On Some Subclasses of Harmonic Functions Defined by Fractional Calculus

Abstract

The purpose of this paper is to study subclasses of normalized harmonic functions with positive real part using fractional derivative. Sharp estimates for coefficients and distortion theorems are given.

1. Introduction

A continuous function 𝑓=𝑢+𝑖𝑣 is a complex-valued harmonic function in a complex domain 𝐶 if both 𝑢 and 𝑣 are real harmonic in 𝐶. In any simply connected domain 𝐷⊆𝐶, we can write 𝑓=ℎ+𝑔, where ℎ and 𝑔 are analytic in 𝐷. We call ℎ the analytic part and 𝑔 the coanalytic part of 𝑓. A necessary and sufficient condition for 𝑓 to be locally univalent and orientation-preserving in 𝐷 is that |𝑔^′(𝑧)|<|ℎ^′(𝑧)| in 𝐷, see [1].

Denote by 𝐻 the class of functions 𝑓=ℎ+𝑔 which are harmonic univalent and orientation-preserving in the open unit disk 𝑈={𝑧∶|𝑧|<1} so that 𝑓=ℎ+𝑔 is normalized by 𝑓(0)=ℎ(0)=𝑓_𝑧(0)−1=0. Therefore, for 𝑓=ℎ+𝑔∈𝐻, we can express ℎ and 𝑔 by the following power series expansion:ℎ(𝑧)=𝑧+^∞_𝑛=2𝑎_𝑛𝑧^𝑛,𝑔(𝑧)=^∞_𝑛=1𝑏_𝑛𝑧^𝑛,|𝑏₁|<1. (1.1)Observe that 𝐻 reduces 𝑆, the class of normalized univalent analytic functions, if the coanalytic part of 𝑓 is zero.

For 𝑓=ℎ+𝑔 given by (1.1) and 𝑛>−1, Murugusundaramoorthy [2] defined the Ruscheweyh derivative of the harmonic function 𝑓=ℎ+𝑔 in 𝐻 by𝐷^𝑛𝑓(𝑧)=𝐷^𝑛ℎ(𝑧)+𝐷^𝑛𝑔(𝑧), (1.2)where the Ruscheweh derivative of a power series 𝑓(𝑧)=𝑧+∑^𝑛_𝑛=2𝑎_𝑛𝑧^𝑛 is given by 𝐷^𝑛𝑧𝑓(𝑧)=(1−𝑧)^𝑛+1∗𝑓. (1.3) The operator ∗ stands for the Hadamard product or convolution of two power series𝑓(𝑧)=^∞_𝑛=1𝑎_𝑛𝑧^𝑛,𝑔(𝑧)=^∞_𝑛=1𝑏_𝑛𝑧^𝑛 (1.4)defined by(𝑓∗𝑔)(𝑧)=^∞_𝑛=1𝑎_𝑛𝑏_𝑛𝑧^𝑛. (1.5)In [3], Owa introduced the following definition.

Definition 1.1. Let the function 𝑓(𝑧) be analytic in a simply connected domain of the 𝑧-plane containing the origin and let 0≤𝜆<1. The fractional derivative of 𝑓 of order 𝜆 is defined by𝐷^𝜆_𝑧1𝑓(𝑧)∶=𝑑Γ(1−𝜆)𝑑𝑧¹₀𝑓(𝜁)(𝑧−𝜁)^𝜆𝑑𝜁(0≤𝜆<1), (1.6)where the multiplicity of (𝑧−𝜁)^−𝜆 is removed by requiring log(𝑧−𝜁) to be real when 𝑧−𝜁>0.

