# On a Subclass of Harmonic Convex Functions of Complex Order

Authors: N. Magesh; S. Mayilvaganan
Journal: International Journal of Mathematics and Mathematical Sciences
 (2012)

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## Abstract

We introduce and study a subclass of harmonic convex functions of complex order. Coefficient bounds, extreme points, distortion bounds, convolution conditions, and convex combination are determined for functions in this class. Further, we obtain the closure property of this class under integral operator.


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## Body

## 1. Introduction

A continuous functionf=u+iv is a complex-valued harmonic function in a complex domain Ω if both u and v are real and harmonic in Ω. In any simply connected domain D⊂Ω, 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 |h′(z)|>|g′(z)| in D (see [1]).Denote by𝒮ℋ the family of functions f=h+g¯, which are harmonic, univalent, and orientation preserving in the open unit disc 𝒰={z:|z|<1} so that f is normalized by f(0)=h(0)=fz(0)-1=0. Thus, for f=h+g¯∈𝒮ℋ, the functions h and g analytic in 𝒰 can be expressed in the following forms:(1.1)h(z)=z+∑k=2∞akzk,g(z)=∑k=1∞bkzk(|b1|<1),
and f(z) is then given by(1.2)f(z)=z+∑k=2∞akzk+∑k=1∞bkzk¯(|b1|<1).We note that the family𝒮ℋ of orientation preserving, normalized harmonic univalent functions reduces to the well-known class 𝒮 of normalized univalent functions if the coanalytic part of f is identically zero (g≡0).Also, we denote byT𝒮ℋ the subfamily of 𝒮ℋ consisting of harmonic functions of the form f=h+g¯ such that h and g are of the form(1.3)h(z)=z-∑k=2∞|ak|zk,g(z)=∑k=1∞|bk|zk.In [1], Clunie and Sheil-Small investigated the class 𝒮ℋ as well as its geometric subclasses and its properties. Since then, there have been several studies related to the class 𝒮ℋ and its subclasses. Following Clunie and Sheil-Small [1], Frasin [2], Jahangiri et al. [3–6], Silverman [7], Silverman and Silvia [8], Yalçin and Öztürk [9], and others have investigated various subclasses of 𝒮ℋ and its properties. In particular, Avcı and Złotkiewicz [10] proved that the coefficient condition(1.4)∑k=2∞k2(|ak|+|bk|)≤1(b1=0)
is sufficient for functions f=h+g¯∈𝒮ℋ to be harmonic convex. Also, Silverman [7] studied that this coefficient condition is also necessary if ak and bk  (k=2,3,…) in (1.2) are negative. Further, Jahangiri [3] showed that if f=h+g¯ is given by (1.2) and if(1.5)∑k=1∞(k(k-γ)1-γ|ak|+k(k+γ)1-γ|bk|)≤2(0≤γ<1,a1=1),
then, f is harmonic, univalent, and convex of order γ in 𝒰. This condition is proved to be also necessary if h and g are of the form (1.3). Furthermore, Yalçin and Öztürk [11] have considered a class T𝒮ℋ*(γ) of harmonic starlike functions of complex order based on a corresponding study of Nasr and Aouf [12] for the analytic case.Motivated by the earlier works given in the literature [9, 11] now we define the class of harmonic convex functions of complex order as follows.Definition 1.1.
For0≤γ<1, 0≤λ≤γ/(1+γ) or λ≥1/(1+γ), and b∈ℂ∖{0} with |b|<1, let 𝒮Cℋ(b,γ,λ) denote the family of harmonic functions f∈𝒮ℋ of the form (1.2) that satisfy the following condition:
(1.6)R{1+1b(λ(z3h′′′(z)-z3g′′′(z)¯)+(2λ+1)z2h′′(z)λ(z2h′′(z)+z2g′′(z)¯)+zh′(z)+(2λ-1)zg′(z)¯+(1-4λ)z2g′′(z)¯+zh′(z)+(1-2λ)zg′(z)¯λ(z2h′′(z)+z2g′′(z)¯)+zh′(z)+(2λ-1)zg′(z)¯-1)}≥γ,
for z∈𝒰. Further, we define the subclass T𝒮Cℋ(b,γ,λ) of 𝒮Cℋ(b,γ,λ) consisting of functions f=h+g¯ of the form (1.3).We observe that forb=1 the class 𝒮Cℋ(1,γ,λ)=𝒮Cℋ(γ,λ) was introduced and studied by Yalçin and Öztürk [9], the class 𝒮Cℋ(1,γ,0)=𝒮Cℋ(γ) is given in [3, 4], and the class 𝒮Cℋ(1,0,0)=𝒮Cℋ is studied in [10].In this paper, we investigate coefficient conditions, extreme points, and distortion bounds for the function classT𝒮Cℋ(b,γ,λ). We also examine their convolution and convex combination properties and the closure property of this class under integral operator. We remark that the results obtained for these general families can be viewed as extensions and generalizations for various subclasses of 𝒮ℋ as listed previously in this section.


