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.
© Kyungpook Mathematical Journal. All rights reserved.

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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 [5]). Recently, Jahangiri and Ahuja [9] 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+kp+1akzk,g(z)=kpbkzk,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+kp+1akzk,gn(z)=(-1)nkpbkzk,bp<1.

The class ℋ1 = ℋ of harmonic univalent functions studied by Jahangiri et al. [10] (see also [6], [12]). For complex parameters α1, …, αq and β1,,βs(βj0-={0,-1,-2,},j=1,2,,s), n ∈ ℕ0 = ℕ ∪ {0}, ℕ = {1, 2, …}, ℓ, λ ≥ 0, the operator Ip,q,s,λn,(α1)f(z) is defined as follows (see El-Ashwah and Aouf [8]):

Ip,q,s,λn,(α1)f(z)=Ip,q,s,λn,(α1)h(z)+(-1)nIp,q,s,λn,(α1)g(z),Ip,q,s,λn,(α1)h(z)=zp+kp+1(p+λ(k-p)+p+)nΓk(α1)akzk,Ip,q,s,λn,(α1)g(z)=(-1)nkp(p+λ(k-p)+p+)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*={0}),θ(θ+1)(θ+ν-1)   (ν;θ).

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

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

Let Hp,q,s-(n,,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,,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,,λ,α1,γ)=H1-(n,,λ,γ)

={fH:{In(λ,)h(z)+(-1)nIn(λ,)gn(z)¯z}<γ,1<γ<2,,λ0,n0,zU}

where In(λ, ℓ) is the modified Cata’s operator (see [14]).

(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)

={fH:{Dnh(z)+(-1)nDngn(z)¯z}<γ,1<γ<2,n0,zU},

where Dn is the modified Salagean operator (see [11]), the differential opertor Dn was introduced by Salagean (see [15]);

(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)

={fH:{Inh(z)+(-1)nIngn(z)¯z}<γ,1<γ<2,n={0,±1,±2,},zU},

where In is the modified Uralegaddi-Somanatha operator (see [16]), 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,,1,α1,γ)=H1-(n,,γ)

={fH:{Inh(z)+(-1)nIngn(z)¯z}<γ,1<γ<2,n,>-1,zU},

where In is the modified Cho-Kim operator [3] (also see [4]), defined as follows:Inf(z)=Inh(z)+(-1)nIngn(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)

={fH:{Dλnh(z)+(-1)nDλngn(z)¯z}<γ,1<γ<2,λ0,n0,zU},

where Dλn is the modified Al-Oboudi operator (see [1]), 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

kp+1(p+λ(k-p)+p+)nΓk(α1)ak+kp(p+λ(k-p)+p+)nΓk(α1)bkγ-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,(α1)h(z)+(-1)nIp,q,s,λn,(α1)g(z)¯zp-1Ip,q,s,λn,(α1)h(z)+(-1)nIp,q,s,λn,(α1)g(z)¯zp-(2γ-1)|<1.

We have

|Ip,q,s,λn,(α1)h(z)+(-1)nIp,q,s,λn,(α1)g(z)¯zp-1Ip,q,s,λn,(α1)h(z)+(-1)nIp,q,s,λn,(α1)g(z)¯zp-(2γ-1)|=|kp+1(p+λ(k-p)+p+)nΓk(α1)akzk-p+(-1)nkp(p+λ(k-p)+p+)nΓk(α1)bkzk-p¯-2(γ-1)+kp+1(p+λ(k-p)+p+)nΓk(α1)akzk-p+(-1)nkp(p+λ(k-p)+p+)nΓk(α1)bkzk-p¯|[kp+1(p+λ(k-p)+p+)nΓk(α1)akzk-p+kp(p+λ(k-p)+p+)nΓk(α1)bkzk-p][2(γ-1)-kp+1(p+λ(k-p)+p+)nΓk(α1)akzk-p-kp(p+λ(k-p)+p+)nΓk(α1)bkzk-p]<[kp+1(p+λ(k-p)+p+)nΓk(α1)ak+kp(p+λ(k-p)+p+)nΓk(α1)bk][2(γ-1)-kp+1(p+λ(k-p)+p+)nΓk(α1)ak-kp(p+λ(k-p)+p+)nΓk(α1)bk]1,

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

Theorem 2.2

Letfn=h+gn¯where h and gn are of the form (1.3). ThenfnHp,q,s-(n,,λ,α1,γ)if and only if

kp+1(p+λ(k-p)+p+)nΓk(α1)ak+kp(p+λ(k-p)+p+)nΓk(α1)bkγ-1

where ap = 1.

