧|^2(𝑘−1)ℎ_𝑘∗𝜙_𝑘(𝑧)^′+𝜎²_𝑘=1|𝑧|^2(𝑘−1)𝑔_𝑘∗𝜙_𝑘(𝑧)^′,Ψ(𝑧)=𝑧^′2_𝑘=1|𝑧|^2(𝑘−1)ℎ_𝑘∗𝜙_𝑘(𝑧)+𝜎𝑔_𝑘∗𝜙_𝑘(𝑧), (1.14)𝜙_𝑘(𝑧)=𝑧+∑^∞_𝑗=2𝜙_𝑘,𝑗𝑧^𝑗 are analytic in 𝔻 for 𝑘∈{1,2}, 𝜎∈ℝ is a constant, 𝑎=𝑝+𝜌𝑒^𝑖𝛾, 𝑏=𝑞+𝜌𝑒^𝑖𝛾, 𝑝,𝑞,𝜌∈[0,+∞) are constants with 𝑎−𝑏>0, 𝛾∈ℝ, and 𝑧=𝑟𝑒^𝑖𝜃. Here and in what follows, “′” always stands for “𝜕/𝜕𝜃”.

Obviously, if 𝜙₂=0, 𝑎=1 and 𝑏=𝛼, then 𝐵𝐻⁰(𝜙_𝑘;𝜎,𝑎,𝑏) reduces to 𝑆⁰_𝐻(𝜙,𝜎,𝛼), and if 𝜙₂=0, 𝑎=1+𝑒^𝑖𝛾 and 𝑏=𝛼+𝑒^𝑖𝛾, then 𝐵𝐻⁰(𝜙_𝑘;𝜎,𝑎,𝑏) reduces to 𝑆𝑃⁰_𝐻(𝜙,𝜎,𝛼).

Further, we use 𝑇𝐵𝐻⁰(𝜙_𝑘;𝜎,𝑎,𝑏) to denote the class consisting of all mappings 𝐹 in 𝐵𝐻⁰(𝜙_𝑘;𝜎,𝑎,𝑏) with the form 𝐹(𝑧)=²_𝑘=1|𝑧|^2(𝑘−1)ℎ_𝑘(𝑧)+𝑔_𝑘(𝑧), (1.15) where ℎ_𝑘(𝑧)=𝑎_𝑘,1𝑧−^∞_𝑗=2𝑎_𝑘,𝑗𝑧^𝑗,𝑎_𝑘,𝑗≥0,𝑎_1,1=1,𝑎_2,1𝑔=0,_𝑘(𝑧)=𝜎^∞_𝑗=1𝑏_𝑘,𝑗𝑧^𝑗,𝑏_𝑘,𝑗≥0,𝑏_1,1=𝑏_2,1=0. (1.16)

The object of this paper is to generalize the discussions in [16] to the setting of 𝐵𝐻⁰(𝜙_𝑘;𝜎,𝑎,𝑏) and 𝑇𝐵𝐻⁰(𝜙_𝑘;𝜎,𝑎,𝑏) in a unified way. The organization of this paper is as follows. In Section 2, we get a convolution characterization for 𝐵𝐻⁰(𝜙_𝑘;𝜎,𝑎,𝑏). As a corollary, we derive a sufficient coefficient condition for mappings in 𝐵𝐻⁰(𝔻) to belong to 𝐵𝐻⁰(𝜙_𝑘;𝜎,𝑎,𝑏). The main results are Theorems 2.1 and 2.3. In Section 3, first, we get a coefficient characterization for 𝑇𝐵𝐻⁰(𝜙_𝑘;𝜎,𝑎,𝑏), and then find the extreme points of 𝑇𝐵𝐻⁰(𝜙_𝑘;𝜎,𝑎,𝑏). The corresponding results are Theorems 3.1 and 3.6.

2. A Convolution Characterization

We begin with a convolution characterization for 𝐵𝐻⁰(𝜙_𝑘;𝜎,𝑎,𝑏).