## 2. Coefficient Inequalities

Our first theorem gives a sufficient condition for functions in𝒮Cℋ(b,γ,λ).Theorem 2.1.
Letf=h+g¯ be so that h and g are given by (1.2). If
(2.1)∑k=1∞k(kλ-λ+1)[(k-1)+|b|(1-γ)](1-γ)|b||ak|+∑k=1∞k(kλ+λ-1)[(k+1)-|b|(1-γ)](1-γ)|b||bk|≤2,
where a1=1, 0≤γ<1, and b(|b|<1) is a nonzero complex number,  0≤λ≤γ/(1+γ) or λ≥1/(1+γ). Then, f∈𝒮Cℋ(b,γ,λ) and f is sense preserving, univalent, and harmonic in 𝒰.Proof.
We show thatf∈𝒮Cℋ(b,γ,λ). We only need to show that if (2.1) holds, then condition (1.6) is satisfied. In view of (1.2), condition (1.6) takes the form
(2.2)R((1-γ)+∑k=2∞((k(kλ-λ+1)[(k-1)+b(1-γ)])/b)|ak|(zk/z)1+∑k=2∞k(kλ-λ+1)|ak|(zk/z)+∑k=1∞k(kλ+λ-1)|bk|(z¯k/z)-∑k=1∞((k(kλ+λ-1)[(k+1)-b(1-γ)])/b)|bk|(z¯k/z)1+∑k=2∞k(kλ-λ+1)|ak|(zk/z)+∑k=1∞k(kλ+λ-1)|bk|(z¯k/z))=R1+A(z)1+B(z).
Setting
(2.3)1+A(z)1+B(z)=1+ω(z)1-ω(z),
we will have ℜ((1+A(z))/(1+B(z)))>0 if |ω(z)|<1,
(2.4)ω(z)=A(z)-B(z)2+A(z)+B(z)-γ+∑k=2∞k(kλ-λ+1)[[(k-1)+b(1-γ)]b-1]|ak|zk-1=-∑k=1∞k(kλ+λ-1)[([(k+1)-b(1-γ)]/b)+1]|bk|(z̅k/z)2-γ+∑k=2∞k(kλ-λ+1)[([(k-1)+b(1-γ)]/b)+1]|ak|zk-1-∑k=1∞k(kλ+λ-1)[[(k+1)-b(1-γ)]b-1]|bk|z̅kz,
so that
(2.5)|ω(z)|<γ+∑k=2∞k(kλ-λ+1)[(A/|b|)-1]|ak|+∑k=1∞k(kλ+λ-1)[(B/|b|)+1]|bk|2-γ-∑k=2∞k(kλ-λ+1)[(A/|b|)+1]|ak|-∑k=1∞k(kλ+λ-1)[(B/|b|)-1]|bk|,
where 𝔄 denotes [(k-1)+|b|(1-γ)] and 𝔅 denotes [(k+1)-|b|(1-γ)].This last expression is bounded above by 1 if and only if
(2.6)∑k=2∞k(kλ-λ+1)[(k-1)+|b|(1-γ)]|b||ak|+∑k=1∞k(kλ+λ-1)[(k+1)-|b|(1-γ)]|b||bk|≤(1-γ).
Or, equivalently
(2.7)∑k=1∞k(kλ-λ+1)[(k-1)+|b|(1-γ)](1-γ)|b||ak|+∑k=1∞k(kλ+λ-1)[(k+1)-|b|(1-γ)](1-γ)|b||bk|≤2.
In order to show thatf is univalent in 𝒰, we show that f(z1)≠f(z2) whenever z1≠z2. Since 𝒰 is simply connected and convex, we have z(t)=(1-t)z1+tz2∈𝒰, where 0≤t≤1, and if z1,z2∈𝒰 so that z1≠z2. Then, we write
(2.8)f(z2)-f(z1)=∫01[(z2-z1)h′(z(t))+(z2-z1)g′(z(t))¯]dt.
Dividing the above equation byz2-z1≠0 and taking the real part, we obtain
(2.9)R{f(z2)-f(z1)z2-z1}=∫01R[h′(z(t))+z2-z1¯z2-z1g′(z(t))¯]dt>∫01[Rh′(z(t))-|g′(z(t))|]dt.
On the other hand, for |b|<1,  λ≥1/(1+γ) or 0≤λ≤γ/(1+γ), we have
(2.10)Rh′(z)-|g′(z)|≥Rh′(z)-∑k=1∞k|bk|≥1-∑k=2∞k|ak|-∑k=1∞k|bk|≥1-∑k=2∞k(kλ-λ+1)[(k-1)+|b|(1-γ)](1-γ)|b||ak|-∑k=1∞k(kλ+λ-1)[(k+1)-|b|(1-γ)](1-γ)|b||bk|≥0,using(2.1).