Proof

Since Hp,q,s-(n,,λ,α1,γ)Hp,q,s(n,,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

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

is equivalent to

{1+kp+1(p+λ(k-p)+p+)nΓk(α1)akzk-p+(-1)2nkp(p+λ(k-p)+p+)nΓk(α1)bkzk-p¯}1+kp+1(p+λ(k-p)+p+)nΓk(α1)akzk-p+kp(p+λ(k-p)+p+)nΓk(α1)bkzk-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,,λ,α1,γ) denoted by Hp,q,s-(n,,λ,α1,γ).

Theorem 3.1

Letfn=h+gn¯where h and gn are of the form (1.3). ThenfnclcoHp,q,s-(n,,λ,α1,γ)if and only if

fn(z)=kpμkhk(z)+ηkgkn(z),

where

hp(z)=zp,hk(z)=zp+(γ-1)Γk(α1)[(p+)p+λ(k-p)+]nzk         (kp+1,n0)

and

gkn(z):=zp+(-1)n(γ-1)Γk(α1)   [(p+)p+λ(k-p)+]nz¯k         (kp,n0)μk,ηk0,         μp=1-kp+1μk-kpηk.

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

Proof

Suppose that

fn(z)=kp(μkhk(z)+ηkgkn(z))=zp+kp+1(γ-1)Γk(α1)   [(p+)p+λ(k-p)+]nμkzk+(-1)nkp(γ-1)Γk(α1)   [(p+)p+λ(k-p)+]nηkz¯k.

Then

kp+1[p+λ(k-p)+]nΓk(α1)(γ-1)   [p+]n   ((γ-1)   [p+]n[p+λ(k-p)+]nΓk(α1)μk)+kp[p+λ(k-p)+]nΓk(α1)(γ-1)   [p+]n   ((γ-1)   [p+]n[p+λ(k-p)+]nΓk(α1)ηk)=kp+1μk+kpηk=1-μp1

and so fnclcoHp,q,s-(n,,λ,α1,γ).

Conversely, if fnclcoHp,q,s-(n,,λ,α1,γ). Set

μk=Γk(α1)[p+λ(k-p)+]n(γ-1)[p+]nak,(kp+1),

and

ηk=Γk(α1)[p+λ(k-p)+]n(γ-1)[p+]nbk,(kp).

Then note that by Theorem 2.2, 0 ≤ μk ≤ 1, (kp + 1), and 0 ≤ ηk ≤ 1, (kp). Let μp=1-kp+1μk-kpηk and μp ≥ 0.

The required representation is obtained as

fn(z)=zp+kp+1akzk+(-1)nkpbkz¯k=zp+kp+1(γ-1)   [p+]n[p+λ(k-p)+]nΓk(α1)μkzk+(-1)nkp(γ-1)[p+]n[p+λ(k-p)+]nΓk(α1)ηkz¯k=zp+kp+1(hk(z)-zp)μk+kp(gk(z)-zp)ηk=(1-kp+1μk-kpηk)zp+kp+1hk(z)μk+kp(gk(z)ηk=kp(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,,λ,α1,γ) which yields a covering result for this class.

Theorem 3.2

LetfnHp,q,s-(n,,λ,α1,γ)with |bp| < γ − 1. Then for |z| = r < 1, we have

(1-bp)rp-[p+]nΓp+1(α1)[p+λ+]n{γ-1-bp}rp+1fn(z)(1+bp)rp+[p+]nΓp+1(α1)[p+λ+]n{γ-1-bp}rp+1
Proof

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

fn(z)(1+bp)rp+kp+1[ak+bk]rk(1+bp)rp+rp+1kp+1[ak+bk]=(1+bp)rp+(γ-1)[p+]n[p+λ+]nΓp+1(α1)rp+1kp+1[p+λ+]nΓp+1(α1)(γ-1)[p+]n[ak+bk](1+bp)rp+(γ-1)[p+]n[p+λ+]nΓp+1(α1)rp+1{kp+1[p+λ(k-p)+]n(γ-1)[p+]nΓk(α1)ak+kp+1[p+λ(k-p)+]n(γ-1)[p+]nΓk(α1)bk}(1+bp)rp+(γ-1)[p+]n[p+λ+]nΓp+1(α1){1-bp(γ-1)}rp+1=(1+bp)rp+[p+]n[p+λ+]nΓp+1(α1){γ-1-bp}rp+1.

