# Subclass of Multivalent Harmonic Functions Defined by Wright Generalized Hypergeometric Functions

Authors: M. K. Aouf; A. O. Moustafa; E. A. Adwan
Journal: Journal of Complex Analysis
 (2013)

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## Abstract

We introduce a new class of multivalent harmonic functions defined by Wright generalized hypergeometric function. Coefficient estimates, extreme points, distortion bounds, and convex combination for functions belonging to this class are obtained.


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## Body

## 1. Introduction

A continuous complex-valued functionf=u+iv defined in a simply connected complex domain D is said to be harmonic in D if both u and v are real harmonic in D. In any simply connected domain the function f(z) can be written in the form
(1)f=h+g¯,
where h and g are analytic in D, h is called the analytic part, and g the coanalytic 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 [1]).Denote bySH the class of functions f of the form (1) that are harmonic univalent and sense preserving in the unit disc U={z:|z|<1} for which f(0)=fz(0)-1=0.Recently, Ahuja and Jahangiri [2] defined the class Hp(p∈ℕ={1,2,…}) consisting of all p-valent harmonic functions f=h+g¯ that are sense-preserving in U and h, g are of the form
(2)h(z)=zp+∑n=1+p∞‍anzn,g(z)=∑n=p∞‍bnzn,|bp|<1.Letα1,A1,…,αq,Aq and β1,B1,…,βs,Bs(q,s∈ℕ0=ℕ∪{0}) be positive real parameters such that
(3)1+∑j=1s‍Bj-∑j=1q‍Aj≥0.
The Wright generalized hypergeometric function [3] (see also [4])
(4)qΨs[(α1,A1),…,(αq,Aq);(β1,B1),…,(βs,Bs);z]=qΨs[(αi,Ai)1,q;(βi,Bi)1,s;z]
is defined by
(5)qΨs[(αi,Ai)1,q;(βi,Bi)1,s;z]=∑n=0∞∏i=1qΓ(αi+nAi)∏i=1sΓ(βi+nBi)·znn!(z∈U).
If Ai=1(i=1,…,q) and Bi=1(i=1,…,s), we have the relationship
(6)ΩqΨs[(αi,1)1,q;(βi,1)1,s;z]=qFs(α1,…,αq;β1,…,βs;z),
where qFs(α1,…,αq;β1,…,βs;z) is the generalized hypergeometric function (see [4]) and
(7)Ω=∏i=1sΓ(βi)∏i=1qΓ(αi).
