# A New Subclass of Harmonic Univalent Functions Associated with Dziok-Srivastava Operator

Authors: A. L. Pathak; K. K. Dixit; R. Agarwal
Journal: International Journal of Mathematics and Mathematical Sciences
 (2012)

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## Abstract

The purpose of the present paper is to study a certain subclass of harmonic univalent functions associated with Dziok-Srivastava operator. We obtain coefficient
conditions, distortion bounds, and extreme points for the above class of harmonic univalent functions belonging to this class and discuss a class preserving integral operator. We also show that class studied in this paper is closed under convolution and convex combination. The results obtained for the class reduced to the corresponding results for several known classes in the literature are briefly indicated.


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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 harmonic in D. In any simply connected domain D we can write f=h+g-, where h and g are analytic in D. A necessary and sufficient condition for f to be locally univalent and sense preserving in D is that |h'(z)|>|g'(z)|,z∈D. See Clunie and Sheil-Small [1].Denote bySH the class of functions f=h+g- that are harmonic univalent and sense preserving in the unit disk U={z:|z|<1} for which f(0)=fz(0)-1=0. Then for f=h+g-∈SH, we may express the analytic function h and g as
(1.1)h(z)=z+∑k=2∞‍akzk,g(z)=∑k=1∞‍bkzk,|b1|<1.Note thatSH, is reduced to  S  the class of normalized analytic univalent functions if the coanalytic part of f=h+g- is identically zero.For more basic results on harmonic univalent functions, one may refer to the following introductory text book by Duren [2] (see also [3–5]).Forαj∈C,(j=1,2,3,…,q) and βj∈C-{0,-1,-2,-3,…},  (j=1,2,3,…,s), the generalized hypergeometric functions are defined by
(1.2)qFs(α1,α2,…,αq;β1,β2,…,βs;z)=∑k=0∞(α1)k(α2)k⋯(αq)kzk(β1)k(β2)k⋯(βs)kk!,(q≤s+1;q,s∈N0={0,1,2,…}),
where (α)k is the Pochhammer symbol defined by
(1.3)(α)k={Γ(α+k)Γ(α)=α(α+1)⋯(α+k-1),ifk=1,2,3,…,1,ifk=0.Corresponding to the function(1.4)h(α1,α2,…,αq;β1,β2,…,βs;z)=zqFs(α1,α2,…,αq;β1,β2,…,βs;z).The Dziok-Srivastava operator [6, 7] Hq,s(α1,α2,…,αq;β1,β2,…,βs) is defined for f∈S by Hq,s(α1,α2,…,αq;β1,β2,…,βs)f(z)=h(α1,α2,…,αq;β1,β2,…,βs;z)*f(z)(1.5)=z+∑k=2∞(α1)k-1(α2)k-1…(αq)k-1(β1)k-1(β2)k-1…(βs)k-1akzk(k-1)!,where * stands for convolution of two power series.To make the notation simple, we write(1.6)Hq,s[α1]f(z)=Hq,s(α1,α2,…,αq;β1,β2,…,βs)f(z).Special cases of the Dziok-Srivastava operator includes the Hohlov operator [8], the Carlson-Shaffer operator L(a,c) [9], the Ruscheweyh derivative operator Dn [10], and the Srivastava-Owa fractional derivative operators ([11–13]).We define the Dziok-Srivastava operator of the harmonic functionsf=h+g- given by (1.1) as
(1.7)Hq,s[α1]f(z)=Hq,s[α1]h(z)+Hq,s[α1]g(z).¯Recently, Porwal [14, Chapter 5] defined the subclass MH(β)⊂SH consisting of harmonic univalent functions f(z) satisfying the following condition:(1.8)MH(β)={f∈SH:Re(zh′(z)-zg′(z)¯h(z)+g(z)¯)<β},(1<β≤43),z∈U.He proved that iff=h+g- is given by (1.1) and if
(1.9)∑k=2∞‍(k-β)(β-1)|ak|+∑k=1∞‍(k+β)(β-1)|bk|≤1,(1<β≤43).Forg≡0 the class of MH(β) is reduced to the class M(β) studied by Uralegaddi et al. [15].Generalizing the classMH(β), we let MH(α1,β) denote the family of functions f=h+g- of form (1.1) which satisfy the condition
(1.10)Re{z(Hq,s[α1]h(z))′-z(Hq,s[α1]g(z))′¯Hq,s[α1]h(z)+Hq,s[α1]g(z)¯}<β,
where (1<β≤4/3) and Γ(α1,k)=|(α1)k-1(α2)k-1⋯(αq)k-1/(β1)k-1(β2)k-1⋯(βs)k-1|.Further, letM-H(α1,β) be the subclass of MH(α1,β) consisting of functions of the form
(1.11)f(z)=z+∑k=2∞‍|ak|zk-∑k=1∞‍|bk|z-k.In this paper, we give a sufficient condition forf=h+g-, given by (1.1) to be in MH(α1,β), and it is shown that this condition is also necessary for functions in M-H(α1,β). We then obtain distortion theorem, extreme points, convolution conditions, and convex combinations and discuss a class preserving integral operator for functions in M-H(α1,β).