In [4], Owa gave the relation between the fractional derivative and Ruscheweyh operator for the function 𝑓(𝑧)=𝑧+∑^∞_𝑛=2𝑎_𝑛𝑧^𝑛 as𝐷^𝜆1𝑓(𝑧)∶=𝐷Γ(1+𝜆)^𝜆𝑧[𝑧^𝜆−1𝐷𝑓(𝑧)],0<𝜆<1,₀𝑓(𝑧)=lim_𝜆→∞𝐷^𝜆𝐷𝑓(𝑧),₁𝑓(𝑧)=lim_𝜆→1𝐷^𝜆𝑓(𝑧). (1.7)

Using (1.2) and the relation between the fractional derivative and Ruscheweyh operator, we define the fractional derivative of order 𝜆, 0≤𝜆<1, for the harmonic function 𝑓=ℎ+𝑔 as𝐷^𝜆_𝑧[𝑧^𝜆−1𝑓(𝑧)]=𝐷^𝜆_𝑧[𝑧^𝜆−1ℎ(𝑧)]+𝐷^𝜆_𝑧[𝑧^𝜆−1𝐷𝑔(𝑧)],0<𝜆<1,_0𝑧𝑓(𝑧)=lim_𝜆→0𝐷^𝜆_𝑧𝐷𝑓(𝑧),_1𝑧𝑓(𝑧)=lim_𝜆→1𝐷^𝜆_𝑧𝑓(𝑧). (1.8)Since 𝐷^𝜆𝑓=𝐷^𝜆ℎ+𝐷^𝜆𝑔, it was proved in [1] that the harmonic function 𝐷^𝜆𝑓 is starlike of order 1/2 if and only if the analytic function 𝐷^𝜆ℎ−𝐷^𝜆𝑔 is starlike of order 1/2, and it was shown in [4, Theorem 3] that 𝐷^𝜆ℎ−𝐷^𝜆𝑔 is starlike of order 1/2 if and only if Re{𝐷_𝑧^𝜆+1[𝑧^𝜆(ℎ−𝑔)]/𝐷^𝜆_𝑧[𝑧^𝜆−1ℎ−𝑔)]}>(1+𝜆)/2 for 0<𝜆<1. Since Re{𝐷_𝑧^𝜆+1[𝑧^𝜆(ℎ−𝑔)]/𝐷^𝜆_𝑧[𝑧^𝜆−1(ℎ−𝑔)]}=Re(Γ(2+𝜆)𝐷^𝜆+1(ℎ−𝑔)/Γ(1+𝜆)𝐷^𝜆(ℎ−𝑔)), then 𝐷^𝜆ℎ−𝐷^𝜆𝑔 is starlike of order (1+𝜆)Γ(1+𝜆)/2Γ(2+𝜆), hence 𝐷^𝜆𝑓=𝐷^𝜆ℎ+𝐷^𝜆𝑔 is starlike of order (1+𝜆)Γ(1+𝜆)/2Γ(2+𝜆). This meansRe𝐷𝐷^𝜆𝑓𝐷^𝜆𝑓>(1+𝜆)Γ(1+𝜆)𝐷2Γ(2+𝜆)⇒Re^𝑧𝜆+1[𝑧^𝜆𝑓]𝐷^𝜆_𝑧[𝑧^𝜆−1>𝑓](1+𝜆)2. (1.9)

Recently, Owa and Srivastava [5] studied the linear Ω^𝜆 defined by operatorΩ^𝜆𝑓(𝑧)∶=Γ(2−𝜆)𝑧^𝜆𝐷^𝜆_𝑧𝑓(𝑧)(0≤𝜆<1), (1.10)where 𝑓 is normalized and analytic function on 𝑈.

It is easily seen thatΩ⁰𝑓=𝑓,Ω¹𝑓=𝑧𝑓^′. (1.11)Analogously, we studied the linear operator Ω^𝜆 defined on the harmonic function 𝑓=ℎ+𝑔 byΩ^𝜆𝑓(𝑧)=Ω^𝜆ℎ(𝑧)+Ω^𝜆𝑔(𝑧), (1.12)whereΩ^𝜆ℎ(𝑧)∶=Γ(2−𝜆)𝑧^𝜆𝐷^𝜆_𝑧ℎ(𝑧)=^∞_𝑛=0Γ(𝑛+2)Γ(2−𝜆)𝑎Γ(𝑛+2−𝜆)^𝑛+1𝑧^𝑛+1,𝑎₁Ω=1,_𝜆𝑔(𝑧)∶=Γ(2−𝜆)𝑧^𝜆𝐷^𝜆_𝑧𝑔(𝑧)=^∞_𝑛=0Γ(𝑛+2)Γ(2−𝜆)𝑏Γ(𝑛+2−𝜆)^𝑛+1𝑧^𝑛+1,𝑏₁=0. (1.13)We will define subclasses of normalized harmonic functions obtained by the Hadamard product and using the fractional derivative.

