# A Subclass of Harmonic Univalent Functions Associated withq-Analogue of Dziok-Srivastava Operator

Authors: Huda Aldweby; Maslina Darus
Journal: ISRN Mathematical Analysis
 (2013)

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## Abstract

We study a class of complex-valued harmonic univalent
functions using a generalized operator involving basic hypergeometric
function. Precisely, we give a necessary and sufficient coefficient condition
for functions in this class. Distortion bounds, extreme points, and
neighborhood of such functions are also considered.


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## Body

## 1. Introduction

Let𝕌={z∈ℂ:|z|<1} be the open unit disc, and let SH denote the class of functions which are complex valued, harmonic, univalent, and sense preserving in 𝕌 normalized by f(0)=fz(0)-1=0. Each f∈SH can be expressed as f=h+g-, where h and g are analytic in 𝕌. We call h 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 𝕌 is that |h′(z)|>|g′(z)| in 𝕌 (see [1]). In [2], there is a more comprehensive study on harmonic univalent functions. Thus, for f=h+g-∈SH, we may write
(1)h(z)=z+∑k=2∞‍akzk,g(z)=∑k=1∞‍bkzk(0≤b1<1).
Note that SH reduces to S, the class of normalized analytic univalent functions, if the coanalytic part of f=h+g- is identically zero.The study of basic hypergeometric series (also calledq-hypergeometric series) essentially started in 1748 when Euler considered the infinite product (q;q)∞-1=∏k=0∞‍(1-qk+1)-1. In the literature, we were told that the development of these functions was much slower until, in 1878, Heine converted a simple observation that limq→1[(1-qa)/(1-q)]=a which returns the theory of2ϕ1 basic hypergeometric series to the famous theory of Gauss’s 2F1 hypergeometric series. The importance of basic hypergeometric functions is due to their application in deriving q-analogue of well-known functions, such as q-analogues of the exponential, gamma, and beta functions. In this paper, we define a class of starlike harmonic functions using basic hypergeometric functions and investigate its properties like coefficient condition, distortion theorem, and extreme points.For complex parametersai, bj, q(i=1,…,r, j=1,…,s, bj∈ℂ∖{0,-1,-2,…}, |q|<1), we define the basic hypergeometric function rΦs(a1,…,ar;b1,…,bs,q,z) by
(2)rΦs(a1…,ar;b1,…,bs,q,z)=∑k=0∞‍(a1,q)k⋯(ar,q)k(q,q)k(b1,q)k⋯(bs,q)kzk,(r=s+1;r,s∈ℕ0=ℕ∪{0};z∈𝕌) where ℕ denote the set of positive integers and (a,q)k is the q-shifted factorial defined by
(3)(a,q)k={1,k=0;(1-a)(1-aq)(1-aq2)⋯(1-aqk-1),k∈ℕ.
We note that
(4)limq→1-[krΦs(qa1,…,qar;qb1,…,qbs,q,(q-1)1+s-rz)]=rFs(a1,…,ar;b1,…,bs,z),
where rFs(a1,…,ar;b1,…,bs,z) is the well-known generalized hypergeometric function. By the ratio test, one observes that for |q|<1 and r=s+1 the series defined in (2) converges absolutely in 𝕌 so that it represented an analytic function in 𝕌. For more mathematical background of basic hypergeometric functions, one may refer to [3, 4].Theq-derivative of a function h(x) is defined by
(5)Dq(h(x))=h(qx)-h(x)(q-1)x,q≠1,x≠0.
For a function h(z)=zk, we can observe that
(6)Dq(h(z))=Dq(zk)=1-qk1-qzk-1=[k]qzk-1.
Then limq→1Dq(h(z))=limq→1[k]qzk-1=kzk-1=h′(z), where h′(z) is the ordinary derivative. For more properties of Dq, see [4, 5].Corresponding to the functionrΦs(a1,…,ar;b1,…,bs,q,z), consider
(7)r𝒢s(a1,…,ar;b1,…,bs,q,z)=zorΦs(a1,…,ar;b1,…,bs,q,z)=z+∑k=2∞‍(a1,q)k-1⋯(ar,q)k-1(q,q)k-1(b1,q)k-1⋯(bs,q)k-1zk.
The authors [6] defined the linear operator Hsr(a1,…,ar;b1,…,bs;q)f:𝒜→𝒜 by
(8)Hsr(a1,…,ar;b1,…,bs;q)f(z)=r𝒢s(a1,…,ar;b1,…,bs,q,z)*f(z)=z+∑k=2∞‍Γ(a1,q,k)akzk,
where (*) stands for convolution and
(9)Γ(a1,q,k)=(a1,q)k-1⋯(ar,q)k-1(q,q)k-1(b1,q)k-1⋯(bs,q)k-1.
To make the notation simple, we write
(10)Hsr[a1,q]f(z)=Hsr(a1,…,ar;b1,…,bs;q)f(z).
We define the operator (8) of harmonic function f=h+g- given by (1) as
(11)Hsr[a1,q]f(z)=Hsr[a1,q]h(z)+Hsr[a1,q]g(z)¯.Definition 1.
For0≤δ<1, let SH*(a1,δ,q) denote the subfamily of starlike harmonic functions f∈SH* of the form (1) such that
(12)∂∂θ(argHsr[a1,q]f)≥δ,|z|=r<1.
Following [7], a function f is said to be in the class VH¯(a1,δ,q)=SH*(a1,δ,q)∩VH if f of the form (1) satisfies the condition that
(13)arg(ak)=θkarg(bk)=ϑk(k≥n+1;n∈ℕ)
and if there exists a real number ρ such that
(14)θk+(k-1)ϕ≡π(mod2π),ϑk+(k-1)ϕ≡0,(k≥n+1;n∈ℕ).
By specializing the parameters of Hsr[a1,q]f, we obtain different classes of starlike harmonic functions, for example,(i)
forr=s+1,a2=b1,…,ar=bs, SH*(q,q,δ)=SH(δ) [8] is the class of sense-preserving harmonic univalent functions f which are starlike of order δ in 𝕌; that is, ∂/∂θ(argf(reiθ))≥δ;(ii)
forr=s+1, a2=b1,…,ar=bs, and a1=qn+1,q→1,  SH*(qn+1,q,δ)=RH(n,α) [9] is the class of starlike harmonic univalent functions with ∂/∂θ(argDnf(z))≥δ, where D is the Ruscheweyh derivative (see [10]);(iii)
fori={1,…,r},j={1,…,s},r=s+1,ai=qαi, and bj=qβj,q→1,SH*(a1,q,δ)=SH*(α1,δ) [11] is the class of starlike harmonic univalent functions with (∂/∂θ)(argHsr[α1]f)≥δ, where Hsr[α1] is the Dziok-Srivastava operator (see [12]).


