# Inclusion Relation between Various Subclasses of Harmonic Univalent Functions Associated with Wright’s Generalized Hypergeometric Functions

Authors: Rajavadivelu Themangani; Saurabh Porwal; Nanjundan Magesh
Journal: Abstract and Applied Analysis
 (2020)

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## Abstract

The purpose of the present paper is to obtain some inclusion relation between various subclasses of harmonic univalent functions by applying certain convolution operators associated with Wright’s generalized hypergeometric functions.


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## Body

## 1. Introduction

A continuous complex-valued functionf=u+iv defined in a simply connected domain D is said to be harmonic in D if both u and v are real harmonic in D. In any simply connected domain D, we can write f=h+g¯, where h and g are analytic in D. In 1984, Clunie and Sheil-Small [1] introduced a class SH of complex-valued harmonic maps f which are univalent and sense-preserving in the open unit disk U=z:z∈ℂandz<1. The function f∈SH can be represented by f=h+g¯, where
(1)hz=z+∑n=2∞hnzn,gz=∑n=1∞gnzn,∣g1∣<1,are analytic in the open unit disk U. They also proved that the function f=h+g¯∈SH is locally univalent and sense-preserving in U, if and only if h′z>g′z,∀z∈U. For more basic studies, one may refer to Duren [2] and Ahuja [3]. It is worthy to note that if gz≡0 in (1), then the class SH reduces to the familiar class S of analytic functions. For this class, fz may be expressed as of the form
(2)fz=z+∑n=2∞hnzn.Further, we supposeSH0 subclass of SH consisting of function f∈SH of the form (1) with g1=0. Now, we let KH0, SH∗,0, and CH0 denote the subclasses of SH0 of harmonic functions which are, respectively, convex, starlike, and close-to-convex in U. Also, let TH0 be the class of sense-preserving, typically real harmonic functions f=h+g¯ in SH. For a detailed study of these classes, one may refer to [1, 2].A functionf=h+g¯ of the form (1) is said to be in the class NHγ, if it satisfy the condition
(3)Rf′zz′≥γ,0≤γ<1,z=reiθ∈U.Similarly, a functionf=h+g¯ of the form (1) is said to be in the class GHγ, if it satisfy the condition
(4)R1+eiαzf′zfz−eiα≥γ,0≤γ<1,α∈ℝ,z=reiθ∈U,where z′=∂/∂θreiθ and f′z=∂/∂θfγeiθ.Now, we define the subclassTSH of SH consisting of functions f=h+g¯, so that h and g are of the form
(5)hz=z−∑n=2∞hnzn,gz=∑n=1∞∣gn∣zn.DefineTNHγ=NHγ∩T and TGHγ=GHγ∩T, where T consists of the functions f=h+g¯ in SH. The classes NHγ, TNHγ, GHγ, and TGHγ, were studied, respectively, by Ahuja and Jahangiri [4] and Rosy et al. [5].Letai∈ℂ, ai/Ai≠0,−1,−2,⋯;i=1,2,⋯,p and bi/Bi≠0,−1,−2,⋯;i=1,2,⋯,q, for Ai>0i=1,⋯,p and Bi>0i=1,⋯,q with
(6)1+∑i=1qBi−∑i=1pAi≥0.Wright’s generalized hypergeometric functions [6] is defined by