By using the generalized hypergeometric function, Dziok and Srivastava [5] introduced a linear operator. Dziok and Raina in [6] and Aouf and Dziok in [7] extended this linear operator by using Wright generalized hypergeometric function.Aouf et al. [8] defined the linear operator θp,q,s[(αi,Ai)1,q;(βi,Bi)1,s] by the Hadamard product as
(8)θp,q,s[(αi,Ai)1,q;(βi,Bi)1,s]f(z)=qΦsp[(αi,Ai)1,q;(βi,Bi)1,s;z]*φ(z),
where qΦsp[(αi,Ai)1,q;(βi,Bi)1,s;z] is given by
(9)qΦsp[(αi,Ai)1,q;(βi,Bi)1,s;z]=ΩzpqΨs[(αi,Ai)1,q;(βi,Bi)1,s;z].
We observe that, for a function φ(z) of the form (2), we have
(10)θp,q,s[(αi,Ai)1,q;(βi,Bi)1,s]φ(z)=zp+∑n=1+p∞‍Ωσn,p(α1)anzn,
where Ω is given by (7) and σn,p(α1) is defined by
(11)σn,p(α1)=Γ(α1+A1(n-p))⋯Γ(αq+Aq(n-p))Γ(β1+B1(n-p))⋯Γ(βs+Bs(n-p))(n-p)!.
For convenience, we write
(12)θp,q,s[α1,A1,B1]f(z)=θp,q,s[(α1,A1),…,(αq,Aq);(β1,B1),…,(βs,Bs)]×φ(z).
Now we can define the modified Wright operator as follows:
(13)θp,q,s[α1,A1,B1]f(z)=θp,q,s[α1,A1,B1]h(z)+θp,q,s[α1,A1,B1]g(z)¯,
where
(14)θp,q,s[α1,A1,B1]h(z)=zp+∑n=1+p∞‍Ωσn,p(α1)anzn,θp,q,s[α1,A1,B1]g(z)=∑n=p∞‍Ωσn,p(α1)bnzn,|bp|<1.Forp=1, θ1,q,s[α1,A1,B1]f(z)=Wqs[α1]f(z), where Wqs[α1]f(z) is the modified Wright generalized hypergeometric functions (see [9]).We note that, forAi=1(i=1,2,…,q) and Bi=1(i=1,2,…,s), we obtain θp,q,s[α1,1,1]f(z)=Hp,q,s[α1]f(z), where Hp,q,s[α1] is the modified Dziok-Srivastava operator (see [10]).For0≤γ<1, p∈ℕ and for all z∈U, let Hp([α1,A1,B1],q,s;γ) denote the family of harmonic p-valent functions f(z)=h(z)+g(z)¯, where h and g are given by (2) and satisfying the analytic criterion
(15)Re{+θp,q,s[α1,A1,B1]g(z)¯)-1(+θp,q,s[α1,A1,B1]g(z)¯)-1z(θp,q,s[α1,A1,B1]h(z))′ffffff-z(θp,q,s[α1,A1,B1]g(z))′¯)ffff×(θp,q,s[α1,A1,B1]h(z)fffffff+θp,q,s[α1,A1,B1]g(z)¯)-1}≥pγ.LetHp¯([α1,A1,B1],q,s;γ) be the subclass of Hp([α1,A1,B1],q,s;γ) consisting of functions f=h+g¯ such that h and g are of the form
(16)h(z)=zp-∑n=1+p∞‍anzn,g(z)=∑n=p∞‍bnzn,|bp|<1.We note that for suitable choices ofq and s, we obtain the following subclasses:(1)
Hp¯([1,1,1],2,1;γ)=SH*(p,γ) (see [2]);(2)
Hp¯([α1,1,1],q,s;γ)=TH*(p,α1,γ) (see [10]);(3)
H1¯([α1,A1,B1],q,s;γ)=WH¯([α1],γ) (see [9]);(4)
H1¯([α1,1,1],q,s;γ)=VH¯(α1,γ) (see [11, 12]);(5)
H1¯([1,1,1],2,1;γ)=SH*(γ) (see [13]).


