KYUNGPOOK Math. J. 2019; 59(4): 651-663
On A Subclass of Harmonic Multivalent Functions Deﬁned by a Certain Linear Operator
Hanan Elsayed Darwish, Abdel Moneim Yousof Lashin and Suliman Mohammed Sowileh∗
Department of Mathematics, Faculty of Science, Mansoura University, Mansoura 35516, Egypt
e-mail : Darwish333@yahoo.com, aylashin@mans.edu.eg and s_soileh@yahoo.com
* Corresponding Author.
Received: November 7, 2016; Revised: February 24, 2018; Accepted: March 9, 2018; Published online: December 23, 2019.

This is an Open Access article distributed under the terms of the Creative Commons Attribution Non-Commercial License (http://creativecommons.org/licenses/by-nc/3.0/) which permits unrestricted non-commercial use, distribution, and reproduction in any medium, provided the original work is properly cited.
Abstract

In this paper, we introduce and study a new subclass of p-valent harmonic functions defined by modified operator and obtain the basic properties such as coefficient characterization, distortion properties, extreme points, convolution properties, convex combination and also we apply integral operator for this class.

Keywords: harmonic, multivalent functions, distortion bounds, extreme points.
1. Introduction

Harmonic mappings have found several applications in many diverse fields such as operations research, engineering, and other allied branches of applied mathematics. A continous function f = u + iv is a complex-valued harmonic function in a complex domain ℂ if both u and v are real harmonic in D. In any simply connected domain D ⊂ ℂ, we can write

$f(z)=h(z)+g(z)¯,$

where h and g are analytic in D. We call h the analytic part and g the co-analytic part of f. A necessary and sufficient condition for f to be locally univalent and sense preserving in D is that |h′(z)| > |g′(z)| in D (see ). Recently, Jahangiri and Ahuja  defined the class ℋp (p ∈ ℕ = {1, 2, 3, …}), consisting of all p–valent harmonic functions f = h + that are sense preserving in the open unit disk U = {z: |z| < 1}, and h, g are of the form:

$h(z)=zp+∑k≥p+1akzk, g(z)=∑k≥pbkzk,∣bp∣<1.$

If g ≡ 0, the harmonic function f = h + reduces to an analytic function f = h. Let $Hp-$ denote the subclass of ℋp consisting of functions $fn=h+gn¯$ such that h and gn given by:

$h(z)=zp+∑k≥p+1akzk, gn(z)=(-1)n∑k≥pbkzk,∣bp∣<1.$

The class ℋ1 = ℋ of harmonic univalent functions studied by Jahangiri et al.  (see also , ). For complex parameters α1, …, αq and $β1,…,βs(βj∉ℤ0-={0,-1,-2,…}, j=1,2,…,s)$, n ∈ ℕ0 = ℕ ∪ {0}, ℕ = {1, 2, …}, ℓ, λ ≥ 0, the operator $Ip,q,s,λn,ℓ(α1)f(z)$ is defined as follows (see El-Ashwah and Aouf ):

$Ip,q,s,λn,ℓ(α1)f(z)=Ip,q,s,λn,ℓ(α1)h(z)+(-1)nIp,q,s,λn,ℓ(α1)g(z),$$Ip,q,s,λn,ℓ(α1)h(z)=zp+∑k≥p+1(p+λ(k-p)+ℓp+ℓ)nΓk(α1)akzk,$$Ip,q,s,λn,ℓ(α1)g(z)=(-1)n∑k≥p(p+λ(k-p)+ℓp+ℓ)nΓk(α1)bkzk,$

where

$Γk(α1)=(α1)k-p…(αq)k-p(β1)k-p…(βs)k-p(1)k-p,$

and (θ)ν is the Pochhammer symbol defined, in terms of the Gamma function Γ, by

$(θ)ν=Γ(θ+ν)Γ(θ)={1, if (ν=0;θ∈ℂ*=ℂ{0}),θ(θ+1)…(θ+ν-1) (ν∈ℕ;θ∈ℂ).$

For 1 < γ < 2, and for all zU, let ℋp,q,s(n, ℓ, λ, α1, γ) denote the family of harmonic p–valent functions f = h + where h and g of the form (1.2) such that

$ℜ{Ip,q,s,λn,ℓ(α1)h(z)+(-1)nIp,q,s,λn,ℓ(α1)g(z)¯zp}<γ,$

Let $Hp,q,s-(n,ℓ,λ,α1,γ)$ be the subclass of ℋp,q,s(n, ℓ, λ, α1, γ) consisting of harmonic functions $fn=h+gn¯$ so that h and gn given by (1.3).