2. Main Results

Let ℎ and 𝑔 be analytic in 𝑈. Let 𝑃_𝐻 stand for harmonic functions 𝑓=ℎ+𝑔 so that Re𝑓>0,𝑧∈𝑈 and 𝑓(0)=1.

If the function 𝑓_𝑧+𝑓_𝑧=ℎ^′+𝑔^′ belongs to 𝑃_𝐻 for the analytic and normalized functionsℎ(𝑧)=𝑧+^∞_𝑛=2𝑎_𝑛𝑧^𝑛,𝑔(𝑧)=^∞_𝑛=2𝑏_𝑛𝑧^𝑛, (2.1)then the class of functions 𝑓=ℎ+𝑔 is denoted by ^0𝑥0𝑃^02𝑑𝑐0_𝐻 [6].

The function𝑡_𝛼1(𝑧)=𝑧+𝑧1+𝛼²1+⋯+𝑧1+(𝑛−1)𝛼^𝑛+⋯ (2.2)is analytic on 𝑈 when 𝛼 is a complex number different from −1,−(1/2),−(1/3),…. For Ω^𝜆𝑓∈𝑃^{0𝑥002𝑑𝑐0}_𝐻, we denote by ^0𝑥0𝑃^02𝑑𝑐_𝐻^𝜆,0(𝛼) the class of functions defined byΩ^𝜆𝐹=Ω^𝜆𝑓∗(𝑡_𝛼+𝑡_𝛼). (2.3)Therefore,Ω^𝜆𝐹=Ω^𝜆𝐻+Ω^𝜆𝑔=𝑧+^∞_𝑛=2Γ(𝑛+1)Γ(2−𝜆)𝑎Γ(𝑛+1−𝜆)[1+(𝑛−1)𝛼]^𝑛𝑧^𝑛+^∞_𝑛=2Γ(𝑛+1)Γ(2−𝜆)𝑏Γ(𝑛+1−𝜆)[1+(𝑛−1)𝛼]^𝑛𝑧^𝑛=𝑧+^∞_𝑛=2𝐴_𝑛𝑧^𝑛+^∞_𝑛=2𝐵_𝑛𝑧^𝑛,𝑧∈𝑈 (2.4)is in ^0𝑥0𝑃^02𝑑𝑐_𝐻^𝜆,0(𝛼). Conversely, if Ω^𝜆𝐹 is in the form (2.4), with 𝑎_𝑛,𝑏_𝑛 being the coefficients of Ω^𝜆𝑓∈𝑃^{0𝑥002𝑑𝑐0}_𝐻, then Ω^𝜆𝐹=𝑃^{0𝑥002𝑑𝑐}_𝐻^𝜆,0(𝛼).

Note that ^0𝑥0𝑃^02𝑑𝑐_𝐻^0,0(𝛼)≡𝑃^{0𝑥002𝑑𝑐0}_𝐻(𝛼) [7] and ^0𝑥0𝑃^02𝑑𝑐_𝐻^0,0(0)≡𝑃^{0𝑥002𝑑𝑐0}_𝐻.

Theorem 2.1. If Ω^𝜆𝐹∈𝑃^{0𝑥002𝑑𝑐}_𝐻^𝜆,0(𝛼), then there exists Ω^𝜆𝑓∈𝑃^{0𝑥002𝑑𝑐0}_𝐻 so that𝛼[𝑧(Ω^𝜆𝐹)_𝑧(𝑧)+𝑧(Ω^𝜆𝐹)_𝑧(𝑧)]+(1−𝛼)Ω^𝜆𝐹(𝑧)=Ω^𝜆𝑓(𝑧). (2.5)Conversely, for any function 𝑓 such that Ω^𝜆𝑓∈𝑃^{0𝑥002𝑑𝑐0}_𝐻, there exists Ω^𝜆𝐹∈𝑃^{0𝑥002𝑑𝑐}_𝐻^𝜆,0(𝛼) satisfying (2.5).