## 2. Main Results

In our first theorem, we introduce a sufficient coefficient bound for harmonic functions inSH*(a1,δ,q).Theorem 2.
Letf=h+g¯ be given by (1). If
(15)∑k=2∞‍([k]q-δ1-δ|ak|+[k]q+δ1-δ|bk|)Γ(a1,q,k)≤1-1+δ1-δ|b1|,
where a1=1,0≤δ<1, and Γ(a1,q,k) is given by (9), then f∈SH*(a1,δ,q).Proof.
To prove thatf∈SH*(a1,δ,q), we only need to show that if (15) holds, then the required condition (12) is satisfied. For (12), we can write
(16)∂∂θ(argHsr[a1,q]f(z))=ℜ{zDq(Hsr[a1,q]g(z))¯Hsr[a1,q]h(z)+Hsr[a1,q]g(z)¯zDq(Hsr[a1,q]h(z))Hsr[a1,q]h(z)+Hsr[a1,q]g(z)¯-zDq(Hsr[a1,q]g(z))¯Hsr[a1,q]h(z)+Hsr[a1,q]g(z)¯}=ℜA(z)B(z).
Using the fact that ℜ(w)≥δ if and only if |1-δ+w|≥|1+δ-w|, it suffices to show that
(17)|A(z)+(1-δ)B(z)|-|A(z)-(1+δ)B(z)|≥0.
Substituting for A(z) and B(z) in (15) yields
(18)|A(z)+(1-δ)B(z)|-|A(z)-(1+δ)B(z)|≥(2-δ)|z|-∑k=2∞‍([k]q+1-δ)Γ(a1,q,k)|ak||z|k-∑k=1∞‍([k]q-1+δ)Γ(a1,q,k)|bk||z|k-δ|z|-∑k=2∞‍([k]q-1-δ)Γ(a1,q,k)|ak||z|k-∑k=1∞‍([k]q+1+δ)Γ(a1,q,k)|bk||z|k≥2(1-δ)|z|{1-∑k=2∞‍[k]q-δ1-δΓ(a1,q,k)|ak|oooooooooooooo-∑k=1∞‍[k]q+δ1-δΓ(a1,q,k)|bk|}=2(1-δ)|z|{∑k=2∞‍1-1+δ1-δ|b1|oooooooooooooo-[∑k=2∞‍([k]q-δ1-δ|ak|+[k]q+δ1-δ|bk|)]=2(1-δ)|zo|oo×Γ(a1,q,k)∑k=2∞‍}.
The last expression is nonnegative by (15), and so, f∈SH*(a1,δ,q).Now, we obtain the necessary and sufficient conditions forf=h+g¯ given by (14).Theorem 3.
Letf=h+g¯ be given by (11). Then, f∈VH¯(a1,δ,q) if and only if
(19)∑k=2∞‍([k]q-δ1-δ|ak|+[k]q+δ1-δ|bk|)Γ(a1,q,k)≤1-1+δ1-δ|b1|,
where a1=1,0≤δ<1, and Γ(a1,q,k) is given by (9).Proof.
SinceVH¯(a1,δ,q)⊂SH*(a1,δ,q), we only to prove the only if part of the theorem. So that for functions f∈VH¯(a1,δ,q), we notice that the condition (∂/∂θ)(argHsr[a1,q]f(z))≥δ is equivalent to
(20)∂∂θ(argHsr[a1,q]f(z))-δ=ℜ{zDq(Hsr[a1,q]h(z))Hsr[a1,q]h(z)+Hsr[a1,q]g(z)¯-zDq(Hsr[a1,q]g(z))¯Hsr[a1,q]h(z)+Hsr[a1,q]g(z)¯-δzDq(Hsr[a1,q]h(z))Hsr[a1,q]h(z)+Hsr[a1,q]g(z)¯}≥0.
That is, (21)ℜ[(1-δ)z+∑k=2∞([k]q-δ)Γ(a1,q,k)|ak|zk-∑k=1∞([k]q+δ)Γ(a1,q,k)¯|bk|z¯kz+∑k=2∞Γ(a1,q,k)|ak|zk+∑k=1∞Γ(a1,q,k)¯|bk|z¯k]≥0.
The previous condition must hold for all values ofz in 𝕌. Upon choosing ϕ according to (14), we must have
