In this paper, we investigate several properties of the harmonic class defined by the modified Dziok-Sirvastava operator, obtain distortion theorem, extreme points, convolution condition, convex combinations and integral operator for this class. Some of our results generalize previously known results

Abdulkarem, F. M.

Aouf, M. K.

El-Ashwah, R. M.

Le Matematiche (Dipartimento di Matematica e Informatica, Università degli Studi di Catania)

LE MATEMATICHEVol. LXIX (2014) – Fasc. I, pp. 197–208doi: 10.4418/2014.69.1.15UNIFIED REPRESENTATION OF A CERTAIN CLASS OFHARMONIC UNIVALENT FUNCTIONS DEFINED BYDZIOK-SRIVASTAVA OPERATORM. K. AOUF - R. M. EL-ASHWAH - F. M. ABDULKAREMIn this paper, we investigate several properties of the harmonic classNH([α1] ,n,γ), defined by the modified Dziok-Sirvastava operator, obtaindistortion theorem, extreme points, convolution condition, convex combi-nations and integral operator for this class. Some of our results generalizepreviously known results.1. IntroductionA continuous complex valued functions f = u+ iv which is define in a simplyconnected complex domain D is said to be harmonic in D if both u and v arereal harmonic in D. In any simply connected domain we can writef (z) = h(z)+g(z) , (1)where h and g are analytic in D. We call h the analytic part and g the co-analyticpart of f . A necessary and sufficient condition for f to be locally univalent andsense-preserving in D is that |h′(z)|> |g′(z)| in D (see [5]).Denote by SH , the class of functions f of the form (2) that are harmonicunivalent and sense preserving in the unit disc U = {z : |z|< 1} for which f (0)=fz(0)−1 = 0. For f = h+g ∈ SH , we may expressEntrato in redazione: 2 aprile 2013AMS 2010 Subject Classification: 30C45.Keywords: Harmonic, Univalent functions, Sense preserving, Convolution.198 M. K. AOUF - R. M. EL-ASHWAH - F. M. ABDULKAREMf (z) = z+∞∑k=2akzk +∞∑k=1bkzk, |b1|< 1, (2)where the analytic functions h and g are of the formh(z) = z+∞∑k=2|ak|zk, g(z) =∞∑k=1|bk|zk , |b1|< 1 . (3)In 1984 Clunie and Sheil-Small [5] investigated the class SH as well as its geo-metric subclasses and obtained some coefficient bounds.For positive real values of α1, . . . ,αq and β1, . . . ,βs (β j /∈Z−0 = {0,−1,−2, . . .} ;j = 1,2, . . . ,s), we now define the generalized hypergeometric function qFs(α1,. . . ,αq;β1, . . . ,βs;z) by (see, for example, [10, p.19])qFs(α1, . . . ,αq;β1, . . . ,βs;z) =∞∑k=0(α1)k . . .(αq)k(β1)k . . .(βs)k.zkk!(4)(q≤ s+1;q,s ∈ N0 = N∪{0},N= {1,2, . . .};z ∈U),where (a)m is the Pochhammer symbol defined by(a)m =Γ(a+m)Γ(a)={1 (m = 0),a(a+1) . . .