Proof. Let Ω^𝜆𝐹∈𝑃^{0𝑥002𝑑𝑐}_𝐻^𝜆,0(𝛼). If Ω^𝜆𝑓∈𝑃^{0𝑥002𝑑𝑐0}_𝐻, then since𝛼𝑧𝑡^′_𝛼(𝑧)+(1−𝛼)𝑡_𝛼(𝑧)=𝑡₀(𝑧) (2.6)as Ω^𝜆𝐹=Ω^𝜆𝑓∗(𝑡_𝛼+𝑡_𝛼), we obtain thatΩ^𝜆𝑓(𝑧)=𝛼[Ω^𝜆𝑓(𝑧)∗(𝑧𝑡^′_𝛼+𝑧𝑡^′_𝛼)]+(1−𝛼)[Ω^𝜆𝑓(𝑧)∗(𝑡_𝛼+𝑡_𝛼)]. (2.7)Therefore,Ω^𝜆𝑓(𝑧)=𝛼[𝑧(Ω^𝜆𝐹)_𝑧(𝑧)+𝑧(Ω^𝜆𝐹)_𝑧(𝑧)]+(1−𝛼)Ω^𝜆𝐹(𝑧). (2.8)Conversely, for Ω^𝜆𝑓∈𝑃^{0𝑥002𝑑𝑐0}_𝐻, from (2.1), (2.2), and (2.5),𝑧+^∞_𝑛=2Γ(𝑛+2)Γ(2−𝜆)𝑎Γ(𝑛+1−𝜆)^𝑛𝑧^𝑛+^∞_𝑛=2Γ(𝑛+1)Γ(2−𝜆)𝑏Γ(𝑛+1−𝜆)^𝑛𝑧^𝑛=𝑧+^∞_𝑛=2[1+(𝑛−1)𝛼]𝐴_𝑛𝑧^𝑛+^∞_𝑛=2[1+(𝑛−1)𝛼]𝐵_𝑛𝑧^𝑛, (2.9)where𝐴_𝑛=Γ(𝑛+1)Γ(2−𝜆)𝑎Γ(𝑛+1−𝜆)[1+(𝑛−1)𝛼]^𝑛,𝐵_𝑛=Γ(𝑛+1)Γ(2−𝜆)𝑏Γ(𝑛+1−𝜆)[1+(𝑛−1)𝛼]^𝑛. (2.10)Therefore,Ω^𝜆𝐹=𝑧+^∞_𝑛=2𝐴_𝑛𝑧^𝑛+^∞_𝑛=2𝐵_𝑛𝑧^𝑛=Ω^𝜆𝑓∗[𝑡_𝛼(𝑧)+𝑡_𝛼(𝑧)]. (2.11)

Theorem 2.2. A function Ω^𝜆𝐹 of the form (2.4) belongs to ^0𝑥0𝑃^02𝑑𝑐_𝐻^𝜆,0(𝛼), if and only ifRe{𝑧(Ω^𝜆𝐻(𝑧))^′′+𝛼(Ω^𝜆𝐺(𝑧))^′′+(Ω^𝜆𝐻(𝑧))^′+(Ω^𝜆𝐺(𝑧))^′}>0,𝑧∈𝑈. (2.12)