## 2. Main Results

First, we give a sufficient coefficient bound for the classMH(α1,β).Theorem 2.1.
Iff=h+g¯∈SH is given by (1.1) and if
(2.1)∑k=2∞‍(k-β)(β-1)Γ(α1,k)(k-1)!|ak|+∑k=1∞‍(k+β)(β-1)Γ(α1,k)(k-1)!|bk|≤1,
then f∈MH(α1,β).Proof.
Let(2.2)∑k=2∞‍(k-β)(β-1)Γ(α1,k)(k-1)!|ak|+∑k=1∞‍(k+β)(β-1)Γ(α1,k)(k-1)!|bk|≤1.
It suffices to show that(2.3)|(z(Hq,s[α1]h(z))′-z(Hq,s[α1]g(z))′¯)/(Hq,s[α1]h(z)+Hq,s[α1]g(z)¯)-1(z(Hq,s[α1]h(z))′-z(Hq,s[α1]g(z))′¯)/(Hq,s[α1]h(z)+Hq,s[α1]g(z)¯)-(2β-1)|<1,z∈U.
We have(2.4)|(z(Hq,s[α1]h(z))′-z(Hq,s[α1]g(z))′¯)/(Hq,s[α1]h(z)+Hq,s[α1]g(z)¯)-1(z(Hq,s[α1]h(z))′-z(Hq,s[α1]g(z))′¯)/(Hq,s[α1]h(z)+Hq,s[α1]g(z)¯)-(2β-1)|≤∑k=2∞(k-1)𝒜|ak||z|k-1+∑k=1∞(k+1)𝒜|bk||z|k-12(β-1)-∑k=2∞(k-2β+1)𝒜|ak||z|k-1-∑k=1∞(k+2β-1)𝒜|bk||z|k-1≤∑k=2∞(k-1)𝒜|ak|+∑k=1∞(k+1)𝒜|bk|2(β-1)-∑k=2∞(k-2β+1)𝒜|ak|-∑k=1∞(k+2β-1)𝒜|bk|,
where 𝒜 denotes (Γ(α1,k)/(k-1)!).
The last expression is bounded above by 1, if(2.5)∑k=2∞‍(k-1)Γ(α1,k)(k-1)!|ak|+∑k=1∞‍(k+1)Γ(α1,k)(k-1)!|bk|≤2(β-1)-∑k=2∞‍(k-2β+1)Γ(α1,k)(k-1)!|ak|-∑k=1∞‍(k+2β-1)Γ(α1,k)(k-1)!|bk|,
which is equivalent to
(2.6)∑k=2∞‍(k-β)(β-1)Γ(α1,k)(k-1)!|ak|+∑k=1∞‍(k+β)(β-1)Γ(α1,k)(k-1)!|bk|≤1.
But (2.6) is true by hypothesis.
Hence(2.7)|(z(Hq,s[α1]h(z))′-z(Hq,s[α1]g(z))′¯)/(Hq,s[α1]h(z)+Hq,s[α1]g(z)¯)-1(z(Hq,s[α1]h(z))′-z(Hq,s[α1]g(z))′¯)/(Hq,s[α1]h(z)+Hq,s[α1]g(z)¯)-(2β-1)|<1,z∈U,
and the theorem is proved.
In the following theorem, it is proved that the condition (2.1) is also necessary for functions f=h+g¯∈M-H(α1,β) that are given by (1.11).Theorem 2.2.
A functionf(z)=z+∑k=2∞‍|ak|zk-∑k=1∞‍|bk|z-k  is in M-H(α1,β), if and only if
(2.8)∑k=2∞‍(k-β)Γ(α1,k)(k-1)!|ak|+∑k=1∞‍(k+β)Γ(α1,k)(k-1)!|bk|≤(β-1).Proof.
SinceM-H(α1,β)⊂MH(α1,β), we only need to prove the “only if” part of the theorem. For this we show that f∉M-H(α1,β) if the condition (2.8) does not hold.
Note that a necessary and sufficient condition forf=h+g¯ given by (1.9) is in M-H(α1,β) if
(2.9)Re{z(Hq,s[α1]h(z))′-z(Hq,s[α1]g(z))′¯Hq,s[α1]h(z)+Hq,s[α1]g(z)¯}<β
is equivalent to
(2.10)Re{(β-1)z-∑k=2∞(k-β)(Γ(α1,k)/(k-1)!)|ak|zk-∑k=1∞(k+β)(Γ(α1,k)/(k-1)!)|bk|z-kz+∑k=2∞(Γ(α1,k)/(k-1)!)|ak|zk-∑k=1∞(Γ(α1,k)/(k-1)!)|bk|z-k}≥0.