(a+m−1) (m ∈ N). (5)Corresponding to the function h(α1, . . . ,αq;β1, . . . ,βs;z) defined byh(α1, . . . ,αq;β1, . . . ,βs;z) = z qFs(α1, . . . ,αq;β1, . . . ,βs;z), (6)which is defined by following Hadamard product (or convolution) forϕ(z) in the form:ϕ(z) = z+∞∑k=2φkzk,H(α1, . . . ,αq;β1, . . . ,βs)ϕ(z) = h(α1, . . . ,αq;β1, . . . ,βs;z)∗ϕ(z). (7)or,H(α1, . . . ,αq;β1, . . . ,βs)ϕ(z) = z+∞∑k=2Γk(α1)φkzk, (8)whereΓk(α1) =(α1)k−1 . . .(αq)k−1(β1)k−1 . . . .(βs)k−1· 1(1)k−1(k ≥ 2) . (9)If, for convenience, we writeHq,s[α1] = H(α1, . . . ,αq;β1, . . . ,βs). (10)UNIFIED REPRESENTATION OF A CERTAIN CLASS . . . 199The linear operator Hq,s[α1]was introduced and studied by Dziok and Srivastava[6].Al-Kharsani and Al-Khal [2] defined the modified Dziok-Srivastava operator ofthe harmonic function f = h+g , where h and g given by (3) as follows:Hq,s[α1] f (z) = Hq,s[α1]h(z)+Hq,s[α1]g(z). (11)Also let SH denote the subclass of SH consisting of functions f = h+g such thatthe functions h and g are of the formh(z) = z−∞∑k=2|ak|zk, g(z) =∞∑k=1|bk|zk , |b1|< 1. (12)By using the modified Dziok-Srivastava operator Hq,s[α1] defined by (11), Al-Khal [1] introduced and studied the class SH(α1,γ), consisting of functions f =h+g such that h and g are given by (3) and f satisfies the conditionℜ{(1+ eiα)z(Hq,s[α1] f (z))′z′Hq,s[α1] f (z)− eiα}≥ γ (0≤ γ < 1;> α ∈ R), (13)where z′ = ∂∂θ(z = reiθ), f ′(z) = ∂∂θ ( f (z) = f (reiθ )), 0 ≤ r < 1, and 0 ≤ θ <2pi .Also, Chandrashekar et al. [4] introduced and studied the class TH([α1,β1],γ),consisting of functions f = h+ g such that h and g are given by (12) andf satisfies the conditionℜ{1+(1+ eiα)z(Hq,s[α1]h(z))′′+2z(Hq,s[α]g(z))′+ z2(Hq,s[α1]g(z))′′z′ (Hq,s[α1] f (z))′}≥ γ,(14)(0≤ γ < 1; α ∈ R),where Hq,s[α1](z) is the modified Dziok-Srivastava operator defined by (11).To prove our results, we need the following lemmas.Lemma 1.1 ([1]). Let f = h+ g such that h and g are given by (12). Thenf (z) ∈ SH(α1,γ) if and only if∞∑k=1(2k−1− γ1− γ |ak|+2k+1+ γ1− γ |bk|)Γk(α1)≤ 2, (15)where a1 = 1, 0≤ γ < 1 and Γk(α1) is given by (9).200 M. K. AOUF - R. M. EL-ASHWAH - F. M. ABDULKAREMLemma 1.2 ([4]). Let f = h+ g such that h and g are given by (12). Thenf (z) ∈ TH([α1,β1],γ) if and only if∞∑k=1k(2k−1− γ1− γ |ak|+2k+1+ γ1− γ |bk|)Γk(α1)≤ 2, (16)where a1 = 1, 0≤ γ < 1 and Γk(α1) is given by (9).In view of Lemma 1.1 and Lemma 1.2, we introduce and study an interestingunification of the classes SH(α1,γ) and TH([α1,β1],γ).Definition 1.3. Let the class NH([α1],n,γ) (n∈N0 =N∪{0} and 0≤ γ < 1) bethe class of functions f = h+g such that h and g are given by (12), which satisfythe condition∞∑k=1kn(2k−1− γ1− γ |ak|+2k+1+ γ1− γ |bk|)Γk(α1)≤ 2, (17)where Γk(α1) is given by (9).Specializing the parameters n, α1 and γ, we obtain the following subclassesstudied by various authors:(i) NH([1] ,1,γ) =HLV(k,γ) [with k = 1] (see Kim et al. [8]);(ii) NH([α1] ,0,γ) = SH(α1,γ) (see Al-Khal [1]);(iii) NH([α1] ,1,γ) = TH([α1,β1],γ) (see Chandrashekar et al. [4]);(iv) NH([α1] ,0,γ) = HGN(k,α,γ) [with k = 1,γ = 0] (see El-Ashwah et al.