## 2. Coefficient Estimates

Unless otherwise mentioned, we will assume in the reminder of this paper that the parametersα1,A1,…,αq,Aq and β1,B1,…,βs,Bs(q,s∈ℕ) are positive real numbers, 0≤γ<1, p∈ℕ, z∈U, Ω is defined by (7), and σn,p(α1) is defined by (11).Theorem 1.
Letf=h+g¯ be such that h(z) and g(z) are given by (2). Furthermore, let
(17)∑n=1+p∞‍(n-pγ)Ωσn,p(α1)|an|+∑n=p∞‍(n+pγ)Ωσn,p(α1)|bn|≤p(1-γ).
Then f(z) is orientation preserving in U and f(z)∈Hp([α1,A1,B1],q,s;γ).Proof.
The inequality|h′(z)|≥|g′(z)| is enough to show that f(z)  is orientation preserving. Note that
(18)|h′(z)|≥p|z|p-1-∑n=1+p∞‍n|an||z|n-1>p-∑n=1+p∞‍n|an|≥p-∑n=1+p∞‍(n-pγ)Ωσn,p(α1)p(1-γ)|an|≥∑n=p∞‍(n+pγ)Ωσn,p(α1)p(1-γ)|bn|≥∑n=p∞‍n|bn|>∑n=p∞‍n|bn||z|n-1≥|g′(z)|.
Now we will show that f(z)∈Hp([α1,A1,B1],q,s;γ). We only need to show that if (17) holds, then condition (15) is satisfied. Using the fact that Re{w}≥pγ if and only if |p(1-γ)+w|≥|p(1+γ)-w|, it suffices to show that
(19)|A(z)+p(1-γ)B(z)|-|A(z)-p(1+γ)B(z)|≥0,
where
(20)A(z)=z(θp,q,s[α1,A1,B1]h(z))′-z(θp,q,s[α1,A1,B1]g(z))′¯,B(z)=θp,q,s[α1,A1,B1]h(z)+θp,q,s[α1,A1,B1]g(z)¯.
Substituting for A(z) and B(z) in (17) yields
(21)|A(z)+p(1-γ)B(z)|-|A(z)-p(1+γ)B(z)|=|(2p-pγ)zp-∑n=1+p∞‍[n+p(1-γ)]Ωσn,p(α1)anznffGf-∑n=p∞‍[n-p(1-γ)]Ωσn,p(α1)bnzn|-|-pγzp+∑n=1+p∞‍[n-p(1+γ)]Ωσn,panznFgggFF-∑n=p∞‍[n+p(1+γ)]Ωσn,p(α1)bnzn|≥(2p-pγ)|z|p-∑n=1+p∞‍[n+p(1-γ)]Ωσn,p(α1)|an||z|n-∑n=p∞‍[n+p(γ-1)]Ωσn,p(α1)|bn||z|n-pγ|z|p-∑n=1+p∞‍[n-p(1+γ)]Ωσn,p(α1)|an||z|n-∑n=p∞‍[n+p(1+γ)]Ωσn,p(α1)|bn||z|n=2p(1-γ)|z|p-2∑n=1+p∞‍(n-pγ)Ωσn,p(α1)|an||z|n-2∑n=p∞‍(n+pγ)Ωσn,p(α1)|bn||z|n=2p(1-γ)|z|p(1-∑n=1+p∞‍(n-pγ)Ωσn,p(α1)p(1-γ)|an||z|n-pffffffffffffffff-∑n=p∞‍(n+pγ)Ωσn,p(α1)p(1-γ)|bn||z|n-p)≥2p(1-γ)|z|p{1-(∑n=1+p∞‍(n-pγ)Ωσn,p(α1)p(1-γ)|an|ffffffffffffffffff+∑n=p∞‍(n+pγ)Ωσn,p(α1)p(1-γ)|bn|)}.
The last expression is nonnegative by (17). This completes the proof of Theorem 1. The harmonic p-valent function
(22)f(z)=zp+∑n=1+p∞‍p(1-γ)(n-pγ)Ωσn,p(α1)Xnzn+∑n=p∞‍p(1-γ)(n+pγ)Ωσn,p(α1)Ynzn¯,
where ∑n=1+p∞|Xn|+∑n=p∞|Yn|=1, shows that the coefficient bound given by (17) is sharp. It is worthy to note that the function of the form (22) belongs to the class Hp([α1,A1,B1],q,s;γ) for all ∑n=2∞|Xn|+∑n=1∞|Yn|≤1 because coefficient inequality (17) holds.Theorem 2.
A functionf(z) is in the class Hp([α1,A1,B1],q,s;γ), if and only if