Theorem 2.1. Let 𝐹∈𝐵𝐻⁰(𝔻). Then 𝐹∈𝐵𝐻⁰(𝜙_𝑘;𝜎,𝑎,𝑏) if and only if ²_𝑘=1|𝑧|^2(𝑘−1)ℎ_𝑘∗𝜙_𝑘(𝑧)∗𝑧+((𝑎𝑥−𝑎+2𝑏)/(2𝑎−2𝑏))𝑧²(1−𝑧)²−𝜎²_𝑘=1|𝑧|^2(𝑘−1)𝑔_𝑘∗𝜙_𝑘(𝑧)∗((𝑎𝑥+𝑏)/(𝑎−𝑏))𝑧−((𝑎𝑥−𝑎+2𝑏)/(2𝑎−2𝑏))𝑧²(1−𝑧)²≠0, (2.1) for all 𝑧∈𝔻⧵{0} and all 𝑥∈ℂ with |𝑥|=1.

Proof. By definition, a necessary and sufficient condition for a mapping 𝐹 in 𝐵𝐻⁰(𝔻) to be in 𝐵𝐻⁰(𝜙_𝑘;𝜎,𝑎,𝑏) is given by (1.13). Let 𝐺(𝑧)=1𝑎Φ(𝑧)𝑎−𝑏Ψ(𝑧)−𝑏. (2.2) Then 𝐺(0)=1, and so the condition (1.13) is equivalent to 𝐺(𝑧)≠𝑥−1,𝑥+1 (2.3) for all 𝑧∈𝔻⧵{0} and all 𝑥∈ℂ with |𝑥|=1 and 𝑥≠−1. Obviously, (2.3) holds if and only if 𝑎(𝑥+1)Φ(𝑧)−𝑏(𝑥+1)Ψ(𝑧)−(𝑎−𝑏)(𝑥−1)Ψ(𝑧)≠0. (2.4)
Straightforward computations show that 𝑎()Φ(𝑧)()Ψ(𝑧)−()Ψ(𝑧)()𝑥+1−𝑏𝑥+1𝑎−𝑏)(𝑥−1=𝑎𝑥+1𝑧′²_𝑘=1|𝑧|^2(𝑘−1)𝑧+^∞_𝑗=2𝑗𝑎_𝑘,𝑗𝜙_𝑘,𝑗𝑧^𝑗−𝜎^∞_𝑗=2𝑗𝑏_𝑘,𝑗𝜙_𝑘,𝑗𝑧^𝑗−(𝑎𝑥−𝑎+2𝑏)𝑧′²_𝑘=1|𝑧|^2(𝑘−1)𝑧+^∞_𝑗=2𝑎_𝑘,𝑗𝜙_𝑘,𝑗𝑧^𝑗+𝜎^∞_𝑗=2𝑏_𝑘,𝑗𝜙_𝑘,𝑗𝑧^𝑗=𝑧^′2_𝑘=1|𝑧|^2(𝑘−1)ℎ_𝑘∗𝜙_𝑘𝑧)∗2()()𝑧(𝑎−𝑏𝑧+𝑎𝑥−𝑎+2𝑏²(1−𝑧)²−𝜎𝑧′²_𝑘=1|𝑧|^2(𝑘−1)𝑔_𝑘∗𝜙_𝑘𝑧)∗2()(𝑎𝑥+𝑏𝑧−(𝑎𝑥−𝑎+2𝑏)𝑧²(1−𝑧)², (2.5) from which we see that (2.3) is true if and only if so is (2.1). The proof is complete.

Remark 2.2. If ℎ₂=𝑔₂=0,𝑎=1 and 𝑏=𝛼, then Theorem 2.1 coincides with Theorem 2.1 in [16], and if ℎ₂=𝑔₂=0, 𝑎=1+𝑒^𝑖𝛾, and 𝑏=𝛼+𝑒^𝑖𝛾, then Theorem 2.1 coincides with Theorem 2.3 in [16].

As an application of Theorem 2.1, we derive a sufficient condition for mappings in 𝐵𝐻⁰(𝔻) to be in 𝐵𝐻⁰(𝜙_𝑘;𝜎,𝑎,𝑏) in terms of their coefficients.

Theorem 2.3. Let 𝐹∈𝐵𝐻⁰(𝔻). Then 𝐹∈𝐵𝐻⁰(𝜙_𝑘;𝜎,𝑎,𝑏) if ^2∞_𝑘=1𝑗=2𝑗‖𝑎‖_max−‖𝑏‖_max𝜙𝑎−𝑏^𝑘,𝑗𝑎_𝑘,𝑗+|𝜎|^2∞_𝑘=1𝑗=2𝑗‖𝑎‖_max+‖𝑏‖_max𝜙𝑎−𝑏^𝑘,𝑗𝑏_𝑘,𝑗≤1, (2.6) here and in the following, ‖𝑧‖_max=max_𝛾∈𝑅{|𝑥+𝑦𝑒^𝑖𝛾|}=𝑥+𝑦, where 𝑧=𝑥+𝑦𝑒^𝑖𝛾, 𝑥 and 𝑦∈[0,+∞) are constants.