The above condition must hold for all values ofz, |z|=r<1. Upon choosing the values of z on the positive real axis where 0≤z=r<1, we must have
(2.11)(β-1)-∑k=2∞(k-β)(Γ(α1,k)/(k-1)!)|ak|rk-1-∑k=1∞(k+β)(Γ(α1,k)/(k-1)!)|bk|rk-11+∑k=2∞(Γ(α1,k)/(k-1)!)|ak|rk-1-∑k=1∞(Γ(α1,k)/(k-1)!)|bk|rk-1≥0.
If the condition (2.8) does not hold then the numerator of (2.11) is negative for r and sufficiently close to 1. Thus there exists a z0=r0 in (0,1) for which the quotient in (2.11) is negative. This contradicts the required condition for f∈M-H(α1,β) and so the proof is complete.
Next, we determine the extreme points of the closed convex hulls ofM-H(α1,β), denoted by clcoM-H(α1,β).Theorem 2.3.
f
∈
clco
M
-
H
(
α
1
,
β
), if and only if
(2.12)f(z)=∑k=1∞‍(xkhk(z)+ykgk(z)),
where
(2.13)h1(z)=z,hk(z)=z+(β-1)(k-β)(k-1)!Γ(α1,k)zk,k=(2,3,…),gk(z)=z-(β-1)(k+β)(k-1)!Γ(α1,k)z-k,k=(1,2,3,…),∑k=1∞‍(xk+yk)=1,xk≥0, and yk≥0. In particular the extreme points of M-H(α1,β) are {hk} and {gk}.Proof.
For functionsf of the form (2.12), we have
(2.14)f(z)=∑k=1∞‍(xkhk(z)+ykgk(z))=z+∑k=2∞‍(β-1)(k-β)(k-1)!Γ(α1,k)xkzk-∑k=1∞‍(β-1)(k+β)(k-1)!Γ(α1,k)ykz-k.
Then(2.15)∑k=2∞‍(k-β)(β-1)Γ(α1,k)(k-1)!((β-1)(k-β)(k-1)!Γ(α1,k)xk)+∑k=1∞‍(k+β)(β-1)Γ(α1,k)(k-1)!((β-1)(k+β)(k-1)!Γ(α1,k)yk)=∑k=2∞‍xk+∑k=1∞‍yk=1-x1≤1,
and sof∈clcoM-H(α1,β).
Conversely, suppose thatf∈clcoM-H(α1,β). Set xk=((k-β)/(β-1))((k-1)!/Γ(α1,k))|ak|,  (k=2,3,4,…) and yk=((k+β)/(β-1))((k-1)!/Γ(α1,k))|bk|,  (k=1,2,3,…).
Then note that by Theorem2.2, 0≤xk≤1,(k=2,3,4,…) and 0≤yk≤1,(k=1,2,3,…). We define x1=1-∑k=2∞xk-∑k=1∞yk, and by Theorem 2.2, x1≥0.
Consequently, we obtainf(z)=∑k=1∞‍(xkhk(z)+ykgk(z))  as required.Theorem 2.4.
Iff∈M-H(α1,β), then
(2.16)|f(z)|≤(1+|b1|)r+1Γ(α1,2)((β-1)(2-β)-(β+1)(2-β)|b1|)r2,|z|=r<1,|f(z)|≥(1-|b1|)r-1Γ(α1,2)((β-1)(2-β)-(β+1)(2-β)|b1|)r2,|z|=r<1.Proof.
We only prove the right hand inequality. The proof for left hand inequality is similar and will be omitted. Letf∈M-H(α1,β). Taking the absolute value of f, we have