We note that by the special choices of αi (i = 1, 2, …, q) and βj (j = 1, 2, …, s), n, ℓ and λ our class $Hp,q,s-(n,ℓ,λ,α1,γ)$ gives rise the following new subclasses of the class ℋp:

(i) For p = 1, q = s + 1, αi = 1(i = 1, …, s + 1), βj = 1 (j = 1, 2, …, s), we get $H1,s+1,s-(n,ℓ,λ,α1,γ)=H1-(n,ℓ,λ,γ)$

$={f∈H:ℜ{In(λ,ℓ)h(z)+(-1)nIn(λ,ℓ)gn(z)¯z}<γ,1<γ<2,ℓ,λ≥0,n∈ℕ0,z∈U}$

where In(λ, ℓ) is the modified Cata’s operator (see ).

(ii) For p = 1, λ = 1, ℓ = 0, q = s + 1, αi = 1(i = 1, …, s + 1), βj = 1 (j = 1, 2, …, s), we get $H1,s+1,s-(n,0,1,α1,γ)=H1-(n,γ)$

$={f∈H:ℜ{Dnh(z)+(-1)nDngn(z)¯z}<γ,1<γ<2,n∈ℕ0,z∈U},$

where Dn is the modified Salagean operator (see ), the differential opertor Dn was introduced by Salagean (see );

(iii) For p = 1, λ = 1, ℓ = 1, q = s + 1, αi = 1(i = 1, …, s + 1), βj = 1 (j = 1, 2, …, s), we get $H1,s+1,s-(n,1,1,α1,γ)=H1-(n,γ)$

$={f∈H:ℜ{Inh(z)+(-1)nIngn(z)¯z}<γ,1<γ<2,n∈ℤ={0,±1,±2,…},z∈U},$

where In is the modified Uralegaddi-Somanatha operator (see ), defined as follows:

$Inf(z)=Inh(z)+(-1)nIngn(z)¯.$

(iv) For p = 1, λ = 1, q = s + 1, αi = 1(i = 1, …, s + 1), βj = 1 (j = 1, 2, …, s), we get $H1,s+1,s-(n,ℓ,1,α1,γ)=H1-(n,ℓ,γ)$

$={f∈H:ℜ{Iℓnh(z)+(-1)nIℓngn(z)¯z}<γ,1<γ<2,n∈ℝ,ℓ>-1,z∈U},$

where $Iℓn$ is the modified Cho-Kim operator  (also see ), defined as follows:$Iℓnf(z)=Iℓnh(z)+(-1)nIℓngn(z)¯$.

(v) For p = 1, ℓ = 0, q = s + 1, αi = 1(i = 1, …, s + 1), βj = 1 (j = 1, 2, …, s), we get $H1,s+1,s-(n,0,λ,α1,γ)=H1-(n,λ,γ)$

$={f∈H:ℜ{Dλnh(z)+(-1)nDλngn(z)¯z}<γ,1<γ<2,λ≥0,n∈ℕ0,z∈U},$

where $Dλn$ is the modified Al-Oboudi operator (see ), defined as follows:

$Dλnf(z)=Dλnh(z)+(-1)nDλngn(z).$
2. Coefficient Characterization

Unless otherwise mentioned, we assume throughout this article that 1 < γ ≤ 2, ℓ > −p, p ∈ ℕ, λ ≥ 0, n ∈ ℕ0 and Γk(α1) is given by (1.7). In our first theorem, we introduce a sufficient condition for the coefficient bounds of harmonic functions in ℋp,q,s(n, ℓ, λ, α1, γ).

### Theorem 2.1

Let f = h + ḡ where h and g are of the form (1.2). Then f ∈ ℋp,q,s(n, ℓ, λ, α1, γ) if

$∑k≥p+1(p+λ(k-p)+ℓp+ℓ)n∣Γk(α1)ak∣+∑k≥p(p+λ(k-p)+ℓp+ℓ)n∣Γk(α1)bk∣≤γ-1$

where ap = 1.