Proof. For 𝐹 given by (1.4), we see that 𝐿(𝑧)≜²_𝑘=1|𝑧|^2(𝑘−1)ℎ_𝑘∗𝜙_𝑘(𝑧)∗𝑧+((𝑎𝑥−𝑎+2𝑏)/(2𝑎−2𝑏))𝑧²(1−𝑧)²−𝜎²_𝑘=1|𝑧|^2(𝑘−1)𝑔_𝑘∗𝜙_𝑘(𝑧)∗((𝑎𝑥+𝑏)/(𝑎−𝑏))𝑧−((𝑎𝑥−𝑎+2𝑏)/(2𝑎−2𝑏))𝑧²(1−𝑧)²=𝑧+²_𝑘=1|𝑧|^{∞2(𝑘−1)}_𝑗=2𝑗+𝑗−1𝑎𝑥−𝑎+2𝑏𝜙2𝑎−2𝑏^𝑘,𝑗𝑎_𝑘,𝑗𝑧^𝑗−𝜎²_𝑘=1|𝑧|^{∞2(𝑘−1)}_𝑗=2𝑗𝑎𝑥+𝑏−𝑎−𝑏𝑗−1𝑎𝑥−𝑎+2𝑏2𝑎−2𝑏𝜙_𝑘,𝑗𝑏_𝑘,𝑗𝑧^𝑗. (2.7) If 𝐹 is the identity, obviously, 𝐿(𝑧)=|𝑧|. If 𝐹 is not the identity, then 𝐿(𝑧)>|𝑧|1−^2∞_𝑘=1𝑗=2𝑗‖𝑎‖_max−‖𝑏‖_max𝜙𝑎−𝑏^𝑘,𝑗𝑎_𝑘,𝑗−|𝜎|^2∞_𝑘=1𝑗=2𝑗‖𝑎‖_max+‖𝑏‖_max𝜙𝑎−𝑏^𝑘,𝑗𝑏_𝑘,𝑗. (2.8)Hence the assumption implies that 𝐿(𝑧)>0 for all 𝑧∈𝔻⧵{0} and all 𝑥∈ℂ with |𝑥|=1. It follows from Theorem 2.1 that 𝐹∈𝐵𝐻⁰(𝜙_𝑘;𝜎,𝑎,𝑏).

Remark 2.4. If ℎ₂=𝑔₂=0,𝑎=1 and 𝑏=𝛼, then Theorem 2.3 coincides with Theorem 2.2 in [16], and if ℎ₂=𝑔₂=0,𝑎=1+𝑒^𝑖𝛾 and 𝑏=𝛼+𝑒^𝑖𝛾, then Theorem 2.3 coincides with Theorem 2.4 in [16].

3. A Coefficient Characterization and Extreme Points

We start with a coefficient characterization for 𝑇𝐵𝐻⁰(𝜙_𝑘;𝜎,𝑎,𝑏).

Theorem 3.1. Let 𝜙_𝑘(𝑧)=𝑧+∑^∞_𝑗=2𝜙_𝑘,𝑗𝑧^𝑗 with 𝜙_𝑘,𝑗≥0, and let 𝐹 be of the form (1.15). Then 𝐹∈𝑇𝐵𝐻⁰(𝜙_𝑘;𝜎,𝑎,𝑏) if and only if ^2∞_𝑘=1𝑗=2𝑗‖𝑎‖_max−‖𝑏‖_max𝜙𝑎−𝑏^𝑘,𝑗𝑎_𝑘,𝑗+𝜎^22∞_𝑘=1𝑗=2𝑗‖𝑎‖_max+‖𝑏‖_max𝜙𝑎−𝑏^𝑘,𝑗𝑏_𝑘,𝑗≤1. (3.1)