Proof. If Ω^𝜆𝐹=Ω^𝜆𝐻+Ω^𝜆𝐺∈𝑃^{0𝑥002𝑑𝑐}_𝐻^𝜆,0(𝛼), then from Theorem 2.1,𝛼[𝑧(Ω^𝜆𝐻)^′+𝑧(Ω^𝜆𝐺)^′]+(1−𝛼)[Ω^𝜆𝐻+Ω^𝜆𝐺]=Ω^𝜆ℎ+Ω^𝜆𝑔∈𝑃^{0𝑥002𝑑𝑐0}_𝐻 (2.13)and (Ω^𝜆ℎ)^′+(Ω^𝜆𝑔)^′∈𝑃_𝐻. Hence0<Re{(Ω^𝜆ℎ)^′+(Ω^𝜆𝑔)^′}×Re{𝛼𝑧(Ω^𝜆𝐻)^′′+𝛼(Ω^𝜆𝐻)^′+(1−𝛼)(Ω^𝜆𝐻)^′+𝛼𝑧(Ω^𝜆𝐺)^′′+𝛼(Ω^𝜆𝐺)^′+(1−𝛼)(Ω^𝜆𝐺)^′}×Re{𝑧(𝛼(Ω^𝜆𝐻)^′′+𝛼(Ω^𝜆𝐺)^′′)+(Ω^𝜆𝐻)^′+(Ω^𝜆𝐺)^′}. (2.14)Conversely, if the function Ω^𝜆𝐹=Ω^𝜆𝐻+Ω^𝜆𝐺 of the form (2.4) satisfies (2.10), then by Theorem 2.1  (Ω^𝜆ℎ)^′+(Ω^𝜆𝑔)^′∈𝑃_𝐻 and the following function holds:Ω^𝜆𝑓=Ω^𝜆ℎ+Ω^𝜆𝑔=𝛼[𝑧(Ω^𝜆𝐻)^′+𝑧(Ω^𝜆𝐺)^′]+(1−𝛼)[Ω^𝜆𝐻+Ω^𝜆𝐺]∈𝑃^{0𝑥002𝑑𝑐0}_𝐻. (2.15)Then by Theorem 2.1, Ω^𝜆𝐹=Ω^𝜆𝐻+Ω^𝜆𝐺∈𝑃^{0𝑥002𝑑𝑐}_𝐻^𝜆,0(𝛼).

Theorem 2.3. ^0𝑥0𝑃^02𝑑𝑐_𝐻^𝜆,0(𝛼) is convex and compact.

Proof. Let Ω^𝜆𝐹₁=Ω^𝜆𝐻₁+Ω^𝜆𝐺₁,Ω^𝜆𝐹₂=Ω^𝜆𝐻₂+Ω^𝜆𝐺₂∈𝑃^{0𝑥002𝑑𝑐}_𝐻^𝜆,0(𝛼) and let 𝜇∈[0,1]. ThenRe{𝑧[𝛼(𝜇(Ω^𝜆𝐻₁(𝑧))^′′+(1−𝜇)(Ω^𝜆𝐻₂(𝑧))^′′)+𝛼(Ω^𝜆𝐺₁(𝑧))^′′+(1−𝜇)(Ω^𝜆𝐺₂(𝑧))^′′)]+𝜇[(Ω^𝜆𝐻₁(𝑧))^′+(Ω^𝜆𝐺₁(𝑧))^′]+(1−𝜇)[(Ω^𝜆𝐻₂(𝑧))^′+(Ω^𝜆𝐺₂(𝑧))^′]}=𝜇Re{𝑧[𝛼(Ω^𝜆𝐻₁(𝑧))^′′+𝛼(Ω^𝜆𝐺₁(𝑧))^′′]+(Ω^𝜆𝐻₁(𝑧))^′+(Ω^𝜆𝐺₁(𝑧))^′}+(1−𝜇)Re{𝑧[𝛼(Ω^𝜆𝐻₂(𝑧))^′′+𝛼(Ω^𝜆𝐺₂(𝑧))^′′]+(Ω^𝜆𝐻₂(𝑧))^′+(Ω^𝜆𝐺₂(𝑧))^′}>0. (2.16)Hence from Theorem 2.2, 𝜇Ω^𝜆𝐹₁+(1−𝜇)Ω^𝜆𝐹₂∈𝑃^{0𝑥002𝑑𝑐}_𝐻^𝜆,0(𝛼). Therefore, ^0𝑥0𝑃^02𝑑𝑐_𝐻^𝜆,0(𝛼) is convex.
Now, let Ω^𝜆𝐹_𝑛=Ω^𝜆𝐻_𝑛+Ω^𝜆𝐺_𝑛∈𝑃^{0𝑥002𝑑𝑐}_𝐻^𝜆,0(𝛼) and let Ω^𝜆𝐹_𝑛→Ω^𝜆𝐹=Ω^𝜆𝐻+Ω^𝜆𝐺. By Theorem 2.2,𝛼[𝑧(Ω^𝜆𝐻_𝑛)^′+𝑧(Ω^𝜆𝐺_𝑛)^′]+(1−𝛼)[Ω^𝜆𝐻_𝑛+Ω^𝜆𝐺_𝑛]∈𝑃^{0𝑥002𝑑𝑐0}_𝐻. (2.17)Since ^0𝑥0𝑃^02𝑑𝑐0_𝐻 is compact, see [6],𝛼[𝑧(Ω^𝜆𝐻)^′+𝑧(Ω^𝜆𝐺)^′]+(1−𝛼)[Ω^𝜆𝐻+Ω^𝜆𝐺]∈𝑃^{0𝑥002𝑑𝑐0}_𝐻. (2.18)Hence by Theorem 2.1, Ω^𝜆𝐹∈𝑃^{0𝑥002𝑑𝑐}_𝐻^𝜆,0(𝛼), therefore ^0𝑥0𝑃^02𝑑𝑐_𝐻^𝜆,0(𝛼) is compact.