Proof

Using the fact that ℜ{w(z)} < γ iff |w(z) − 1| < |w(z) – (2γ − 1)|, it suffices to show that

$|Ip,q,s,λn,ℓ(α1)h(z)+(-1)nIp,q,s,λn,ℓ(α1)g(z)¯zp-1Ip,q,s,λn,ℓ(α1)h(z)+(-1)nIp,q,s,λn,ℓ(α1)g(z)¯zp-(2γ-1)|<1.$

We have

$|Ip,q,s,λn,ℓ(α1)h(z)+(-1)nIp,q,s,λn,ℓ(α1)g(z)¯zp-1Ip,q,s,λn,ℓ(α1)h(z)+(-1)nIp,q,s,λn,ℓ(α1)g(z)¯zp-(2γ-1)|=|∑k≥p+1(p+λ(k-p)+ℓp+ℓ)nΓk(α1)akzk-p+(-1)n∑k≥p(p+λ(k-p)+ℓp+ℓ)nΓk(α1)bkzk-p¯-2(γ-1)+∑k≥p+1(p+λ(k-p)+ℓp+ℓ)nΓk(α1)akzk-p+(-1)n∑k≥p(p+λ(k-p)+ℓp+ℓ)nΓk(α1)bkzk-p¯|≤[∑k≥p+1(p+λ(k-p)+ℓp+ℓ)n∣Γk(α1)ak∣ ∣zk-p∣+∑k≥p(p+λ(k-p)+ℓp+ℓ)n∣Γk(α1)bk∣ ∣zk-p∣][2(γ-1)-∑k≥p+1(p+λ(k-p)+ℓp+ℓ)n∣Γk(α1)ak∣ ∣zk-p∣-∑k≥p(p+λ(k-p)+ℓp+ℓ)n∣Γk(α1)bk∣ ∣zk-p∣]<[∑k≥p+1(p+λ(k-p)+ℓp+ℓ)n∣Γk(α1)ak∣+∑k≥p(p+λ(k-p)+ℓp+ℓ)n∣Γk(α1)bk∣][2(γ-1)-∑k≥p+1(p+λ(k-p)+ℓp+ℓ)n∣Γk(α1)ak∣-∑k≥p(p+λ(k-p)+ℓp+ℓ)n∣Γk(α1)bk∣]≤1,$

which is bounded above by 1 by using (2.1). This completes the proof of Theorem 2.1.

### Theorem 2.2

Let$fn=h+gn¯$where h and gn are of the form (1.3). Then$fn∈Hp,q,s-(n,ℓ,λ,α1,γ)$if and only if

$∑k≥p+1(p+λ(k-p)+ℓp+ℓ)n∣Γk(α1)ak∣+∑k≥p(p+λ(k-p)+ℓp+ℓ)n∣Γk(α1)bk∣≤γ-1$

where ap = 1.

Proof

Since $Hp,q,s-(n,ℓ,λ,α1,γ)⊆Hp,q,s(n,ℓ,λ,α1,γ)$, we only need to prove the ”only if” part of this theorem. For functions fn(z) of the form (1.3), the condition

$ℜ{Ip,q,s,λn,ℓ(α1)h(z)+(-1)nIp,q,s,λn,ℓ(α1)g(z)¯zp}<γ$

is equivalent to

$ℜ{1+∑k≥p+1(p+λ(k-p)+ℓp+ℓ)nΓk(α1)akzk-p+(-1)2n∑k≥p(p+λ(k-p)+ℓp+ℓ)nΓk(α1)bkzk-p¯}≤1+∑k≥p+1(p+λ(k-p)+ℓp+ℓ)n∣Γk(α1)ak∣∣zk-p∣+∑k≥p(p+λ(k-p)+ℓp+ℓ)n∣Γk(α1)bk∣∣zk-p∣¯<γ.$

Letting z → 1, we obtain the inequality (2.1), and so the proof of Theorem 2.2 is completed.

### Remark 2.1

(i) If p = 1, q = s + 1, αi = 1(i = 1, …, s + 1) and βj = 1 (j = 1, 2, …, s) in Theorem 2.2, then we get the result obtained by Mostafa et al. [14, Theorem 2].

(ii) If λ = 1, ℓ = 0, p = 1, q = s + 1, αi = 1(i = 1, …, s + 1), βj = 1 (j = 1, 2, …, s) and n = 1, in Theorem 2.2, then we get the result obtained by Dixit and Porwal [7, Theorem 2.1].

3. Extreme Points and Distortion Theorem

In the following theorem we give the extreme points of the closed convex hulls of the class $Hp,q,s-(n,ℓ,λ,α1,γ)$ denoted by $Hp,q,s-(n,ℓ,λ,α1,γ)$.