(22)(1-δ)-(1+δ)|b1|1+|b1|+∑k=2∞(|ak|+|bk|)Γ(a1,q,k)rk-1-∑k=2∞(([k]q-δ)|ak|+([k]q+δ)|bk|)Γ(a1,q,k)rk-11+|b1|+∑k=2∞(|ak|+|bk|)Γ(a1,q,k)rk-1≥0.
If condition (19) does not hold, then the numerator in (22) is negative for r sufficiently close to 1. Hence, there exist z0=r0 in (0,1) for which the quotient of (22) is negative. This contradicts the fact that f∈VH¯(a1,δ,q), and this completes the proof.The following theorem gives the distortion bounds for functions inVH¯(a1,δ,q) which yield a covering result for this class.Theorem 4.
Iff∈VH¯(a1,δ,q), then
(23)|f(z)|≤(1+|b1|)r+1Γ(a1,q,2)(1-δ(q+1)-δ-1+δ(q+1)-δ)r2|z|=r<1,|f(z)|≥(1+|b1|)r-1Γ(a1,q,2)(1-δ(q+1)-δ-1+δ(q+1)-δ)r2|z|=r<1,
where
(24)Γ(a1,q,2)=(1-a1)⋯(1-ar)(1-q)(1-b1)⋯(1-bs),[2]q=(q+1).Proof.
We will only prove the right hand inequality. The proof for the left hand inequality is similar.
letf∈VH¯(a1,δ,q). Taking the absolute value of f, we obtain
(25)|f(z)|≤(1+|b1|)r+∑k=2∞‍(|ak|+|bk|)rk|f(z)|≤(1+|b1|)r+∑k=2∞‍(|ak|+|bk|)r2.
That is,
(26)|f(z)|≤(1+|b1|)r+1-δΓ(a1,q,2)([2]q-δ)×∑k=2∞‍([2]q-δ1-δ|ak|+[2]q-δ1-δ|bk|)×Γ(a1,q,2)r2≤(1+|b1|)r+1-δΓ(a1,q,2)((q+1)-δ)×[1-1+δ1-δ|b1|]r2=(1+|b1|)r+1Γ(a1,q,2)×[1-δ(q+1)-δ-1+δ(q+1)-δ|b1|)r2.Corollary 5.
Letf be of the form (1) so that f∈VH¯(a1,δ,q). Then,
(27){w:|w|<2Γ(a1,q,2)-1-(Γ(a1,q,2)-1)δ((q+1)-δ)Γ(a1,q,2)-2Γ(a1,q,2)-1-(Γ(a1,q,2)-1)δ((q+1)+δ)Γ(a1,q,2)|b1|}⊂f(𝕌).
Next, one determines the extreme points of closed convex hull of VH¯(a1,δ,q) denoted by clcoVH¯(a1,δ,q).Theorem 6.
Set(28)λk=1-δ([k]q-δ)Γ(a1,q,k),μk=1-δ([k]q+δ)Γ(a1,q,k).
For b1 fixed, the extreme points for clcoVH¯(a1,δ,q) are
(29){z+λkxzk+b1z¯}∪{z+b1z+μkxzk¯},
where k≥2 and |x|=1-|b1|.Proof.
Any functionf∈clcoVH¯(a1,δ,q) may be expressed as
(30)f(z)=z+∑k=2∞‍|ak|eiαkzk+b1z¯+∑k=2∞‍|bk|eiβkzk¯,
where the coefficients satisfy the inequality (15). Set h1(z)=z, g1(z)=b1z,hk(z)=z+λkeiαkzk, andgk(z)=b1(z)+eiβkzk, for k=2,3,…. Writing Xk=|ak|/λk,Yk=|bk|/μk, k=2,3,… and X1=1-∑k=2∞‍Xk;Y1=1-∑k=2∞‍Yk,  we have
(31)f(z)=∑k=1∞‍(Xkhk(z)+Ykgk(z)).
In particular, set
(32)f1(z)=z+b1z¯,fk(z)=z+λkxzk+b1z¯+μkyzk,¯ppppppppppppppipppppp(k≥2,|x|+|y|=1-|b1|).