This along with inequality (2.9) leads to the univalence of f. Note that f is sense preserving in 𝒰, for 0≤λ≤γ/(1+γ) or λ≥1/(1+γ). This is because
(2.11)|h′(z)|≥1-∑k=2∞k|ak||z|k-1>1-∑k=2∞k|ak|≥1-∑k=2∞k(kλ-λ+1)[(k-1)+|b|(1-γ)](1-γ)|b||ak|≥∑k=1∞k(kλ+λ-1)[(k+1)-|b|(1-γ)](1-γ)|b||bk|>∑k=1∞k(kλ+λ-1)[(k+1)-|b|(1-γ)](1-γ)|b||bk||z|k-1≥∑k=1∞k|bk||z|k-1≥|g′(z)|.
The function
(2.12)f(z)=z+∑k=2∞(1-γ)|b|k(kλ-λ+1)[(k-1)+|b|(1-γ)]xkzk+∑k=1∞(1-γ)|b|k(kλ+λ-1)[(k+1)-|b|(1-γ)]ykzk¯,
where ∑k=2∞|xk|+∑k=1∞|yk|=1, shows that the coefficient bound given by (2.1) is sharp. The functions of the form (2.12) are in 𝒮Cℋ(b,γ,λ) because
(2.13)∑k=1∞(k(kλ-λ+1)[(k-1)+|b|(1-γ)](1-γ)|b||ak|+k(kλ+λ-1)[(k+1)-|b|(1-γ)](1-γ)|b||bk|)=1+∑k=2∞|xk|+∑k=1∞|yk|=2.The next theorem shows that condition (2.1) is necessary for f∈T𝒮Cℋ(b,γ,λ).Theorem 2.2.
Letf=h+g¯ be so that h and g are given by (1.3). Then, f∈T𝒮Cℋ(b,γ,λ) if and only if
(2.14)∑k=1∞k(kλ-λ+1)[(k-1)+|b|(1-γ)](1-γ)|b||ak|+∑k=1∞k(kλ+λ-1)[(k+1)-|b|(1-γ)](1-γ)|b||bk|≤2,
where a1=1, 0≤γ<1, 0≤λ≤γ/(1+γ) or  λ≥1/(1+γ), and b∈ℂ∖{0}.Proof.
The “if part” follows from Theorem2.1 upon noting that T𝒮Cℋ(b,γ,λ)⊂𝒮Cℋ(b,γ,λ). For the “only if” part, we show that f∈T𝒮Cℋ(b,γ,λ). Then, for z=reiθ in 𝒰, we obtain
(2.15)R{1+1b(λ(z3h′′′(z)-z3g′′′(z)¯)+(2λ+1)z2h′′(z)λ(z2h′′(z)+z2g′′(z)¯)+zh′(z)+(2λ-1)zg′(z)¯+(1-4λ)z2g′′(z)¯+zh′(z)+(1-2λ)zg′(z)¯λ(z2h′′(z)+z2g′′(z)¯)+zh′(z)+(2λ-1)zg′(z)¯-1)-γ}=R{(1-γ)z-∑k=2∞((k(kλ-λ+1)[(k-1)+b(1-γ)])/b)|ak|zkz-∑k=2∞k(kλ-λ+1)|ak|zk+∑k=1∞k(kλ+λ-1)|bk|z¯k-∑k=1∞((k(kλ+λ-1)[(k+1)-b(1-γ)])/b)|bk|z¯kz-∑k=2∞k(kλ-λ+1)|ak|zk+∑k=1∞k(kλ+λ-1)|bk|z¯k}≥(1-γ)-∑k=2∞((k(kλ-λ+1)[(k-1)+b(1-γ)])/b)|ak|rk-1z-∑k=2∞k(kλ-λ+1)|ak|rk-1+∑k=1∞k(kλ+λ-1)|bk|rk-1-∑k=1∞((k(kλ+λ-1)[(k+1)-b(1-γ)])/b)|bk|rk-1z-∑k=2∞k(kλ-λ+1)|ak|rk-1+∑k=1∞k(kλ+λ-1)|bk|rk-1>0.
The above inequality must hold for all z∈𝒰. In particular, letting z=r→1- yields the required condition.As special cases of Theorem2.2, we obtain the following two corollaries.Corollary 2.3.
A functionf=h+g¯ belongs to T𝒮Cℋ(b,γ,0) if and only if
(2.16)∑k=1∞k[(k-1)+|b|(1-γ)](1-γ)|b||ak|+∑k=1∞k[(k+1)-|b|(1-γ)](1-γ)|b||bk|≤2.Corollary 2.4.
A functionf=h+g¯ belongs to T𝒮Cℋ(b,γ,1) if and only if
(2.17)∑k=1∞k2[(k-1)+|b|(1-γ)](1-γ)|b||ak|+∑k=1∞k2[(k+1)-|b|(1-γ)](1-γ)|b||bk|≤2.