### Theorem 3.1

Let$fn=h+gn¯$where h and gn are of the form (1.3). Then$fn∈clcoHp,q,s-(n,ℓ,λ,α1,γ)$if and only if

$fn(z)=∑k≥pμkhk(z)+ηkgkn(z),$

where

$hp(z)=zp,$$hk(z)=zp+(γ-1)∣Γk(α1)∣ [(p+ℓ)p+λ(k-p)+ℓ]nzk (k≥p+1,n∈ℕ0)$

and

$gkn(z):=zp+(-1)n(γ-1)∣Γk(α1)∣ [(p+ℓ)p+λ(k-p)+ℓ]nz¯k (k≥p,n∈ℕ0)μk,ηk≥0, μp=1-∑k≥p+1μk-∑k≥pηk.$

In particular, the extreme points of the class$Hp,q,s-(n,ℓ,λ,α1,γ)$are {hk(z)} and {gkn(z)}.

Proof

Suppose that

$fn(z)=∑k≥p(μkhk(z)+ηkgkn(z))=zp+∑k≥p+1(γ-1)∣Γk(α1)∣ [(p+ℓ)p+λ(k-p)+ℓ]nμkzk+(-1)n∑k≥p(γ-1)∣Γk(α1)∣ [(p+ℓ)p+λ(k-p)+ℓ]nηkz¯k.$

Then

$∑k≥p+1[p+λ(k-p)+ℓ]n∣Γk(α1)∣(γ-1) [p+ℓ]n ((γ-1) [p+ℓ]n[p+λ(k-p)+ℓ]n∣Γk(α1)∣μk)+∑k≥p[p+λ(k-p)+ℓ]n∣Γk(α1)∣(γ-1) [p+ℓ]n ((γ-1) [p+ℓ]n[p+λ(k-p)+ℓ]n∣Γk(α1)∣ηk)=∑k≥p+1μk+∑k≥pηk=1-μp≤1$

and so $fn∈clcoHp,q,s-(n,ℓ,λ,α1,γ)$.

Conversely, if $fn∈clcoHp,q,s-(n,ℓ,λ,α1,γ)$. Set

$μk=∣Γk(α1)∣ [p+λ(k-p)+ℓ]n(γ-1) [p+ℓ]nak, (k≥p+1),$

and

$ηk=∣Γk(α1)∣ [p+λ(k-p)+ℓ]n(γ-1) [p+ℓ]nbk, (k≥p).$

Then note that by Theorem 2.2, 0 ≤ μk ≤ 1, (kp + 1), and 0 ≤ ηk ≤ 1, (kp). Let $μp=1-∑k≥p+1μk-∑k≥pηk$ and μp ≥ 0.

The required representation is obtained as

$fn(z)=zp+∑k≥p+1akzk+(-1)n∑k≥pbkz¯k=zp+∑k≥p+1(γ-1) [p+ℓ]n[p+λ(k-p)+ℓ]n∣Γk(α1)∣μkzk+(-1)n∑k≥p(γ-1) [p+ℓ]n[p+λ(k-p)+ℓ]n∣Γk(α1)∣ηkz¯k=zp+∑k≥p+1(hk(z)-zp)μk+∑k≥p(gk(z)-zp)ηk=(1-∑k≥p+1μk-∑k≥pηk)zp+∑k≥p+1hk(z)μk+∑k≥p(gk(z)ηk=∑k≥p(μkhk(z)+ηkgk(z)).$

This completes the proof of Theorem 3.1.

The following theorem gives the distortion bounds for functions in the class $Hp,q,s-(n,ℓ,λ,α1,γ)$ which yields a covering result for this class.

### Theorem 3.2

Let$fn∈Hp,q,s-(n,ℓ,λ,α1,γ)$with |bp| < γ − 1. Then for |z| = r < 1, we have

$(1-∣bp∣)rp-[p+ℓ]n∣Γp+1(α1)∣ [p+λ+ℓ]n{γ-1-∣bp∣}rp+1≤∣fn(z)∣≤(1+∣bp∣)rp+[p+ℓ]n∣Γp+1(α1)∣ [p+λ+ℓ]n{γ-1-∣bp∣}rp+1$
Proof