Theorem 2.4. If Ω^𝜆𝐹=Ω^𝜆𝐻+Ω^𝜆𝐺∈𝑃^{0𝑥002𝑑𝑐}_𝐻^𝜆,0(𝛼) and |𝑧|=𝑟<1, then−𝑟+2In(1+𝑟)≤Re{𝛼[𝑧(Ω^𝜆𝐻_𝑛)^′+𝑧(Ω^𝜆𝐺_𝑛)^′]+(1−𝛼)[Ω^𝜆𝐻_𝑛+Ω^𝜆𝐺_𝑛]}≤−𝑟−2In(1−𝑟). (2.19)Equality is obtained for the function (2.3) whereΩ^𝜆𝑓=2𝑧+ln(1−𝑧)−3𝑧−3ln(1−𝑧),𝑧∈𝑈. (2.20)

Proof. From Theorem 2.1, if Ω^𝜆𝐻+Ω^𝜆𝐺∈𝑃^{0𝑥002𝑑𝑐}_𝐻^𝜆,0(𝛼), then there exists Ω^𝜆𝑓=Ω^𝜆ℎ+Ω^𝜆𝑔∈𝑃^{0𝑥002𝑑𝑐0}_𝐻 so that𝛼[𝑧(Ω^𝜆𝐻)^′+𝑧(Ω^𝜆𝐺)^′]+(1−𝛼)[Ω^𝜆𝐻+Ω^𝜆𝐺]=Ω^𝜆𝑓. (2.21)Since by [5, Proposition 2.2]−𝑟+2ln(1+𝑟)≤Re(Ω^𝜆𝑓)≤−𝑟−2ln(1−𝑟), (2.22)this completes the proof.

Theorem 2.5. If Ω^𝜆𝐹=Ω^𝜆𝐻+Ω^𝜆𝐺∈𝑃^{0𝑥002𝑑𝑐}_𝐻^𝜆,0(𝛼) and Re𝛼>0, then there exists Ω^𝜆𝑓∈𝑃^{0𝑥002𝑑𝑐0}_𝐻 so thatΩ^𝜆1𝐹=𝛼¹₀𝜁^1/𝛼−2(Ω^𝜆𝑓)(𝑧𝜁)𝑑𝜁,𝑧∈𝑈. (2.23)

Proof. Since𝑡_𝛼1(𝑧)=𝛼¹₀𝜁^1/𝛼−1𝑧1−𝑧𝜁𝑑𝜁,|𝜁|≤1,Re𝛼>0, (2.24)and for Ω^𝜆𝑓=Ω^𝜆ℎ+Ω^𝜆𝑔∈𝑃^{0𝑥002𝑑𝑐0}_𝐻,(Ω^𝜆𝑧ℎ)(𝑧)∗=1−𝑧𝜁(Ω^𝜆ℎ)(𝑧𝜁)𝜁,(Ω^𝜆𝑧𝑔)(𝑧)∗=1−𝑧𝜁(Ω^𝜆𝑔)(𝑧𝜁)𝜁, (2.25)we have(Ω^𝜆𝐻)(𝑧)=(Ω^𝜆ℎ)(𝑧)∗𝑡_𝛼=1𝛼¹₀𝜁^1/𝛼−2(Ω^𝜆ℎ)(𝑧𝜁)𝑑𝜁,(Ω^𝜆𝐺)(𝑧)=(Ω^𝜆𝑔)(𝑧)∗𝑡_𝛼=1𝛼¹₀𝜁^1/𝛼−2(Ω^𝜆𝑔)(𝑧𝜁)𝑑𝜁. (2.26)Hence Ω^𝜆𝐹 is type (2.23).