[7]).In this paper, we have obtained some properties for the class NH([α1] ,n,γ).Further distortion theorems and extreme points are also obtained for functionsin the class NH([α1],n,γ).2. Main ResultsUnless otherwise mentioned, we assume in the reminder of this paper that0≤ γ < 1, q≤ s+1, q,s ∈ N0, for positive real α1, . . . ,αq and β1, . . . ,βs (β j /∈Z−0 = {0,−1,−2, . . .} ; j = 1,2, . . . ,s)and n ∈ N0.We begain with proving thatthe functions in the class NH([α1],n,γ) is sense-preserving, harmonic univalent.Theorem 2.1. Let f = h+ g, where h and g are given by (12)and f (z) ∈NH([α1] ,n,γ). Then f (z) is sense-preserving, harmonic univalent in U.UNIFIED REPRESENTATION OF A CERTAIN CLASS . . . 201Proof. If z1 6= z2,∣∣∣∣ f (z2)− f (z1)h(z2)−h(z1)∣∣∣∣≥ 1− ∣∣∣∣g(z2)−g(z1)h(z2)−h(z1)∣∣∣∣= 1−∣∣∣∣∣∣∣∣∞∑k=1bk(zk2− zk1)(z2− z1)−∞∑k=2ak(zk2− zk1)∣∣∣∣∣∣∣∣≥ 1−∞∑k=1k |bk|1−∞∑k=2k |ak|,by using (17), we have∣∣∣∣ f (z2)− f (z1)h(z2)−h(z1)∣∣∣∣≥ 1−∞∑k=1kn(2k+1+ γ1− γ)Γk(α1) |bk|1−∞∑k=2kn(2k−1− γ1− γ)Γk(α1) |ak|> 0,which proves the univalence. Also f is sense-preserving in U since|h′(z)| ≥ 1−∞∑k=2k |ak| |z|k−1 > 1−∞∑k=2k |ak|≥ 1−∞∑k=1kn(2k−1− γ1− γ)|ak|Γk(α1)≥∞∑k=1kn(2k+1+ γ1− γ)|bk|Γk(α1)≥∞∑k=1k |bk| |z|k−1 ≥ |g′(z)|.The harmonic univalent functionsf (z) = z+∞∑k=2(1− γ)kn(2k−1− γ)Γk(α1)xkzk+∞∑k=1(1− γ)kn(2k+1+ γ)Γk(α1)ykzk, (18)where∞∑k=2|xk|+∞∑k=1|yk| = 1, shows that the coefficient bound given by (17) issharp. The functions of the form (18) are in the class NH([α1] ,n,γ) because∞∑k=1kn[2k−1− γ1− γ |ak|+2k+1+ γ1− γ |bk|]Γk(α1) = 1+∞∑k=2|xk|+∞∑k=1|yk|= 2.This completes the proof of Theorem 2.1.Next, we obtain the distortion theorems and extreme points of the closedconvex hull for functions in the class NH([α1] ,n,γ).202 M. K. AOUF - R. M. EL-ASHWAH - F. M. ABDULKAREMTheorem 2.2. Let the function f = h+g, where h and g are given by (12) be inthe class NH([α1] ,n,γ), then| f (z)| ≥ (1−|b1|)r− 12nΓ2(α1)[(1− γ)(3− γ) −(3+ γ)(3− γ) |b1|]r2, (19)and| f (z)| ≤ (1+ |b1|)r+ 12nΓ2(α1)[(1− γ)(3− γ) −(3+ γ)(3− γ) |b1|]r2. (20)The equalities in (19) and (20) are attained for the functions f given byf (z) = (1−b1)z− 12nΓ2(α1)[(1− γ)(3− γ) −(3+ γ)(3− γ) |b1|]2z2, (21)andf (z) = (1+b1)z+12nΓ2(α1)[(1− γ)(3− γ) −(3+ γ)(3− γ) |b1|]z2, (22)where |b1| ≤ 1− γ3+ γ .Proof. Let f (z) ∈ NH([α1] ,n,γ), then we have| f (z)| ≥ (1−|b1|)r−∞∑k=2(|ak|+ |bk|)rk≥ (1−|b1|)r− r2∞∑k=2(|ak|+ |bk|)≥ (1−|b1|)r− (1− γ)2n(3− γ)Γ2(α1)r2..∞∑k=2kn[(3− γ)(1− γ) |ak|+(3− γ)(1− γ) |bk|]Γk(α1)≥ (1−|b1|)r− (1− γ)2n(3− γ)Γ2(α1)r2..∞∑k=2kn((2k− γ−1)(1− γ) |ak|+(2k+ γ+1)(1− γ) |bk|)Γk(α1)≥ (1−|b1|)r− (1− γ)2n(3− γ)Γ2(α1)[1− (3+ γ)(1− γ) |b1|]r2≥ (1−|b1|)r− 12nΓ2(α1)[(1− γ)(3− γ) −(3+ γ)(3− γ) |b1|]r2,which proves the asserion (19) of Theorem 2.2. The proof of the assertion (20)is similar, thus, we omit it.UNIFIED REPRESENTATION OF A CERTAIN CLASS . . . 