Let $fn(z)∈Hp,q,s-(n,ℓ,λ,α1,γ)$. Taking the absolute value of fn(z) we have

$∣fn(z)∣≤(1+∣bp∣)rp+∑k≥p+1[∣ak∣+∣bk∣]rk≤(1+∣bp∣)rp+rp+1∑k≥p+1[∣ak∣+∣bk∣]=(1+∣bp∣)rp+(γ-1) [p+ℓ]n[p+λ+ℓ]n ∣Γp+1(α1)∣rp+1∑k≥p+1[p+λ+ℓ]n∣Γp+1(α1)∣(γ-1) [p+ℓ]n[∣ak∣+∣bk∣]≤(1+∣bp∣)rp+(γ-1) [p+ℓ]n[p+λ+ℓ]n ∣Γp+1(α1)∣rp+1{∑k≥p+1[p+λ(k-p)+ℓ]n(γ-1) [p+ℓ]n∣Γk(α1)ak∣+∑k≥p+1[p+λ(k-p)+ℓ]n(γ-1) [p+ℓ]n∣Γk(α1)bk∣}≤(1+∣bp∣)rp+(γ-1) [p+ℓ]n[p+λ+ℓ]n ∣Γp+1(α1)∣ {1-∣bp∣(γ-1)}rp+1=(1+∣bp∣)rp+[p+ℓ]n[p+λ+ℓ]n ∣Γp+1(α1)∣ {γ-1-∣bp∣}rp+1.$

Similarly we can prove

$∣f(z)∣≥(1-∣bp∣)rp-[p+ℓ]n∣Γp+1(α1)∣ [p+λ+ℓ]n{γ-1-∣bp∣}rp+1.$

### Remark 2.2

The bounds given in Theorem 3.2 for functions $fn=h+gn¯$, where h and gn are given by (1.3), also hold for functions of the form f = h + g, where h and g are given by (1.2) if the coefficient condition (2.1) is satisfied. The upper bound given for $f(z)∈Hp,q,s-(n,ℓ,λ,α1,γ)$ is sharp and the equality occurs for the functions

$∣f(z)∣=zp+∣bp∣z¯p+[p+ℓ]n∣Γp+1(α1)∣ [p+λ+ℓ]n{γ-1-∣bp∣}z¯p+1$

and

$∣f(z)∣=zp-∣bp∣z¯p-[p+ℓ]n∣Γp+1(α1)∣ [p+λ+ℓ]n{γ-1-∣bp∣}zp+1,$

showing that the bounds given in Theorem 3.2 are sharp.

4. Closure Property of the Class $Hp,q,s-(n,ℓ,λ,α1,γ)$

In the next two theorems, we prove that the class $Hp,q,s-(n,ℓ,λ,α1,γ)$ is invariant under convolution and convex combinations of its members. The convolution of two harmonic functions,

$fn(z)=zp+∑k≥p+1akzk+(-1)n∑k≥pbkz¯k,$

and

$Fn(z)=zp+∑k≥p+1Akzk+(-1)n∑k≥pBkz¯k,$

is defined as

$(fn*Fn) (z)=(Fn*fn) (z)=zp+∑k≥p+1akAkzk+(-1)n∑k≥pbkBkz¯k.$

Using this definition, the next theorem shows that the class $Hp,q,s-(n,ℓ,λ,α1,γ)$ is closed under convolution.

### Theorem 4.1

For 1 < βγ ≤ 2, Let$fn∈Hp,q,s-(n,ℓ,λ,α1,γ)$and$Fn∈Hp,q,s-(n,ℓ,λ,α1,β)$. Then

$fn*Fn∈Hp,q,s-(n,ℓ,λ,α1,γ)⊆Hp,q,s-(n,ℓ,λ,α1,β).$
Proof

Let the functions fn(z) defined by (4.1) be in $Hp,q,s-(n,ℓ,λ,α1,γ)$ and the functions Fn(z) defined by (4.2) be in $Hp,q,s-(n,ℓ,λ,α1,β)$. Then the convolution fn *Fn is given by (4.3). We wish to show that the coefficients of fn *Fn satisfy the required condition given in Theorem 2.2. For $Fn(z)∈Hp,q,s-(n,ℓ,λ,α1,β)$, we note that |Ak| < 1 and |Bk| < 1. Now for the convolution function fn * Fn, we obtain

$∑k≥p+11β-1(p+λ(k-p)+ℓp+ℓ)n∣Γk(α1)ak∣ ∣Ak∣+∑k≥p(p+λ(k-p)+ℓp+ℓ)n∣Γk(α1)bk∣ ∣Bk∣≤∑k≥p+11β-1(p+λ(k-p)+ℓp+ℓ)n∣Γk(α1)ak∣+∑k≥p(p+λ(k-p)+ℓp+ℓ)n∣Γk(α1)bk∣≤∑k≥p+11γ-1(p+λ(k-p)+ℓp+ℓ)n∣Γk(α1)ak∣+∑k≥p(p+λ(k-p)+ℓp+ℓ)n∣Γk(α1)bk∣≤1,$

since 1 < βγ ≤ 2, and $fn∈Hp,q,s-(n,ℓ,λ,α1,γ)$.