Similarly we can prove

f(z)(1-bp)rp-[p+]nΓp+1(α1)[p+λ+]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,,λ,α1,γ) is sharp and the equality occurs for the functions

f(z)=zp+bpz¯p+[p+]nΓp+1(α1)[p+λ+]n{γ-1-bp}z¯p+1

and

f(z)=zp-bpz¯p-[p+]nΓp+1(α1)[p+λ+]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,,λ,α1,γ)

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

fn(z)=zp+kp+1akzk+(-1)nkpbkz¯k,

and

Fn(z)=zp+kp+1Akzk+(-1)nkpBkz¯k,

is defined as

(fn*Fn)(z)=(Fn*fn)(z)=zp+kp+1akAkzk+(-1)nkpbkBkz¯k.

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

Theorem 4.1

For 1 < βγ ≤ 2, LetfnHp,q,s-(n,,λ,α1,γ)andFnHp,q,s-(n,,λ,α1,β). Then

fn*FnHp,q,s-(n,,λ,α1,γ)Hp,q,s-(n,,λ,α1,β).
Proof

Let the functions fn(z) defined by (4.1) be in Hp,q,s-(n,,λ,α1,γ) and the functions Fn(z) defined by (4.2) be in Hp,q,s-(n,,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,,λ,α1,β), we note that |Ak| < 1 and |Bk| < 1. Now for the convolution function fn * Fn, we obtain

kp+11β-1(p+λ(k-p)+p+)nΓk(α1)akAk+kp(p+λ(k-p)+p+)nΓk(α1)bkBkkp+11β-1(p+λ(k-p)+p+)nΓk(α1)ak+kp(p+λ(k-p)+p+)nΓk(α1)bkkp+11γ-1(p+λ(k-p)+p+)nΓk(α1)ak+kp(p+λ(k-p)+p+)nΓk(α1)bk1,

since 1 < βγ ≤ 2, and fnHp,q,s-(n,,λ,α1,γ).

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

Theorem 4.2

The familyHp,q,s-(n,,λ,α1,γ)is closed under convex combination.

Proof

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

fni(z)=zp+kp+1akizk+(-1)nkpbkiz-k.

Then by (2.2), we have

kp+11(γ-1)(p+λ(k-p)+p+)nΓk(α1)aki+kp1(γ-1)(p+λ(k-p)+p+)nΓk(α1)bki1.

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

i=1tifi=zp+kp+1(i=1tiaki)zk+(-1)nkp(i=1tibki)z-k.

Using the inequality (4.4), we have

kp+11(γ-1)(p+λ(k-p)+p+)nΓk(α1)(i=1tiaki)+kp1(γ-1)(p+λ(k-p)+p+)nΓk(α1)(i=1tibki)=i=1ti(kp+11(γ-1)(p+λ(k-p)+p+)nΓk(α1)aki+kp1(γ-1)(p+λ(k-p)+p+)nΓk(α1)bki)i=1ti=1,

which is the required coefficient condition.

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

Ic(f)=c+pzc0ztc-1f(t)dt,c>-p.

Theorem 4.3

LetfnHp,q,s-(n,,λ,α1,γ). Then

Ic(fn(z))Hp,q,s-(n,,λ,α1,γ).
Proof

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

Ic(fn(z))=c+pzc0ztc-1[tp+kp+1aktk+(-1)nkpbktk¯]dt=zp+kp+1Φkzk+(-1)nkpΨkzk¯,

where

Φk=(c+pc+k)ak         and         Ψk=(c+pc+k)bk.

Therefore, we have

kp+1(p+λ(k-p)+p+)nΓk(α1)Φk+kp(p+λ(k-p)+p+)nΓk(α1)Ψk=kp+1(p+λ(k-p)+p+)nΓk(α1)(c+pc+k)ak+kp(p+λ(k-p)+p+)nΓk(α1)(c+pc+k)bkkp+1(p+λ(k-p)+p+)nΓk(α1)ak+kp(p+λ(k-p)+p+)nΓk(α1)bk(γ-1)by(2.2).

Hence by Theorem 2.2, Ic(fn(z))Hp,q,s-(n,,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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