(23)θp,q,s[α1,A1,B1]h(z)*[2p(1-γ)zp+(ζ-2p+2pγ+1)zp+1(1-z)2]-θp,q,s[α1,A1,B1]g(z)′¯*[2p(ζ+γ)zp¯+(ζ-2pζ-2pγ+1)zp+1¯(1-z¯)2]≠0,ffffffffffffffffffGffffffffffff|ζ|=1.Proof.
From (15), we have the necessary and sufficient condition for f(z)∈Hp([α1,A1,B1],q,s;γ) that is,
(24)Re{1p(1-γ)fffff×[(-z(θp,q,s[α1,A1,B1]g(z))′¯z(θp,q,s[α1,A1,B1]h(z))′fffgffffff-z(θp,q,s[α1,A1,B1]g(z))′¯)ffffff×(θp,q,s[α1,A1,B1]h(z)ffffffffff+θp,q,s[α1,A1,B1]g(z)¯)-1-pγ]}≥0.
Hence we have the equivalent condition
(25)1p(1-γ)[(z(θp,q,s[α1,A1,B1]g(z))′¯z(θp,q,s[α1,A1,B1]h(z))′fffffgfff-z(θp,q,s[α1,A1,B1]g(z))′¯)ffffffff×(θp,q,s[α1,A1,B1]g(z)¯θp,q,s[α1,A1,B1]h(z)fffffffffff+θp,q,s[α1,A1,B1]g(z)¯)-1-pγz(θp,q,s[α1,A1,B1]g(z))′¯]≠ζ-1ζ+1,|ζ|=1, ζ≠-1,0<|z|<1. Simple algebraic manipulation of (25) yields
(26)0≠[(z(θp,q,s[α1,A1,B1]h(z))′ffgf-z(θp,q,s[α1,A1,B1]g(z))′¯)fff×(θp,q,s[α1,A1,B1]h(z)ffffff+θp,q,s[α1,A1,B1]g(z)¯)-1-pγ]-p(1-γ)(ζ-1ζ+1)=θp,q,s[α1,A1,B1]h(z)*[2p(1-γ)zp+(ζ-2p+2pγ+1)zp+1(1-z)2]-θp,q,s[α1,A1,B1]g(z)¯*[2p(ζ+γ)zp¯+(ζ-2pζ-2pγ+1)zp+1¯(1-z¯)2].
This completes the proof of Theorem 2.Theorem 3.
A functionf(z)=h+g¯, where h(z) and g(z) are given by (16), is in the class Hp¯([α1,A1,B1],q,s;γ), if and only if
(27)∑n=1+p∞‍(n-pγ)Ωσn,p(α1)|an|+∑n=p∞‍(n+pγ)Ωσn,p(α1)|bn|≤p(1-γ).Proof .
SinceHp¯([α1,A1,B1],q,s;γ)⊂Hp([α1,A1,B1],q,s;γ), we only need to prove the “only if” part of this theorem. To this end, for functions f(z)=h+g¯, whereh(z)andg(z)given by (16), we notice that condition
(28)Re{(z(θp,q,s[α1,A1,B1]h(z))′ffff-z(θp,q,s[α1,A1,B1]g(z))′¯)ffff×(θp,q,s[α1,A1,B1]h(z)ffffff+θp,q,s[α1,A1,B1]g(z)¯)-1}≥pγ
is equivalent to
(29)Re{(p(1-γ)zp-∑n=1+p∞‍(n-pγ)Ωσn,p(α1)anzngggggg-∑n=p∞‍(n+pγ)Ωσn,p(α1)bnzn¯)gggg×(zp-∑n=1+p∞‍Ωσn,p(α1)anznggggggg-∑n=p∞‍Ωσn,p(α1)bnzn¯)-1}≥0.
The above condition must hold for all z, |z|=r<1. Choosing the values of z on the positive real axis where 0≤r<1, we must have
(30)(p(1-γ)-∑n=1+p∞‍(n-pγ)Ωσn,p(α1)anrn-1ff-∑n=p∞‍(n+pγ)Ωσn,p(α1)bnrn-1)×(1-∑n=1+p∞‍Ωσn,p(α1)anrn-1fffff-∑n=p∞‍Ωσn,p(α1)bnrn-1)-1≥0.
If condition (27) does not hold, then the numerator in (30) is negative for r sufficiently close to 1. Hence there exist z0=r0 in (0,1) for which the quotient in (30) is negative. This contradicts the required condition for f(z)∈Hp¯([α1,A1,B1],q,s;γ). This completes the proof of Theorem 3.