## 3. Extreme Points and Distortion Bounds

In this section, our first theorem gives the extreme points of the closed convex hulls ofT𝒮Cℋ(b,γ,λ).Theorem 3.1.
A functionf=h+g¯  belongs to T𝒮Cℋ(b,γ,λ) if and only if f can be expressed as
(3.1)f(z)=∑k=1∞(Xkhk(z)+Ykgk(z)),z∈U,
where h1(z)=z, hk(z)=z-((1-γ)|b|/k(kλ-λ+1)[(k-1)+|b|(1-γ)])zk(k=2,3,…), and gk(z)=z+((1-γ)|b|/k(kλ+λ-1)[(k+1)-|b|(1-γ)])z¯k(k=1,2,3,…),  ∑k=1∞(Xk+Yk)=1, Xk≥0, Yk≥0. In particular, the extreme points of T𝒮Cℋ(b,γ,λ) are {hk} and {gk}.Proof.
For functionsf of the form (3.1), we have
(3.2)f(z)=∑k=1∞(Xkhk(z)+Ykgk(z))=∑k=1∞(Xk+Yk)z-∑k=2∞(1-γ)|b|k(kλ-λ+1)[(k-1)+|b|(1-γ)]Xkzk+∑k=1∞(1-γ)|b|k(kλ+λ-1)[(k+1)-|b|(1-γ)]Ykz¯k.
Then,
(3.3)∑k=2∞k(kλ-λ+1)[(k-1)+|b|(1-γ)](1-γ)|b|((1-γ)|b|k(kλ-λ+1)[(k-1)+|b|(1-γ)])Xk+∑k=1∞k(kλ+λ-1)[(k+1)-|b|(1-γ)](1-γ)|b|((1-γ)|b|k(kλ+λ-1)[(k+1)-|b|(1-γ)])Yk=∑k=2∞Xk+∑k=1∞Yk=1-X1≤1,
and so f∈cl coT𝒮Cℋ(b,γ,λ).
Conversely, suppose thatf∈cl coT𝒮Cℋ(b,γ,λ). Letting
(3.4)X1=1-∑k=2∞Xk-∑k=1∞Yk,
where Xk=(k(kλ-λ+1)[(k-1)+|b|(1-γ)]/(1-γ)|b|)|ak|, k=2,3,…,  and Yk=(k(kλ+λ-1)[(k+1)-|b|(1-γ)]/(1-γ)|b|)|bk|, k=1,2,…, we obtain the required representation since
(3.5)f(z)=z-∑k=2∞|ak|zk+∑k=1∞|bk|z¯k=z-∑k=2∞(1-γ)|b|Xkk(kλ-λ+1)[(k-1)+|b|(1-γ)]zk+∑k=1∞(1-γ)|b|Ykk(kλ+λ-1)[(k+1)-|b|(1-γ)]z¯k=z-∑k=2∞(z-hk(z))Xk-∑k=1∞(z-gk(z))Yk=(1-∑k=2∞Xk-∑k=1∞Yk)z+∑k=2∞hk(z)Xk+∑k=1∞gk(z)Yk=∑k=1∞(Xkhk(z)+Ykgk(z)).The following theorem gives the distortion bounds for functions inT𝒮Cℋ(b,γ,λ), which yields a covering result for this family.Theorem 3.2.
Iff∈T𝒮Cℋ(b,γ,λ) then for z=reiθ, one has
(3.6)|f(z)|≤(1+|b1|)r+r2((1-γ)|b|2(λ+1)[1+b(1-γ)]-(2λ-1)[2-b(1-γ)]2(λ+1)[1+b(1-γ)]|b1|),|f(z)|≥(1-|b1|)r-r2((1-γ)|b|2(λ+1)[1+b(1-γ)]-(2λ-1)[2-b(1-γ)]2(λ+1)[1+b(1-γ)]|b1|).Proof.
Letf∈T𝒮Cℋ(b,γ,λ). Taking the absolute value of f and then by Theorem 2.2, we obtain
(3.7)|f(z)|≤(1+|b1|)r+∑k=2∞(|ak|+|bk|)rk≤(1+|b1|)r+(1-γ)|b|2(λ+1)[1+b(1-γ)](∑k=2∞k(kλ-λ+1)[(k-1)+|b|(1-γ)](1-γ)|b||ak|+k(kλ+λ-1)[(k+1)-|b|(1-γ)](1-γ)|b||bk|)r2≤(1+|b1|)r+((1-γ)|b|2(λ+1)[1+b(1-γ)]-(2λ-1)[2-b(1-γ)]2(λ+1)[1+b(1-γ)]|b1|)r2.
Similarly,
(3.8)|f(z)|≥(1-|b1|)r-((1-γ)|b|2(λ+1)[1+b(1-γ)]-(2λ-1)[2-b(1-γ)]2(λ+1)[1+b(1-γ)]|b1|)r2.
The upper and lower bounds given in Theorem3.2 are, respectively, attained for the following functions
(3.9)f(z)=z+|b1|z¯+1Γ(2)((1-γ)|b|2(λ+1)[1+b(1-γ)]-(2λ-1)[2-b(1-γ)]2(λ+1)[1+b(1-γ)]|b1|)z¯2,f(z)=(1-|b1|)z-1Γ(2)((1-γ)|b|2(λ+1)[1+b(1-γ)]-(2λ-1)[2-b(1-γ)]2(λ+1)[1+b(1-γ)]|b1|)z2.