Theorem 2.6. If Re𝛼>0, then ^0𝑥0𝑃^02𝑑𝑐₀^𝜆,0(𝛼)⊂𝑃^{0𝑥002𝑑𝑐0}_𝐻.

Proof. Let Ω^𝜆𝐹∈𝑃^{0𝑥002𝑑𝑐}_𝐻^𝜆,0(𝛼) and Re𝛼>0. Then there exists Ω^𝜆𝑓∈𝑃^{0𝑥002𝑑𝑐0}_𝐻 so thatΩ^𝜆𝐹=Ω^𝜆𝐻+Ω^𝜆𝐺=Ω^𝜆𝑓∗(𝑡_𝛼+𝑡_𝛼)=(Ω^𝜆ℎ∗𝑡_𝛼)+(Ω^𝜆𝑔∗𝑡_𝛼). (2.27)Hence 0<Re{(Ω^𝜆ℎ)^′+(Ω^𝜆𝑔)^′}=Re{(Ω^𝜆ℎ)^′+(Ω^𝜆𝑔)^′} and since Re𝛼>0, Re{(Ω^𝜆𝐻)^′+(Ω^𝜆𝐺)^′}>0 and Ω^𝜆𝐻(0)=0,(Ω^𝜆𝐻)^′(0)=1,Ω^𝜆𝐺(0)=0,(Ω^𝜆𝐺)^′(0)=0. And hence Ω^𝜆𝐹∈𝑃^{0𝑥002𝑑𝑐0}_𝐻.

Theorem 2.7. Let Ω^𝜆𝐹=Ω^𝜆𝐻+Ω^𝜆𝐺∈𝑃^{0𝑥002𝑑𝑐}_𝐻^𝜆,0(𝛼). Then
(i) ‖𝐴_𝑛|−|𝐵_𝑛‖≤2Γ(𝑛+1)Γ(2−𝜆)𝑛Γ(𝑛+1−𝜆)|(1+(𝑛−1)𝛼|,𝑛≥1, (2.28)
(ii) if Ω^𝜆𝐹 is sense-preserving, then|𝐴_𝑛|≤2𝑛−1𝑛Γ(𝑛+1)Γ(2−𝜆)Γ(𝑛+1−𝜆)|1+(𝑛−1)𝛼|,𝑛=1,2,…,|𝐵_𝑛|≤2𝑛−3𝑛Γ(𝑛+1)Γ(2−𝜆)Γ(𝑛+1−𝜆)|1+(𝑛−1)𝛼|,𝑛=2,3,…. (2.29)

Proof. By (2.10),‖𝐴_𝑛|−|𝐵_𝑛‖=Γ(𝑛+1)Γ(2−𝜆)Γ(𝑛+1−𝜆)|1+(𝑛−1)𝛼|‖𝑎_𝑛|−|𝑏_𝑛‖. (2.30)Also, by [6, Theorem 2.3], we have‖𝑎_𝑛|−|𝑏_𝑛2‖≤𝑛. (2.31)The required results are obtained.
On the other hand, from (2.10), it is known [6, Corollary 2.5] that|𝑎_𝑛|≤2𝑛−1𝑛,|𝑏_𝑛|≤2𝑛−3𝑛. (2.32)Then we get the coefficient inequalities for ^0𝑥0𝑃^02𝑑𝑐₀^𝜆,0(𝛼).

Remark 2.8. Taking 𝜆=0 in Theorems 2.1–2.7, we get the similar results in [7].

