ISRN.MATHEMATICAL.ANALYSIS ISRN Mathematical Analysis 2090-4665 Hindawi Publishing Corporation 382312 10.1155/2013/382312 382312 Research Article A Subclass of Harmonic Univalent Functions Associated with q-Analogue of Dziok-Srivastava Operator Aldweby Huda Darus Maslina Ólafsson G. Zhou D.-X. School of Mathematical Sciences Faculty of Science and Technology Universiti Kebangsaan Malaysia 43600 Bangi, Selangor Malaysia ukm.my 2013 9 9 2013 2013 26 06 2013 01 08 2013 2013 Copyright © 2013 Huda Aldweby and Maslina Darus. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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.

1. Introduction

Let 𝕌={z:|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 fSH 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 ). In , there is a more comprehensive study on harmonic univalent functions. Thus, for f=h+g-SH, we may write (1)h(z)=z+k=2akzk,g(z)=k=1bkzk(0b1<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 called q-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 limq1[(1-qa)/(1-q)]=a which returns the theory of  2ϕ1 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 parameters ai, bj, q(i=1,,r, j=1,,s, bj{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,s0={0};z𝕌) where 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  . We note that (4)limq1-[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].

The q-derivative of a function h(x) is defined by (5)Dq(h(x))=h(qx)-h(x)(q-1)x,q1,x0. For a function h(z)=zk, we can observe that (6)Dq(h(z))=Dq(zk)=1-qk1-qzk-1=[k]qzk-1. Then limq1Dq(h(z))=limq1[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 function   rΦs(a1,,ar;b1,,bs,q,z), consider (7)  r𝒢s(a1,,ar;b1,,bs,q,z)=zorΦs(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  defined the linear operator Hsr(a1,,ar;b1,,bs;q)f:𝒜𝒜 by (8)Hsr(a1,,ar;b1,,bs;q)f(z)=r𝒢s(a1,,ar;b1,,bs,q,z)*f(z)=z+k=2Γ(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.

For 0δ<1, let SH*(a1,δ,q) denote the subfamily of starlike harmonic functions fSH* of the form (1) such that (12)θ(argHsr[a1,q]f)δ,|z|=r<1. Following , 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(kn+1;n) and if there exists a real number ρ such that (14)θk+(k-1)ϕπ(mod2π),ϑk+(k-1)ϕ0,(kn+1;n). By specializing the parameters of Hsr[a1,q]f, we obtain different classes of starlike harmonic functions, for example,

for r=s+1,a2=b1,,ar=bs, SH*(q,q,δ)=SH(δ)  is the class of sense-preserving harmonic univalent functions f which are starlike of order δ in 𝕌; that is, /θ(argf(reiθ))δ;

for r=s+1, a2=b1,,ar=bs, and a1=qn+1,q1,  SH*(qn+1,q,δ)=RH(n,α)  is the class of starlike harmonic univalent functions with /θ(argDnf(z))δ, where D is the Ruscheweyh derivative (see );

for i={1,,r},j={1,,s},r=s+1,ai=qαi, and bj=qβj,q1,  SH*(a1,q,δ)=SH*(α1,δ)  is the class of starlike harmonic univalent functions with (/θ)(argHsr[α1]f)δ, where Hsr[α1] is the Dziok-Srivastava operator (see ).

2. Main Results

In our first theorem, we introduce a sufficient coefficient bound for harmonic functions in SH*(a1,δ,q).

Theorem 2.

Let f=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=1,0δ<1, and Γ(a1,q,k) is given by (9), then fSH*(a1,δ,q).

Proof.

To prove that fSH*(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))={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)¯}=A(z)B(z). Using the fact that (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|k2(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=21-1+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, fSH*(a1,δ,q).

Now, we obtain the necessary and sufficient conditions for f=h+g¯ given by (14).

Theorem 3.

Let f=h+g¯ be given by (11). Then, fVH¯(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=1,0δ<1, and Γ(a1,q,k) is given by (9).

Proof.

Since VH¯(a1,δ,q)SH*(a1,δ,q), we only to prove the only if part of the theorem. So that for functions fVH¯(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={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)[(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=2Γ(a1,q,k)|ak|zk+k=1Γ(a1,q,k)¯|bk|z¯k]0.

The previous condition must hold for all values of z 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-10. 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 fVH¯(a1,δ,q), and this completes the proof.

The following theorem gives the distortion bounds for functions in VH¯(a1,δ,q) which yield a covering result for this class.

Theorem 4.

If fVH¯(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),q=(q+1).

Proof.

We will only prove the right hand inequality. The proof for the left hand inequality is similar.

let fVH¯(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)(q-δ)×k=2(q-δ1-δ|ak|+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.

Let f be of the form (1) so that fVH¯(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 k2 and |x|=1-|b1|.

Proof.

Any function fclcoVH¯(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=2Xk;Y1=1-k=2Yk,  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(k2,|x|+|y|=1-|b1|).

Therefore, the extreme points of clcoVH¯(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=2Akzk+k=1Bkzk¯:k=2k(|ak-Ak|+|bk-Bk|)+|b1-B1|γ}.

We define the q-γ-neighborhood of a function fSH as follows: (37)Nγq(f)={F=z+k=2Akzk+k=1Bkzk¯:k=2[k]q(|ak-Ak|+|bk-Bk|)+|b1-B1|γk=2}.

In our case, let us define the generalized q-γ-neighborhood of f to be the set (38)Nγq(f)={F:k=2Γ(a1,q,k)[([k]q-δ)|ak-Ak|pppppppppppppippp+([k]q+δ)|bk-Bk|]+(1+δ)|b1-B1|(1-δ)γk=2}.

Theorem 7.

Let f 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,pppppppppppppppppppppipihhpppp0δ<1,γ1-δ(q+1)-δ(1-1+δ1-δ|b1|), then Nγq(f)SH*(a1,δ,q).

Proof.

Let f satisfy (39) and let (40)F(z)=z+B1z¯+k=2(Akzk+Bkzk¯) belong to Nγq(f). We have (41)(1+δ)|B1|+k=2Γ(a1,q,k)[([k]q-δ)|Ak|ppppppppppppppppppppp+([k]q+δ)|Bk|](1+δ)|B1-b1|+(1+δ)|b1|+k=2Γ(a1,q,k)[([k]q-δ)|Ak-ak|pppppppipppppppp+([k]q+δ)|Bk-bk|]+k=2Γ(a1,q,k)[([k]q-δ)|ak|+([k]q+δ)|bk|](1-δ)γ+(1+δ)|b1|+1q-δk=2Γ(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)γ1-δ(q+1)-δ(1-1+δ1-δ|b1|),FSH*(a1,δ,q).

Acknowledgment

The work presented here was partially supported by LRGS/TD/2011/UKM/ICT/03/02 and UKM-DLP-2011-050.

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