(7)pΨqai,Ai1,pbi,Bi1,q;z=∑n=0∞∏i=1pΓai+nAizn∏i=1qΓbi+nBin!,which is analytic for suitable bounded values of ∣z∣ (see also [7, 8]). The generalized Mittag-Leffler, Bessel-Maitland, and generalized hypergeometric functions are some of the important special cases of Wright’s generalized hypergeometric functions, and for their details, one may refer to [8].ForAi>0i=1,⋯,p,Bi>0,bi>0i=1,⋯,q with 1+∑i=1qBi−∑i=1pAi≥0 and Ci>0i=1,⋯,r,Di>0,di>0i=1,⋯,s with 1+∑i=1sDi−∑i=1rCi≥0, we define Wright’s generalized hypergeometric functions:
(8)pΨqai,Ai1,pbi,Bi1,q;z=∑n=0∞∏i=1pΓai+nAizn∏i=1qΓbi+nBin!,rΨsci,Ci1,rdi,Di1,s;z=∑n=0∞∏i=1rΓci+nCizn∏i=1sΓdi+nDin!,with
(9)∏i=1rΓ∣ci∣+nCi/Γ∣ci∣∏i=1sΓdi+nDi/Γdi<1.We consider a harmonic univalent function(10)Wz=Hz+Gz¯∈SH,where
(11)Hz=z∏i=1qΓbi∏i=1pΓaipΨqai,Ai1,pbi,Bi1,q;z=z+∑n=2∞θnzn,Gz=σz∏i=1sΓdi∏i=1rΓcirΨsci,Ci1,rdi,Di1,s;z=σ∑n=1∞ζnzn,∣σ∣<1,and θn and ζn are given by
(12)θn=∏i=1pΓai+n−1Ai/Γai∏i=1qΓbi+n−1Bi/Γbin−1!,ζn=∏i=1rΓci+n−1Ci/Γci∏i=1sΓdi+n−1Di/Γdin−1!.From (12), we have for n∈ℕ=1,2,⋯(13)θn≤∏i=1pΓai+n−1Ai/Γai∏i=1qΓbi+n−1Bi/Γbin−1!=νn,ζn≤∏i=1rΓci+n−1Ci/Γci∏i=1sΓdi+n−1Di/Γdin−1!=ηn.For some fixed value ofj∈ℕ0=ℕ∪0 and for
(14)∏i=1qBiBi≥∏i=1pAiAi,∏i=1sDiDi≥∏i=1rCiCi,we denote
(15)pΨq∣ai∣+jAi,Ai1,p∣bi∣+jBi,Bi1,q;1=pΨqj,rΨs∣ci∣+jCi,Ci1,r∣di∣+jDi,Di1,s;1=rΨsj,provided that
(16)∑i=1qbi−∑i=1pai+p−q2>12+j,∑i=1sdi−∑i=1r∣ci∣+r−s2>12+j.Making use of (13) and (15), we have
(17)∑n=1+j∞n−jjνn=∏i=1qΓbi∏i=1pΓaipΨqj,∑n=1+j∞n−jjηn=∏i=1sΓdi∏i=1rΓcirΨsj,provided that (16) holds true.The convolution of two functionsfz of the form (1) and Fz of the form
(18)Fz=z+∑n=2∞Hnzn+∑n=1∞Gnzn¯,is given by
(19)f∗Fz=fz∗Fz=z+∑n=2∞hnHnzn+∑n=1∞gnGnzn¯.Now, we introduce a convolution operatorΩp,q,r,s as
(20)Ωp,q,r,sfz=fz∗Wz=hz∗Hz+gz∗Gz¯,where f=h+g¯ and Wz=Hz+Gz¯ given by (1) and (10), respectively. Hence
(21)Ωp,q,r,sfz=z+∑n=2∞θnhnzn+∑n=1∞ζngnzn¯.The application of the special functions on the geometric function theory always attracts researchers with various kinds of special functions, for example, hypergeometric functions [9–11], confluent hypergeometric functions [12], generalized hypergeometric functions [6, 13], Bessel functions [14], generalized Bessel functions [15–17], Wright functions [18–21], Fox-Wright functions [6, 22], and Mittag-Leffler functions [23] that have rich applications in analytic and harmonic univalent functions. By using special functions, some researchers introduce operators, for example, Carlson-Shaffer operator [24], Hohlov operator [25], and Dziok-Srivastava operator [26, 27], and obtain interesting results. Motivated with the work of [20], we obtain some inclusion relation between the classes GHγ, KH0, SH∗,0, CH0, and NHβ by applying the convolution operator Ω.