Proof. By similar arguments as in the proof of Theorem 2.3, we see that it suffices to prove the “only if” part. For 𝐹∈𝑇𝐵𝐻⁰(𝜙_𝑘;𝜎,𝑎,𝑏), obviously, (1.13) is equivalent to 𝑃(𝑧)(𝑧)Re−𝑄𝑧−∑²_𝑘=1|𝑧|^2(𝑘−1)∑^∞_𝑗=2𝑎_𝑘,𝑗𝜙_𝑘,𝑗𝑧^𝑗−𝜎²∑^∞_𝑗=2𝑏_𝑘,𝑗𝜙_𝑘,𝑗𝑧^𝑗>0 (3.2) in 𝔻, where 𝑃(𝑧)=(𝑎−𝑏)𝑧−²_𝑘=1|𝑧|^{∞2(𝑘−1)}_𝑗=2𝑎𝑗−𝑏𝑎_𝑘,𝑗𝜙_𝑘,𝑗𝑧^𝑗,𝑄(𝑧)=𝜎²²_𝑘=1|𝑧|^{∞2(𝑘−1)}_𝑗=2𝑎𝑗+𝑏𝑏_𝑘,𝑗𝜙_𝑘,𝑗𝑧^𝑗. (3.3) Letting 𝑧→1⁻ through real values leads to the desired inequality. So the proof is complete.

Remark 3.2. If ℎ₂=𝑔₂=0, 𝑎=1, and 𝑏=𝛼, then Theorem 3.1 coincides with Theorem 3.1 in [16].

It follows from Theorem 3.1 that we have the following.

Corollary 3.3. Let 𝜙_𝑘(𝑧)=𝑧+∑^∞_𝑗=2𝜙_𝑘,𝑗𝑧^𝑗 with 𝜙_𝑘,𝑗≥𝜙_1,2>0(𝑘∈{1,2},𝑗≥2) and |𝜎|≥(2‖𝑎‖_max−‖𝑏‖_max)/(2‖𝑎‖_max+‖𝑏‖_max). If 𝐹∈𝑇𝐵𝐻⁰(𝜙_𝑘;𝜎,𝑎,𝑏), then for |𝑧|=𝑟<1, one has 𝑟−𝑎−𝑏2‖𝑎‖_max−‖𝑏‖_max𝜙_1,2𝑟²≤|𝐹(𝑧)|≤𝑟+𝑎−𝑏2‖𝑎‖_max−‖𝑏‖_max𝜙_1,2𝑟². (3.4) The result is sharp with equality for mappings 𝐹(𝑧)=𝑧−𝑎−𝑏2‖𝑎‖_max−‖𝑏‖_max𝜙_1,2𝑧². (3.5)

Theorem 3.1 and Corollary 3.3 imply the following

Corollary 3.4. Under the hypotheses of Corollary 3.3, one has that 𝑇𝐵𝐻⁰(𝜙_𝑘;𝜎,𝑎,𝑏) is closed under the convex combination.

Definition 3.5. Let 𝑋 be a topological vector space over the field of complex numbers, and let 𝐸 be a subset of 𝑋. A point 𝑥∈𝐸 is called an extreme point of 𝐸 if it has no representation of the form 𝑥=𝑡𝑦+(1−𝑡)𝑧 (0<𝑡<1) as a proper convex combination of two distinct points 𝑦 and 𝑧 in 𝐸 (cf. [17]).

We now determine the extreme points of 𝑇𝐵𝐻⁰(𝜙_𝑘;𝜎,𝑎,𝑏).

Theorem 3.6. Let (1)ℎ₁₁(𝑧)=𝑧, (2)ℎ₂₁(𝑧)=𝑔₁₁(𝑧)=𝑔₂₁(𝑧)=0, (3)ℎ_𝑘𝑗(𝑧)=𝑧−|𝑧|^2(𝑘−1)((𝑎−𝑏)/(𝑗‖𝑎‖_max−‖𝑏‖_max)𝜙_𝑘,𝑗)𝑧^𝑗 for 𝑘∈{1,2} and all 𝑗≥2, (4)𝑔_𝑘𝑗(𝑧)=𝑧+|𝑧|^2(𝑘−1)((𝑎−𝑏)/𝜎(𝑗‖𝑎‖_max+‖𝑏‖_max)𝜙_𝑘,𝑗)𝑧^𝑗 for 𝑘∈{1,2} and all 𝑗≥2. Under the hypotheses of Corollary 3.3, one has that 𝐹∈𝑇𝐵𝐻⁰(𝜙_𝑘;𝜎,𝑎,𝑏) if and only if it can be expressed as 𝐹(𝑧)=^2∞_𝑘=1𝑗=1𝑥_𝑘𝑗ℎ_𝑘𝑗(𝑧)+𝑦_𝑘𝑗𝑔_𝑘𝑗(𝑧), (3.6) where 𝑥₂₁=𝑦₁₁=𝑦₂₁=0, all other 𝑥_𝑘𝑗 and 𝑦_𝑘𝑗 are nonnegative, and ∑²_𝑘=1∑^∞_𝑗=1(𝑥_𝑘𝑗+𝑦_𝑘𝑗)=1.
In particular, the extreme points of 𝑇𝐵𝐻⁰(𝜙_𝑘;𝜎,𝑎,𝑏) are all mappings ℎ_𝑘𝑗 and 𝑔_𝑘𝑗 listed in (1), (3), and (4) above.