203Remark 2.3. (i) Putting n= 1, q= 2,s= 1,α1 =α2 = 1 and β1 = 1 in Theorem2.2, we improve the result obtained by Kim et al. [8, Theorem 4.2, with k = 1],by adding the condition |b1| ≤ 1− γ3+ γ ;(ii) Putting n = 0 in Theorem 2.2, we improve the result obtained by Al-Khal[1, Theorem 2.3], by adding the condition |b1| ≤ 1− γ3+ γ .The following covering result follows the left hand inequality of Theorem 2.2Corollary 2.4. Let f ∈ NH([α1] ,n,γ), then{w : |w|< (3 ·2nΓ2(α1)−1)− (2nΓ2(α1)−1)γ2n(3− γ)Γ2(α1)−3(2nΓ2(α1)−1)− (2nΓ2(α1)+1)γ2n(3− γ)Γ2(α1) |b1|}⊂ f (U),where |b1| ≤(1− γ3+ γ).Remark 2.5. Putting n = 1 in Theorem 2.2, we obtain of the following corol-lary:Corollary 2.6. Let the function f = h+ g, where h and g are given by (12) bein the class TH([α1,β1],γ), then| f (z)| ≥ (1−|b1|)r− 12Γ2(α1)[(1− γ)(3− γ) −(3+ γ)(3− γ) |b1|]r2, (23)and| f (z)| ≤ (1+ |b1|)r+ 12Γ2(α1)[(1− γ)(3− γ) −(3+ γ)(3− γ) |b1|]r2. (24)The equalities in (23) and (24) are attained for the functions f given byf (z) = (1−b1)z− 12Γ2(α1)[(1− γ)(3− γ) −(3+ γ)(3− γ) |b1|]2z2, (25)andf (z) = (1+b1)z+12Γ2(α1)[(1− γ)(3− γ) −(3+ γ)(3− γ) |b1|]z2, (26)where |b1| ≤(1− γ3+ γ).204 M. K. AOUF - R. M. EL-ASHWAH - F. M. ABDULKAREMTheorem 2.7. Let f = h+ g, where h and g are given by (12). Then f (z) ∈clco NH([α1] ,n,γ) if and only iff (z) =∞∑k=1[Xkhk(z)+Ykgk(z)] , (27)whereh1(z) = z, (28)hk(z) = z− (1− γ)kn(2k−1− γ)Γk(α1)zk (k = 2,3, . . .), (29)andgk(z) = z+(1− γ)kn(2k+1+ γ)Γk(α1)zk (k = 1,2, . . .), (30)where∞∑k=1(Xk +Yk) = 1, Xk ≥ 0 and Yk ≥ 0. In particular, the extreme points ofthe class NH([α1] ,n,γ) are {hk}(k ≥ 2)and {gk}(k ≥ 1), respectively.Proof. For a function f (z) of the form (27), we havef (z) =∞∑k=1[Xkhk(z)+Ykgk(z)]=∞∑k=1[Xk(z− (1− γ)kn(2k−1− γ)Γk(α1)zk)+Yk(z+(1− γ)kn(2k+1+ γ)Γk(α1)zk)]= z−∞∑k=2(1− γ)kn(2k−1− γ)Γk(α1)Xkzk+∞∑k=1(1− γ)kn(2k−1− γ)Γk(α1)Ykzk.But,∞∑k=2(kn(2k−1− γ)Γk(α1)(1− γ) ·(1− γ)kn(2k−1− γ)Γk(α1)Xk)+∞∑k=1(kn(2k−1− γ)Γk(α1)(1− γ) ·(1− γ)kn(2k−1− γ)Γk(α1)Yk)=∞∑k=2Xk +∞∑k=1Yk = 1−X1 ≤ 1.UNIFIED REPRESENTATION OF A CERTAIN CLASS . . . 205Thus f (z) ∈ clco NH([α1] ,n,γ).Conversely, assume that f (z) ∈ clco NH([α1] ,n,γ). SetXk =kn(2k−1− γ)Γk(α1)(1− γ) |ak| (k = 2,3, . . .),Yk =kn(2k+1+ γ)Γk(α1)(1− γ) |bk| (k = 1,2, . . .)Then by using (17), we have0≤ Xk ≤ 1 (k = 2,3, . . .) and 0≤ Yk ≤ 1(k = 1,2,3, . . .).Define X1 = 1−∞∑k=2Xk−∞∑k=1Yk. Thus we obtain f (z) =∞∑k=1[Xkhk(z)+Ykgk(z)].This completes the proof of Theorem 2.7.Finally, we discusse the convolution properties, convex combination andintegral operator.Let the functions fm(z) defined byfm(z) = z−∞∑k=2|ak,m|zk +∞∑k=1|bk,m|zk (m = 1,2) (31)are in the class NH([α1] ,n,γ), we denote by ( f1 ∗ f2)(z) the