## 2. Main Results

In order to establish our main results, we shall require the following lemmas.Lemma 1 [1].
Iff=h+g¯∈KH0, where h and g are given by (5) with g1=0, then
(22)∣hn∣≤n+12,∣gn∣≤n−12.Lemma 2 [1].
Letf=h+g¯∈SH∗,0orCH0, where h and g are given by (1) with g1=0. Then
(23)hn≤2n+1n+16,∣gn∣≤2n−1n−16.Lemma 3 [5].
Letf=h+g¯ be given by (5). If 0≤γ<1 and
(24)∑n=2∞2n−1−γhn+∑n=1∞2n+1+γgn≤1−γ,then f is a sense-preserving Goodman-Rønning-type harmonic univalent function in U and f∈GHγ.Remark 4.
In [5], it is also shown that f=h+g¯ given by (5) is in the family TGHγ, if and only if the coefficient condition (24) holds. Moreover, if f∈TGHγ, then
(25)hn=1−γ2n−1−γ,n≥2,∣gn∣=1−γ2n+1+γ,n≥1.Theorem 5.
Let∑i=1qbi−∑i=1p∣ai∣+p−q/2>5/2 and ∑i=1sdi−∑i=1r∣ci∣+r−s/2>5/2, and if the inequality
(26)∏i=1qΓbi∏i=1pΓ∣ai∣2pΨq2+7−γpΨq1+21−γpΨq0−1+σ∏i=1sΓdi∏i=1rΓci2rΨs2+5+γrΨs1≤21−γ,
holds, thenΩKH0⊂GHγ.Proof.
Letf=h+g¯∈KH0, where h and g are given by (1) with g1=0. We have to prove that Ωf∈GHγ, where Ωf is defined by (21). To prove Ωf∈GHγ, in view of Lemma 3, it is sufficient to prove that P1≤1−γ, where
(27)P1=∑n=2∞2n−1−γ∣θnhn∣+∑n=2∞2n+1+γζngn.
By using Lemma1,
(28)P1≤∑n=2∞n+12n−1−γθn+∑n=2∞n−12n+1+γζn=12∑n=2∞2n−1n−2+7−γn−1+21−γνn+σ2∑n=2∞2n−2+5+γηn=12∏i=1qΓbi∏i=1pΓai2+7−γΨpq1+21−γΨpq0−1+σ∏i=1sΓdi∏i=1rΓci2Ψrs2+5+γΨrs1≤1−γ,by the given hypothesis. This completes the proof of Theorem 5.The result is sharp for the function(29)Lz=z+∑n=2∞n+12zn−∑n=2∞n−12z¯n.Theorem 6.
Let∑i=1qbi−∑i=1p∣ai∣+p−q/2>7/2 and ∑i=1sdi−∑i=1r∣ci∣+r−s/2>7/2, and if the inequality
(30)∏i=1qΓbi∏i=1pΓ∣ai∣4pΨq3+28−2γpΨq2+39−9γpΨq1+61−γpΨq0−1+σ∏i=1sΓdi∏i=1rΓci4rΨs3+210+γrΨs2+35+γrΨs1≤61−γ,holds, then ΩSH∗,0⊂GHγ and ΩCH0⊂GHγ.Proof.
Letf=h+g¯∈SH∗,0orCH0, where h and g are given by (1) with g1=0; we need to prove that Ωf∈GHγ, where Ωf is defined by (21). In view of Lemma 3, it is sufficient to prove that P1≤1−γ, where P1 is given by (27). Now using Lemma 2, we have
(31)P1≤16∑n=2∞n+12n+12n−1−γθn+σ∑n=2∞n−12n−12n+1+γζn=16∑n=2∞4n−1n−2n−3+28−2γn−1n−2+39−9γn−1+61−γνn+σ6∑n=2∞4n−1n−2n−3+20+2γn−1n−2+15+3γn−1ηn=16∏i=1qΓbi∏i=1pΓai4Ψpq3+28−2γΨpq2+39−9γΨq1p+61−γΨq0p−1+σ6∏i=1sΓdi∏i=1rΓci4Ψs3r+210+γΨs2r+35+γrΨs1≤1−γ,by the given hypothesis. Thus, the proof of Theorem 6 is established.The result is sharp for the function(32)fz=Hz+Gz¯,where
(33)Hz=z−1/2z2+1/6z31−z3,Gz=1/2z2+1/6z31−z3.In our next theorem, we establish connections betweenTGHγ and GHγ.Theorem 7.
Let∑i=1qbi−∑i=1p∣ai∣+p−q/2>1/2 and ∑i=1sdi−∑i=1r∣ci∣+r−s/2>1/2, and if the inequality
(34)∏i=1qΓbi∏i=1pΓ∣ai∣pΨq0−1+∣σ∣∏i=1sΓdi∏i=1rΓcirΨs0≤1,holds, then ΩTGHγ⊆GHγ.Proof.
Letf=h+g¯∈TGHγ be given by (1). We have to prove that P2≤1−γ, where
(35)P2=∑n=2∞2n−1−γ∣θnhn∣+∣σ∣∑n=1∞2n+1+γζngn.
Now, using Remark4, we have
(36)P2≤1−γ∑n=2∞νn+1−γσ∑n=1∞ηn=1−γ∏i=1qΓbi∏i=1pΓaipΨq0−1+σ∏i=1sΓdi∏i=1rΓcirΨs0≤1−γ,by the given hypothesis. This completes the proof of Theorem 7.The result is sharp for the function(37)fz=z−∑n=2∞1−γ2n−1−γ∣xn∣zn+∑n=1∞1−γ2n+1+γ∣yn∣z¯n,where
(38)∑n=2∞xn+∑n=1∞yn=1.