Now, we show that the class $Hp,q,s-(n,ℓ,λ,α1,γ)$ is closed under convex combination of its members.

### Theorem 4.2

The family$Hp,q,s-(n,ℓ,λ,α1,γ)$is closed under convex combination.

Proof

For i = 1, 2, 3, …, suppose $fni∈Hp,q,s-(n,ℓ,λ,α1,γ)$, where fni is given by

$fni(z)=zp+∑k≥p+1akizk+(-1)n∑k≥pbkiz-k.$

Then by (2.2), we have

$∑k≥p+11(γ-1) (p+λ(k-p)+ℓp+ℓ)n∣Γk(α1)aki∣+∑k≥p1(γ-1) (p+λ(k-p)+ℓp+ℓ)n∣Γk(α1)bki∣≤1.$

For $∑i=1∞ti=1$ 0 ≤ ti ≤ 1, the convex combination of fni may be written as

$∑i=1∞tifi=zp+∑k≥p+1(∑i=1∞tiaki) zk+(-1)n∑k≥p(∑i=1∞tibki)z-k.$

Using the inequality (4.4), we have

$∑k≥p+11(γ-1) (p+λ(k-p)+ℓp+ℓ)n∣Γk(α1)∣ (∑i=1∞ti∣aki∣)+∑k≥p1(γ-1) (p+λ(k-p)+ℓp+ℓ)n∣Γk(α1)∣ (∑i=1∞ti∣bki∣)=∑i=1∞ti(∑k≥p+11(γ-1) (p+λ(k-p)+ℓp+ℓ)n∣Γk(α1)∣ ∣aki∣+∑k≥p1(γ-1) (p+λ(k-p)+ℓp+ℓ)n∣Γk(α1)∣ ∣bki∣)≤∑i=1∞ti=1,$

which is the required coefficient condition.

Finally, we examine the closure property of the class $Hp,q,s-(n,ℓ,λ,α1,γ)$ under the generalized Bernardi-Libera-Livingston integral operator (see [2, 13]), Ic(f) which is defined by,

$Ic(f)=c+pzc∫0ztc-1f(t)dt,c>-p.$

### Theorem 4.3

Let$fn∈Hp,q,s-(n,ℓ,λ,α1,γ)$. Then

$Ic(fn(z))∈Hp,q,s-(n,ℓ,λ,α1,γ).$
Proof

From the representation of Ic (fn(z)), it follows that

$Ic(fn(z))=c+pzc∫0ztc-1[tp+∑k≥p+1aktk+(-1)n∑k≥pbktk¯]dt=zp+∑k≥p+1Φkzk+(-1)n∑k≥pΨkzk¯,$

where

$Φk=(c+pc+k)ak and Ψk=(c+pc+k) bk.$

Therefore, we have

$∑k≥p+1(p+λ(k-p)+ℓp+ℓ)n∣Γk(α1)∣ ∣Φk∣+∑k≥p(p+λ(k-p)+ℓp+ℓ)n∣Γk(α1)∣ ∣Ψk∣=∑k≥p+1(p+λ(k-p)+ℓp+ℓ)n∣Γk(α1)∣ (c+pc+k)∣ak∣+∑k≥p(p+λ(k-p)+ℓp+ℓ)n∣Γk(α1)∣ (c+pc+k)∣bk∣≤∑k≥p+1(p+λ(k-p)+ℓp+ℓ)n∣Γk(α1)∣ ∣ak∣+∑k≥p(p+λ(k-p)+ℓp+ℓ)n∣Γk(α1)∣ ∣bk∣≤(γ-1) by (2.2).$

Hence by Theorem 2.2, $Ic(fn(z))∈Hp,q,s-(n,ℓ,λ,α1,γ)$.

Acknowledgements

The authors wish to acknowledge and thank both the reviewers and editors for job well-done reviewing this article.

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