Proof. It follows from the assumptions that 𝐹(𝑧)=^2∞_𝑘=1𝑗=1𝑥_𝑘𝑗ℎ_𝑘𝑗(𝑧)+𝑦_𝑘𝑗𝑔_𝑘𝑗(𝑧)=𝑧−^2∞_𝑘=1𝑗=2|𝑧|^2(𝑘−1)𝑎−𝑏𝑗‖𝑎‖_max−‖𝑏‖_max𝜙_𝑘,𝑗𝑥_𝑘𝑗𝑧^𝑗+𝜎^2∞_𝑘=1𝑗=2|𝑧|^2(𝑘−1)𝑎−𝑏𝜎²𝑗‖𝑎‖_max+‖𝑏‖_max𝜙_𝑘,𝑗𝑦_𝑘𝑗𝑧^𝑗, (3.7) whence ^2∞_𝑘=1𝑗=2𝑗‖𝑎‖_max−‖𝑏‖_max𝜙𝑎−𝑏^𝑘,𝑗⋅𝑎−𝑏𝑗‖𝑎‖_max−‖𝑏‖_max𝜙_𝑘,𝑗𝑥_𝑘𝑗+𝜎^22∞_𝑘=1𝑗=2𝑗‖𝑎‖_max+‖𝑏‖_max𝜙𝑎−𝑏^𝑘,𝑗⋅𝑎−𝑏𝜎²𝑗‖𝑎‖_max+‖𝑏‖_max𝜙_𝑘,𝑗𝑦_𝑘𝑗=^2∞_𝑘=1𝑗=2𝑥_𝑘𝑗+^2∞_𝑘=1𝑗=2𝑦_𝑘𝑗≤1, (3.8) and so Theorem 3.1 implies that 𝐹∈𝑇𝐵𝐻⁰(𝜙_𝑘;𝜎,𝑎,𝑏).
Conversely, assume 𝐹∈𝑇𝐵𝐻⁰(𝜙_𝑘;𝜎,𝑎,𝑏), and let 𝑥₂₁=𝑦₁₁=𝑦₂₁=0,𝑥₁₁=1−^2∞_𝑘=1𝑗=2𝑥_𝑘𝑗−^2∞_𝑘=1𝑗=2𝑦_𝑘𝑗,𝑥_𝑘𝑗=𝑗‖𝑎‖_max−‖𝑏‖_max𝜙_𝑘,𝑗𝑎_𝑘,𝑗,𝑦𝑎−𝑏_𝑘𝑗=𝜎²𝑗‖𝑎‖_max+‖𝑏‖_max𝜙_𝑘,𝑗𝑏_𝑘,𝑗,𝑎−𝑏 (3.9) for 𝑘∈{1,2} and all 𝑗≥2. Then 𝐹(𝑧)=𝑧−²_𝑘=1|𝑧|^{∞2(𝑘−1)}_𝑗=2𝑎_𝑘,𝑗𝑧^𝑗+𝜎²_𝑘=1|𝑧|^{∞2(𝑘−1)}_𝑗=2𝑏_𝑘,𝑗𝑧^𝑗. (3.10) The proof of the theorem is complete.

Remark 3.7. If ℎ₂=𝑔₂=0, 𝑎=1 and 𝑏=𝛼, then Theorem 3.6 coincides with Theorem 3.2 in [16].

Acknowledgment

The research was partly supported by NSFs of China (No. 11071063).