(2.17)|f(z)|≤(1+|b1|)r+∑k=2∞‍(|ak|+|bk|)rk≤(1+|b1|)r+∑k=2∞‍(|ak|+|bk|)r2=(1+|b1|)r+(β-1)(2-β)1Γ(α1,2)∑k=2∞‍((2-β)Γ(α1,2)(β-1)|ak|+(2-β)Γ(α1,2)(β-1)|bk|)r2≤(1+|b1|)r+(β-1)(2-β)1Γ(α1,2)∑k=2∞‍((k-β)(β-1)Γ(α1,k)(k-1)!|ak|+(k+β)(β-1)Γ(α1,k)(k-1)!|bk|)r2≤(1+|b1|)r+(β-1)(2-β)1Γ(α1,2)(1-(β+1)(β-1)|b1|)r2=(1+|b1|)r+1Γ(α1,2)((β-1)(2-β)-(β+1)(2-β)|b1|)r2.
For our next theorem, we need to define the convolution of two harmonic functions. For harmonic functions of the form(2.18)f(z)=z+∑k=2∞‍|ak|zk-∑k=1∞‍|bk|z-k,F(z)=z+∑k=2∞‍|Ak|zk-∑k=1∞‍|Bk|z-k.
We define the convolution of two harmonic functions f and F as
(2.19)(f*F)(z)=f(z)*F(z)=z+∑k=2∞‍|akAk|zk-∑k=1∞‍|bkBk|z-k.
Using this definition, we show that the classM-H(α1,β) is closed under convolution.Theorem 2.5.
For1<γ≤β≤4/3, let f∈M-H(α1,γ) and F∈M-H(α1,β). Then
(2.20)f*F∈M-H(α1,γ)⊆M-H(α1,β).Proof.
Letf(z)=z+∑k=2∞‍|ak|zk-∑k=1∞‍|bk|z-k be in M-H(α1,γ), and F(z)=z+∑k=2∞‍|Ak|zk-∑k=1∞‍|Bk|z-k be in M-H(α1,β).
Then the convolutionf*F is given by (2.19). We wish to show that the coefficients of f*F satisfy the required condition given in Theorem 2.2. For F(z)∈M-H(α1,β) we note that |Ak|≤1 and |Bk|≤1. Now, for the convolution function f*F, we obtain
(2.21)∑k=2∞‍(k-α)(α-1)Γ(α1,k)(k-1)!|akAk|+∑k=1∞‍(k+α)(α-1)Γ(α1,k)(k-1)!|bkBk|≤∑k=2∞‍(k-α)(α-1)Γ(α1,k)(k-1)!|ak|+∑k=1∞‍(k+α)(α-1)Γ(α1,k)(k-1)!|bk|≤1,(sincef∈M-H(α1,γ)).
Thereforef*F∈M-H(α1,γ)⊆M-H(α1,β).Theorem 2.6.
The classM-H(α1,β) is closed under convex combination.Proof.
Fori=1,2,3,…, let fi(z)∈M-H(α1,β), where fi(z) is given by
(2.22)fi(z)=z+∑k=2∞‍|aki|zk-∑k=1∞‍|bki|z-k.
Then by Theorem2.2,
(2.23)∑k=2∞‍(k-β)(β-1)Γ(α1,k)(k-1)!|aki|+∑k=1∞‍(k+β)(β-1)Γ(α1,k)(k-1)!|bki|≤1.
For∑i-1∞‍ti=1, 0≤ti≤1, the convex combination of fi may be written as
(2.24)∑i-1∞‍tifi(z)=z+∑k=2∞‍(∑i-1∞‍ti|aki|)zk-∑k=1∞‍(∑i-1∞‍ti|bki|)z-k.
Then by (2.23),
(2.25)∑k=2∞‍(k-β)(β-1)Γ(α1,k)(k-1)!(∑i-1∞‍ti|aki|)+∑k=1∞‍(k+β)(β-1)Γ(α1,k)(k-1)!(∑i-1∞‍ti|bki|)=∑i-1∞‍ti(∑k=2∞‍(k-β)(β-1)Γ(α1,k)(k-1)!|aki|+∑k=1∞‍(k+β)(β-1)Γ(α1,k)(k-1)!|bki|)≤∑i-1∞‍ti=1.
This is the condition required by Theorem2.2 and so ∑i-1∞‍tifi(z)∈M-H(α1,β).