## 3. Some Consequences of the Main Results

If we letp=q=r=s=1 and a1=A1=c1=C1=1 in (10), then Wz reduces to a harmonic univalent function Ez involving the following generalized Mittag-Leffler functions as
(39)Ez=zΓb1Eb1,B11,1z+σzΓd1Ed1,D11,1z¯,where
(40)Eb1,B11,1z=1Ψ11,1b1,B1;z=∑n=0∞znΓb1+nB1,Ed1,D11,1z=1Ψ11,1d1,D1;z=∑n=0∞znΓd1+nD1.With these specializations, the convolution operatorΩp,q,r,s reduces to the operator Φb1;B1;d1;D1, which is defined as
(41)Φb1;B1;d1;D1fz=fz∗Ez=hz∗zΓb1Eb1,B11,1z+σgz∗zΓd1Ed1,D11,1z¯.For these specific values ofp=q=r=s=1 and a1=A1=c1=C1=1, Theorems 5–7 yield the following results.Corollary 8.
If the inequality(42)Γb12Eb1+2B1,B13,11+7−γEb1+B1,B12,11+21−γEb1,B11,1−1+σΓd12Ed1+2D1,D13,11+5+γEd1+D1,D12,11≤21−γ,holds, then ΦKH0⊂GHγ.Corollary 9.
If the inequality(43)Γb14Eb1+3B1,B14,11+28−2γEb1+2B1,B13,11+39−9γEb1+B1,B12,11+21−γEb1,B11,1−1+σΓd14Ed1+3D1,D14,11+210+γEd1+2D1,D13,11+35+γEd1+D1,D12,11≤61−γ,holds, then ΦSH∗,0⊂GHγ and ΦCH0⊂GHγ.Corollary 10.
If the inequality(44)Γb1Eb1,B11,1−1+∣σ∣Γd1Ed1,D11,1≤1,holds, then ΦTGHγ⊂GHγ.Remark 11.
If we putp=q=r=s=1, a1=c1=1, A1=C1=0, and σ=1, then
(45)Wz=z+∑n=2∞Γb1Γb1+B1n−1n−1!zn+∑n=1∞Γd1Γd1+D1n−1n−1!zn¯,and results of Theorems 5–7 reduce to corresponding results of Maharana and Sahoo [28].Remark 12.
If we putp=r=2, q=s=1, A1=A2=B1=C1=C2=D1=1, and σ=1, then
(46)Wz=z+∑n=2∞a1n−1a2n−1b1n−1n−1!zn+∑n=2∞c1n−1c2n−1d1n−1n−1!zn¯,and results of Theorems 5–7 reduce to corresponding results of Porwal and Dixit [11].



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