## 4. Convolution and Convex Combinations

In this section we show that the classT𝒮Cℋ(b,γ,λ) is closed under convolution and convex combinations. Now we need the following definition of convolution of two harmonic functions. For f(z)=z-∑k=2∞|ak|zk+∑k=1∞|bk|z¯k and F(z)=z-∑k=2∞|Ak|zk+∑k=1∞|Bk|z¯k, we define the convolution of two harmonic functions f and F as(4.1)(f*F)(z)=f(z)*F(z)=z-∑k=2∞|ak||Ak|zk+∑k=1∞|bk||Bk|z¯k.
Using the definition, we show that the class T𝒮Cℋ(b,γ,λ) is closed under convolution.Theorem 4.1.
For0≤δ<γ<1, let f∈T𝒮Cℋ(b,γ,λ) and F∈T𝒮Cℋ(b,δ,λ). Then, f*F∈T𝒮Cℋ(b,γ,λ)⊂T𝒮Cℋ(b,δ,λ).Proof.
Letf(z)=z-∑k=2∞|ak|zk+∑k=1∞|bk|z¯k and F(z)=z-∑k=2∞|Ak|zk+∑k=1∞|Bk|z¯k be in T𝒮Cℋ(b,δ,λ). Then, f*F∈T𝒮Cℋ(b,δ,λ). We note that |Ak|≤1 and |Bk|≤1. In view of Theorem 2.2 and the inequality 0≤δ≤γ<1, we have
(4.2)∑k=2∞k(kλ-λ+1)[(k-1)+|b|(1-δ)](1-δ)|b||ak||Ak|+∑k=1∞k(kλ+λ-1)[(k+1)-|b|(1-δ)](1-δ)|b||bk||Bk|≤∑k=2∞k(kλ-λ+1)[(k-1)+|b|(1-δ)](1-δ)|b||ak|+∑k=1∞k(kλ+λ-1)[(k+1)-|b|(1-δ)](1-δ)|b||bk|≤∑k=2∞k(kλ-λ+1)[(k-1)+|b|(1-γ)](1-γ)|b||ak|+∑k=1∞k(kλ+λ-1)[(k+1)-|b|(1-γ)](1-γ)|b||bk|≤1,
by Theorem 2.2, f∈T𝒮Cℋ(b,γ,λ). By the same token, we then conclude that f*F∈T𝒮Cℋ(b,γ,λ)⊂T𝒮Cℋ(b,δ,λ).Next, we show that the classT𝒮Cℋ(b,γ,λ) is closed under convex combination of its members.Theorem 4.2.
The classT𝒮Cℋ(b,γ,λ) is closed under convex combinations.Proof.
Suppose thatfi(z)∈T𝒮Cℋ(b,γ,λ), where fi is given by
(4.3)fi(z)=z-∑k=2∞|ai,k|zk+∑k=1∞|bi,k|z¯k,i=1,2,3,….