## 3.  A Family of Class Preserving Integral Operator

Letf(z)=h(z)+g(z)¯ be defined by (1.1); then F(z) defined by the relation
(3.1)F(z)=c+1zc∫0z‍tc-1h(t)dt+c+1zc∫0z‍tc-1g(t)dt¯,(c>-1).Theorem 3.1.
Letf(z)=h(z)+g(z)¯∈SH be given by (1.11) and f∈M-H(α1,β), then F(z) is defined by (3.1) also belong to M-H(α1,β).Proof.
Letf(z)=z+∑k=2∞‍|ak|zk-∑k=1∞‍|bk|z-k be in M-H(α1,β), then by Theorem 2.2, we have
(3.2)∑k=2∞‍(k-β)(β-1)Γ(α1,k)(k-1)!|ak|+∑k=1∞‍(k+β)(β-1)Γ(α1,k)(k-1)!|bk|≤1.
By definition ofF(z), we have
(3.3)F(z)=z+∑k=2∞‍(c+1)(c+k)|ak|zk-∑k=1∞‍(c+1)(c+k)|bk|z-k.
Now we have(3.4)∑k=2∞(k-β)(β-1)Γ(α1,k)(k-1)!((c+1)(c+k)|ak|)+∑k=1∞(k+β)(β-1)Γ(α1,k)(k-1)!((c+1)(c+k)|bk|)≤∑k=2∞(k-β)(β-1)Γ(α1,k)(k-1)!|ak|+∑k=1∞(k+β)(β-1)Γ(α1,k)(k-1)!|bk|≤1.
ThusF(z)∈M-H(α1,β).



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