Theorem 2.9. Let Ω^𝜆𝐹=Ω^𝜆𝐻=Ω^𝜆𝐺∈𝑃^{0𝑥002𝑑𝑐}_𝐻^𝜆,0(𝛼) and sense-preserving in 𝑈, then for |𝑧|=𝑟<1,|𝛼𝑧(Ω^𝜆𝐻)^′+(1−𝛼)Ω^𝜆𝐻|≤2𝑟|1−𝑟+ln(1−𝑟),𝛼𝑧(Ω^𝜆𝐺)^′+(1−𝛼)Ω^𝜆𝐺|≤3𝑟−𝑟²1−𝑟+3ln(1−𝑟). (2.33)

Proof. From Theorems 2.1 and 2.2, if Ω^𝜆𝐹=Ω^𝜆𝐻+Ω^𝜆𝐺∈𝑃^{0𝑥002𝑑𝑐}_𝐻^𝜆,0(𝛼), then there exists Ω^𝜆𝑓=Ω^𝜆ℎ+Ω^𝜆𝑔∈𝑃^{0𝑥002𝑑𝑐0}_𝐻 such that𝛼𝑧(Ω^𝜆𝐻)^′+(1−𝛼)Ω^𝜆𝐻=Ω^𝜆ℎ,𝛼𝑧(Ω^𝜆𝐺)^′+(1−𝛼)Ω^𝜆𝐺=Ω^𝜆𝑔. (2.34)By [6, Theorem 3.5], we obtain the results.

Remark 2.10. Taking 𝜆=0 and 𝛼=0 in Theorem 2.9, we get [6, Theorem 2.4].

3. Positive Order

We say that the harmonic function 𝑓=ℎ+𝑔 of the form (2.1) is in the class 𝑃_𝐻(𝛽),0≤𝛽<1 for |𝑧|=𝑟 if Re𝑓>𝛽 and 𝑓(0)=1.

If the function 𝑓_𝑧+𝑓_𝑧=ℎ^′+𝑔^′ belongs to 𝑃_𝐻(𝛽) for the analytic and normalized functions ℎ and 𝑔 of the form (2.1), then the class of functions 𝑓=ℎ+𝑔 is denoted by 𝑃⁰_𝐻(𝛽).

Denote by 𝑃_𝐻^𝜆,0(𝛽,𝛼) the class of functions defined By (2.3) where Ω^𝜆𝑓∈𝑃⁰_𝐻(𝛽).

Many of our results can be rewritten for functions in the class 𝑃_𝐻^𝜆,0(𝛽,𝛼). For instance, see the following theorems.

Theorem 3.1. If Ω^𝜆𝐹∈𝑃_𝐻^𝜆,0(𝛽,𝛼), then there exists Ω^𝜆𝑓∈𝑃⁰_𝐻(𝛽) so that𝛼[𝑧(Ω^𝜆𝐹)_𝑧(𝑧)+𝑧(Ω^𝜆𝐹)_𝑧(𝑧)]+(1−𝛼)Ω^𝜆𝐹(𝑧)=Ω^𝜆𝑓(𝑧). (3.1)Conversely, for any function 𝑓 such that Ω^𝜆𝑓∈𝑃⁰_𝐻(𝛽), there exists Ω^𝜆𝐹∈𝑃_𝐻^𝜆,0(𝛽,𝛼) satisfying (3.1).

Theorem 3.2. A function Ω^𝜆𝐹 belongs to 𝑃_𝐻^𝜆,0(𝛽,𝛼) if and only ifRe{𝑧(Ω^𝜆𝐻(𝑧))^′′+𝛼(Ω^𝜆𝐺(𝑧))^′′+(Ω^𝜆𝐻(𝑧))^′+(Ω^𝜆𝐺(𝑧))^′}>𝛽,𝑧∈𝑈. (3.2)

Theorem 3.3. If Ω^𝜆𝐹∈𝑃_𝐻^𝜆,0(𝛽,𝛼) and Re𝛼>0, then there exists Ω^𝜆𝑓∈𝑃⁰_𝐻(𝛽) so thatΩ^𝜆1𝐹=𝛼¹₀𝜁^1/𝛼−2(Ω^𝜆𝑓)(𝑧𝜁)𝑑𝜁,𝑧∈𝑈. (3.3)

Theorem 3.4. If Re𝛼>0, then 𝑃_𝐻^𝜆,0(𝛽,𝛼)⊂𝑃⁰_𝐻(𝛽).