convolution (orHadamard Product) of the functions f1(z) and f2(z), that is,( f1 ∗ f2)(z) = z−∞∑k=2|ak,1| |ak,2|zk +∞∑k=1|bk,1| |bk,2|zk. (32)We first show that the class NH([α1] ,n,γ) is closed under convolution.Theorem 2.8. For 0 ≤ δ ≤ γ < 1, let the functions f1 ∈ NH([α1] ,n,γ) andf2 ∈ NH([α1] ,n,δ ). Then( f1 ∗ f2)(z) ∈ NH([α1] ,n,γ)⊂ NH([α1] ,n,δ ). (33)Proof. Let fm(z) (m = 1,2) are given by (31), where f1(z) be in the classNH([α1] ,n,γ) and f2(z) be in the class NH([α1] ,n,δ ). We wish to show that thecoefficients of ( f1 ∗ f2)(z) satisfy the required condition given in (17). For f2 ∈NH([α1] ,n,δ ), we note that |ak,2| < 1 and |bk,2| < 1. Now for the convolution206 M. K. AOUF - R. M. EL-ASHWAH - F. M. ABDULKAREMfunctions ( f1 ∗ f2)(z), we obtain∞∑k=1kn(2k−1−δ1−δ |ak,1| |ak,2|+2k+1+δ1−δ |bk,1| |bk,2|)Γk(α1)≤∞∑k=1kn(2k−1−δ1−δ |ak,1|+2k+1+δ1−δ |bk,1|)Γk(α1)≤∞∑k=1kn(2k−1− γ1− γ |ak,1|+2k+1+ γ1− γ |bk,1|)Γk(α1)≤ 2,since 0≤ δ ≤ γ < 1 and f1 ∈NH([α1] ,n,γ). Thus ( f1∗ f2)(z)∈NH([α1] ,n,γ)⊂NH([α1] ,n,δ ). This completes the proof of Theorem 2.8.Theorem 2.9. The class NH([α1] ,n,γ) is closed under convex combinations.Proof. For i = 1,2, . . . , let fi ∈ NH([α1] ,n,γ), wherefi(z) = z−∞∑k=2|ak,i|zk +∞∑k=1|bk,i|zk,then from (17), for∞∑i=1ti = 1, 0 ≤ ti < 1, the convex combination of fi can bewritten as∞∑i=1ti fi(z) = z−∞∑k=2(∞∑i=1ti |ak,i|)zk +∞∑k=1(∞∑i=1ti |bk,i|)zk. (34)Then by (17), we have∞∑k=1kn[(2k−1− γ)1− γ(∞∑i=1ti |ak,i|)+(2k+1+ γ)1− γ(∞∑i=1ti |bk,i|)]Γk(α1)=∞∑i=1ti(∞∑k=1kn[(2k−1− γ)1− γ |ak,i|+(2k+1+ γ)1− γ |bk,i|])Γk(α1)≤ 2∞∑i=1ti = 2.This completes the proof of Theorem 2.9.We examine a closure property of the class NH([α1] ,n,γ) under the general-ized Bernardi-Libera-Livingston integral operator (see [3, 9]) Lc( f (z)) which isdefined byLc( f (z)) =c+1zcz∫0tc−1 f (t)dt, c >−1. (35)UNIFIED REPRESENTATION OF A CERTAIN CLASS . . . 207Theorem 2.10. Let f (z) ∈ NH([α1] ,n,γ). Then Lc( f (z)) ∈ NH([α1] ,n,γ).Proof. From (35), it follows thatLc( f (z)) =c+1zcz∫0tc−1[h(t)+g(t)]dt=c+1zc z∫0tc−1(t−∞∑k=2aktk)dt+z∫0tc−1(∞∑k=2bktk)dt= z−∞∑k=2Akzk +∞∑k=1Bkzk,whereAk =c+1c+ kak, Bk =c+1c+ kbk.Therefore,∞∑k=1kn[2k−1− γ1− γ(c+1c+ k)|ak|+ 2k+1+ γ1− γ(c+1c+ k)|bk|]Γk(α1)≤∞∑k=1kn[2k−1− γ1− γ |ak|+2k+1+ γ1− γ |bk|]Γk(α1)≤ 2.Since f (z) ∈ NH([α1] ,n,γ), by Theorem 2.1, Lc( f (z)) ∈ NH([α1] ,n,γ). Thiscompletes the proof of Theorem 2.10.Remark 2.11. (i) Putting n = 1, q = 2,s = 1,α1 = α2 = 1 and β1 = 1 in theabove results, we obtain the corresponding results obtained by Kim et al. [8,with k = 1];(ii) Putting n = 0 in all the above results, we obtain the corresponding resultsobtained by Al-Khal [1];(iii) Putting n = 1 in all the above results, we obtain the corresponding resultsobtained by Chandrashekar et al. [4];(iv) Putting n = 0 in all the above results, we obtain the correspondingresults obtained by El-Ashwah et al. [7, with k = 1 and γ = 0].3. AcknowledgmentsThe authors thank the referees for their valuable suggestions which led to theimprovement of this paper.208 M. K. AOUF - R. M. EL-ASHWAH - F. M. ABDULKAREMREFERENCES[1] R. A. Al-Khal, Goodman-Ronning-type harmonic univalent functions based onDziok-Srivastava operator, Appl. Math. Sci. 5 (12) (2011), 573–584.