## 3. Distortion Theorem

Theorem 4.
Let the functionf(z)=h+g¯, where h(z) and g(z) are given by (16), belong to the class Hp¯([α1,A1,B1],q,s;γ). Then for |z|=r<1, we have
(31)|f(z)|≤(1+|bp|)rp+1Ωσ1+p,p(α1)×(p(1-γ)[1+p(1-γ)]-p(1+γ)[1+p(1-γ)]|bp|)r1+p,(32)|f(z)|≥(1-|bp|)rp-1Ωσ1+p,p(α1)×(p(1-γ)[1+p(1-γ)]-p(1+γ)[1+p(1-γ)]|bp|)r1+p
for |bp|≤(1-γ)/(1+γ). The results are sharp with equality for the functions f(z) defined by
(33)f(z)=(1+|bp|)zp¯-1Ωσ1+p,p(α1)×(p(1-γ)[1+p(1-γ)]-p(1+γ)[1+p(1-γ)]|bp|)z¯1+p,f(z)=(1-|bp|)zp¯-1Ωσ1+p,p(α1)×(p(1-γ)[1+p(1-γ)]-p(1+γ)[1+p(1-γ)]|bp|)z1+p.Proof.
We only prove the first inequality. The proof for the second inequality is similar and will be omitted. Letf(z)∈Hp¯([α1,A1,B1],q,s;γ). Taking the absolute value of f, we have
(34)|f(z)|≤(1+|bp|)rp+∑n=1+p∞‍(|an|+|bn|)rn≤(1+|bp|)rp+∑n=1+p∞‍(|an|+|bn|)r1+p=(1+|bp|)rp+p(1-γ)[1+p(1-γ)]Ωσ1+p,p(α1)×∑n=1+p∞‍([1+p(1-γ)]Ωσ1+p,p(α1)p(1-γ)|an|gggggggg+[1+p(1-γ)]Ωσ1+p,p(α1)p(1-γ)|bn|)r1+p≤(1+|bp|)rp+p(1-γ)[1+p(1-γ)]Ωσ1+p,p(α1)×∑n=1+p∞‍((n-pγ)Ωσn,p(α1)p(1-γ)|an|ffffffffff+(n+pγ)Ωσn,p(α1)p(1-γ)|bn|)r1+p≤(1+|bp|)rp+p(1-γ)[1+p(1-γ)]Ωσ1+p,p(α1)×(1-(1+γ)(1-γ)|bp|)r1+p=(1+|bp|)rp+1Ωσ1+p,p(α1)×(p(1-γ)[1+p(1-γ)]-p(1+γ)[1+p(1-γ)]|bp|)r1+p.
This completes the proof of Theorem 4.PuttingAi=1(i=1,2,…,q) and Bi=1(i=1,2,…,s) in Theorem 4, we obtain the following corollary which modifies the result obtained by Omar and Halim [10, Theorem 2.6].Corollary 5.
Let the functionf(z)=h+g¯, where h(z) and g(z) are given by (16), belong to the class Hp¯([α1],q,s;γ). Then, for |z|=r<1, we have
(35)|f(z)|≤(1+|bp|)rp+1Ψ1+p,p(α1)×(p(1-γ)[1+p(1-γ)]-p(1+γ)[1+p(1-γ)]|bp|)r1+p,|f(z)|≥(1-|bp|)rp-1Ψ1+p,p(α1)×(p(1-γ)[1+p(1-γ)]-p(1+γ)[1+p(1-γ)]|bp|)r1+p
for |bp|≤(1-γ)/(1+γ). The results are sharp with equality for the functions f(z) defined by
(36)f(z)=(1+|bp|)zp¯-1Ψ1+p,p(α1)×(p(1-γ)[1+p(1-γ)]-p(1+γ)[1+p(1-γ)]|bp|)z¯1+p,f(z)=(1-|bp|)zp¯-1Ψ1+p,p(α1)×(p(1-γ)[1+p(1-γ)]-p(1+γ)[1+p(1-γ)]|bp|)z1+p,
where Ψ1+p,p(α1)=Ωσ1+p,p(α1) with Ai=1(i=1,2,…,q) and Bi=1(i=1,2,…,s).