Therefore, the extreme points ofclcoVH¯(a1,δ,q) are contained in {fk(z)}. To see that f1 is not an extreme point, note that f1 may be written as a convex linear combination of functions in clcoVH¯(a1,δ,q) as follows:
(33)f1(z)=12{f1(z)+λ2(1-|b1|)z2}+12{f1(z)-λ2(1-|b1|)z2}.
If both |x|≠0 and |y|≠0, we will show that it can also be expressed as a convex linear combination of functions in clcoVH¯(a1,δ,q). Without loss of generality, assume that |x|≥|y|. Choose ϵ>0 small enough so that ϵ<|x|/|y|. Set A=1+ϵ and B=1-|ϵx/y|. We then see that both
(34)t1(z)=z+λkAxzk+b1z¯+μkyBzk¯,t2(z)=z+λk(2-A)xzk+b1z¯+μky(2-B)zk,¯
are in clcoVH¯(a1,δ,q) and note that
(35)fn(z)=12{t1(z)+t2(z)}.
The extremal coefficient bound shows that the functions of the form (29) are extreme points for clcoVH¯(a1,δ,q), and so the proof is complete.Following the earlier works in [2, 13], we refer to the γ-neighborhood of the functions f defined by (1) to be the set of functions F for which
(36)Nγ(f)={F=z+∑k=2∞‍Akzk+∑k=1∞‍Bkzk¯:∑k=2∞‍k(|ak-Ak|+|bk-Bk|)+|b1-B1|≤γ}.We define theq-γ-neighborhood of a function f∈SH as follows:
(37)Nγq(f)={F=z+∑k=2∞‍Akzk+∑k=1∞‍Bkzk¯:∑k=2∞‍[k]q(|ak-Ak|+|bk-Bk|)+|b1-B1|≤γ∑k=2∞‍}.In our case, let us define the generalizedq-γ-neighborhood of f to be the set
(38)Nγq(f)={F:∑k=2∞‍Γ(a1,q,k)[([k]q-δ)|ak-Ak|pppppppppppppippp+([k]q+δ)|bk-Bk|]+(1+δ)|b1-B1|≤(1-δ)γ∑k=2∞‍}.Theorem 7.
Letf be given by (1). If f satisfies the conditions
(39)∑k=2∞‍[k]q([k]q-δ)|ak|Γ(a1,q,k)+∑1=2∞‍[k]q([k]q+δ)|bk|Γ(a1,q,k)≤1-δ,pppppppppppppppppppppipihhpppp0≤δ<1,γ≤1-δ(q+1)-δ(1-1+δ1-δ|b1|),
then Nγq(f)⊂SH*(a1,δ,q).Proof.
Letf satisfy (39) and let
(40)F(z)=z+B1z¯+∑k=2∞‍(Akzk+Bkzk¯)
belong to Nγq(f). We have
(41)(1+δ)|B1|+∑k=2∞‍Γ(a1,q,k)[([k]q-δ)|Ak|ppppppppppppppppppppp+([k]q+δ)|Bk|]≤(1+δ)|B1-b1|+(1+δ)|b1|+∑k=2∞‍Γ(a1,q,k)[([k]q-δ)|Ak-ak|pppppppipppppppp+([k]q+δ)|Bk-bk|]+∑k=2∞‍Γ(a1,q,k)[([k]q-δ)|ak|+([k]q+δ)|bk|]≤(1-δ)γ+(1+δ)|b1|+1[2]q-δ∑k=2∞‍Γ(a1,q,k)[[k]q([k]q-δ)|ak|pppppppppppppppppppp+[k]q([k]q+δ)|bk|]≤(1-δ)γ+(1+δ)|b1|+1(q+1)-δ×[(1-δ)-(1+δ)|b1|]≤1-δ.
Hence,
(42)γ≤1-δ(q+1)-δ(1-1+δ1-δ|b1|),F∈SH*(a1,δ,q).



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