[2] H. A. Al-Kharsani - R. A. Al-Khal, Univalent harmonic functions, J. Inequal. PureAppl. Math. 8 (2) (2007), Art. 59, 1–8.[3] S. D. Bernardi, Convex and starlike univalent functions, Trans. Amer. Math. Soc.135 (1969), 429–446.[4] R. Chandrashekar - G. Murugusundaramoorthy - S. K. Lee - K. G. Subramanian,A class of complex-valued harmonic functions defined by Dziok-Srivastava oper-ator, Chamchuri J. Math. 1 (2) (2009), 31–42.[5] J. Clunie - T. Sheil-Small, Harmonic univalent functions, Ann. Acad. Sci. Fenn.Ser. A. I. Math. 9 (1984), 3–25.[6] J. Dziok - H. M. Srivastava, Classes of analytic functions with the generalizedhypergeometric function, Appl. Math. Comput. 103 (1999), 1–13.[7] R. M. El-Ashwah - M. K. Aouf - A. A. M. Hassan - A. H. Hassan, A unified rep-resentation of some starlike and convex harmonic functions with negative coeffi-cients, Opuscula Math. 33 (2) (2013), 273–281.[8] Y. C. Kim - J. M. Jahangiri - J. H. Choi, Certain convex harmonic functions, Inter-nat. J. Math. Math. Sci. 29 (8) (2002), 459–465.[9] R. J. Libera, Some classes of regular univalent functions, Proc. Amer. Math. Soc.16 (1965), 755–758 .[10] H. M. Srivastava - P. W. Karlsson, Multiple gaussian hypergeometric series, EllisHorwood Ltd., Chichester, Halsted Press (John Wiley & Sons), New York, 1985.M. K. AOUFDepartment of Mathematics, Faculty of Science,Mansoura University, Mansoura 35516, Egypte-mail: mkaouf127@yahoo.comR. M. EL-ASHWAHDepartment of Mathematics, Faculty of Science, Damietta University,New Damietta Egypt 34517, Egypte-mail: r elashwah@yahoo.comF. M. ABDULKAREMDepartment of Mathematics, Faculty of Science, Damietta University,New Damietta Egypt 34517, Egypte-mail: fammari76@gmail.com

Unified representation of a certain class of harmonic univalent functions defined by Dziok-Srivastava operator

In this paper, we investigate several properties of the harmonic class defined by the modified Dziok-Sirvastava operator, obtain distortion theorem, extreme points, convolution condition, convex combinations and integral operator for this class. Some of our results generalize previously known results.<br /

M. K. Aouf

R. M. El-Ashwah

F. M. Abdulkarem

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Unified representation of a certain class of harmonic univalent functions defined by Dziok-Srivastava operator

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