## 4. Extreme Points

Theorem 6.
Letf(z)=h+g¯, where h(z) and g(z) are given by (16). Then f(z)∈Hp¯([α1,A1,B1],q,s;γ), if and only if
(37)f(z)=∑n=p∞‍(μnhn(z)+ηngn(z)),
where hp(z)=zp,
(38)hn(z)=zp-p(1-γ)(n-pγ)Ωσn,p(α1)zn(n=1+p,2+p,…),(39)gn(z)=zp+p(1-γ)(n+pγ)Ωσn,p(α1)zn¯(n=p,1+p,…),μn≥0, ηn≥0, ∑n=p∞(μn+ηn)=1. In particular, the extreme points of the class Hp¯([α1,A1,B1],q,s;γ) are {hn} and {gn}.Proof.
Suppose that(40)f(z)=∑n=p∞‍(μnhn(z)+ηngn(z))=zp-∑n=1+p∞‍p(1-γ)(n-pγ)Ωσn,p(α1)μnzn+∑n=p∞‍p(1-γ)(n+pγ)Ωσn,p(α1)ηnzn¯.
Then
(41)∑n=1+p∞‍(n-pγ)Ωσn,p(α1)p(1-γ)(p(1-γ)(n-pγ)Ωσn,p(α1)μn)+∑n=p∞‍(n+pγ)Ωσn,p(α1)p(1-γ)(p(1-γ)(n+pγ)Ωσn,p(α1)ηn)=∑n=1+p∞‍μn+∑n=p∞‍ηn=1-μp≤1
and so f(z)∈Hp¯([α1,A1,B1],q,s;γ).
Conversely, iff(z)∈Hp¯([α1,A1,B1],q,s;γ), then
(42)|an|≤p(1-γ)(n-pγ)Ωσn,p(α1)(n≥1+p),|bn|≤p(1-γ)(n+pγ)Ωσn,p(α1)(n≥p).
Set
(43)μn=(n-pγ)Ωσn,p(α1)p(1-γ)|an|(n=1+p,2+p,…),ηn=(n+pγ)Ωσn,p(α1)p(1-γ)|bn|(n=p,1+p,…).
Since 0≤μn≤1  (n=1+p,2+p,…) and 0≤ηn≤1(n=p,1+p,…), μp=1-∑n=1+p∞μn+∑n=p∞ηn≥0, then we can see that f(z) can be expressed in the form (37). This completes the proof of Theorem 6.Now we show that the classHp¯([α1,A1,B1],q,s;γ) is closed under convex combinations of its members.Theorem 7.
The classHp¯([α1,A1,B1],q,s;γ) is closed under convex combination.Proof.
Fori=1,2,3,…, let fi∈Hp¯([α1,A1,B1],q,s;γ), where fi is given by
(44)fi=z-∑n=1+p∞‍|ani|zn+∑n=p∞‍|bni|zn¯.
Then by using Theorem 3, we have
(45)∑n=1+p∞‍(n-pγ)Ωσn,p(α1)p(1-γ)|ani|zn+∑n=p∞‍(n+pγ)Ωσn,p(α1)p(1-γ)|bni|z¯n≤1.
For ∑n=1∞ti=1, 0≤ti≤1, the convex combination of fi may be written as
(46)∑i=1∞‍tifi(z)=zp-∑n=1+p∞‍(∑i=1∞‍ti|ani|)zn+∑n=p∞‍(∑i=1∞‍ti|bni|)zn¯.
Then by (45), we have
(47)∑n=1+p∞‍(n-pγ)Ωσn,p(α1)p(1-γ)(∑i=1∞‍ti|ani|)+∑n=p∞‍(n+pγ)Ωσn,p(α1)p(1-γ)(∑i=1∞‍ti|bni|)=∑i=1∞‍ti(∑n=1+p∞‍(n-pγ)Ωσn,p(α1)p(1-γ)|ani|ffffffff+∑n=p∞‍(n+pγ)Ωσn,p(α1)p(1-γ)|bni|)≤∑i=1∞‍ti=1.
This is the required condition and so ∑i=1∞tifi(z)∈Hp¯([α1,A1,B1],q,s;γ). This completes the proof of Theorem 7.



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