For ∑i=1∞ti=1, 0≤ti≤1, the convex combinations of fi may be written as
(4.4)∑i=1∞tifi(z)=z-∑k=2∞(∑i=1∞ti|ai,k|)zk+∑k=1∞(∑i=1∞ti|bi,k|)z¯k.
Since,
(4.5)∑k=2∞k(kλ-λ+1)[(k-1)+|b|(1-γ)](1-γ)|b||ai,k|+∑k=1∞k(kλ+λ-1)[(k+1)-|b|(1-γ)](1-γ)|b||bi,k|≤1,
from the above equation we obtain
(4.6)∑k=2∞k(kλ-λ+1)[(k-1)+|b|(1-γ)](1-γ)|b|∑i=1∞ti|ai,k|+∑k=1∞k(kλ+λ-1)[(k+1)-|b|(1-γ)](1-γ)|b|∑i=1∞ti|bi,k|=∑i=1∞ti{∑k=2∞k(kλ-λ+1)[(k-1)+|b|(1-γ)](1-γ)|b||ai,k|+∑k=1∞k(kλ+λ-1)[(k+1)-|b|(1-γ)](1-γ)|b||bi,k|}≤∑i=1∞ti=1.
This is the required condition by (2.14) and so ∑i=1∞tifi(z)∈T𝒮Cℋ(b,γ,λ).


## 5. Class-Preserving Integral Operator

In this section, we consider the closure property of the classT𝒮Cℋ(b,γ,λ) under the Bernardi integral operator ℒc[f(z)], which is defined by(5.1)Lc[f(z)]=c+1zc∫0zξc-1f(ξ)dξ(c>-1).Theorem 5.1.
Letf(z)∈T𝒮Cℋ(b,γ,λ); then ℒc[f(z)]∈T𝒮Cℋ(b,γ,λ).Proof.
From the representation ofℒc[f(z)], it follows that
(5.2)Lc[f(z)]=c+1zc∫0zξc-1h(ξ)dξ+c+1zc∫0zξc-1g(ξ)¯dξ=c+1zc∫0zξc-1(ξ-∑k=2∞|ak|ξk)dξ+c+1zc∫0zξc-1(∑k=1∞|bk|ξk)dξ¯=z-∑k=2∞Akzk+∑k=1∞Bkzk,
where Ak=((c+1)/(c+k))|ak|  and Bk=((c+1)/(c+k))|bk|.  Hence,
(5.3)∑k=2∞k(kλ-λ+1)[(k-1)+|b|(1-γ)](1-γ)|b|(c+1c+k|ak|)+∑k=1∞k|kλ+λ-1|[(k+1)-|b|(1-γ)](1-γ)|b|(c+1c+k|bk|)≤∑k=2∞k(kλ-λ+1)[(k-1)+|b|(1-γ)](1-γ)|b||ak|+∑k=1∞k|kλ+λ-1|[(k+1)-|b|(1-γ)](1-γ)|b||bk|≤1,
and since f∈T𝒮Cℋ(b,γ,λ), by Theorem 2.2, ℒc[f(z)]∈T𝒮Cℋ(b,γ,λ).



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*Source: 496731-2012-05-29.xml*