In this paper we introduce a new class of harmonic univalent functions defined by the Dziok-Srivastava operator. Coefficient estimates, extreme points, distortion bounds and convex combination for functions belonging to this class are obtained and also for a class preserving the integral operator

Adwan, E. A.

Aouf, M. K.

Mostafa, A. O.

Shamandy, A.

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

LE MATEMATICHEVol. LXVIII (2013) – Fasc. I, pp. 165–177doi: 10.4418/2013.68.1.13SUBCLASS OF HARMONIC UNIVALENT FUNCTIONSDEFINED BY DZIOK-SRIVASTAVA OPERATORM. K. AOUF - A. O. MOSTAFA - A. SHAMANDY - E. A. ADWANIn this paper we introduce a new class of harmonic univalent functionsdefined by the Dziok-Srivastava operator. Coefficient estimates, extremepoints, distortion bounds and convex combination for functions belong-ing to this class are obtained and also for a class preserving the integraloperator.1. IntroductionA continuous complex-valued function f = u+ iv is defined in a simply con-nected complex domain D is said to be harmonic in D if both u and v are realharmonic in D. In any simply connected domain we can writef = h+g, (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 [6]).Denote by SH the class of functions f of the form (1) that are harmonicunivalent and sense-preserving in the unit disc U = {z : |z| < 1} for whichEntrato in redazione: 31 maggio 2012AMS 2010 Subject Classification: 30C45.Keywords: Harmonic univalent functions, Dziok-Srivastava operator, Extreme points.166 M. K. AOUF - A. O. MOSTAFA - A. SHAMANDY - E. A. ADWANf (0) = fz(0)− 1 = 0. Then for f = h+ g ∈ SH we may express the analyticfunctions h and g ash(z) = z+∞∑n=2anzn, g(z) =∞∑n=1bnzn, |b1|< 1. (2)In [6] Clunie and Shell-Small investigated the class SH as well as its geo-metric subclasses and obtained some coefficient bounds. Since then, there havebeen several related papers on SH and its subclasses. Denote by VH the subclassof SH consisting of functions of the form f = h+g, whereh(z) = z+∞∑n=2|an|zn,g(z) =∞∑n=1|bn|zn, |b1|< 1. (3)For positive real parameters α1, . . . ,αq and β1, . . . ,βs (β j ∈ C\Z−0 , with Z−0 =0,−1,−2, . . . and j = 1,2, . . . ,s), the generalized hypergeometric function qFsis defined byqFs(α1, . . . ,αq;β1, . . . ,βs;z) =∞∑n=0(α1)n . . .(αq)n(β1)n . . .(βs)nn!zn(q≤ s+1;s,q ∈ N0 = N∪{0}, N= {1,2, . . .}; z ∈U),where (θ)n is the Pochhammer symbol defined in terms of the Gamma functionΓ by(θ)n =Γ(θ +n)Γ(θ)={1 (n = 0)θ(θ +1) . . .(θ +n−1) (n ∈ N).For the function h(α1, . . . ,αq;β1, . . .βs;z) = zqFs(α1, . . . ,αq; β1, . . . ,βs; z),the Dziok-Srivastava linear operator (see [8] and [9]) Hq,s(α1, . . . ,αq; β1, . . .. . . ,βs) is defined by the Hadamard product as follows:Hq,s(α1, . . . ,αq; β1, . . . ,βs) f (z) = h(α1, . . . ,αq;β1, . . .βs;z)∗ f (z)= z+∞∑n=2Ψn (α1)anzn (z ∈U), (4)whereΨn (α1) =(α1)n−1 . . .(αq)n−1(β1)n−1 . . .(βs)n−1 (n−1)! . (5)For brevity, we writeHq,s(α1, . . . ,αq; β1, . . . ,βs;z) f (z) = Hq,s(α1) f (z).SUBCLASS OF HARMONIC UNIVALENT FUNCTIONS 167Al-Kharsani and Al-Khal [2] and Al-Khal [1] defined the modified Dziok-Srivastava operator of the harmonic function f = h+g given by (1) asHq,s(α1) f (z) = Hq,s(α1)h(z)+Hq,s(α1)g(z).For 1 < γ ≤ 2 and for all z ∈U, let SHq,s([α1] ;γ) denote the family of harmonicfunctions f (z) = h(z)+g(z), where h and g are given by (2) and satisfying theanalytic criterionℜ{Hq,s(α1)h(z)+Hq,s(α1)g(z)z}< γ. (6)Let SHq,s([α1] ;γ) be the subclass of SHq,s([α1] ;γ) consisting of functions f =h+g such that h and g given by (3).We note that for suitable choices of q and s, we obtain the following sub-classes:1) Putting q = 2,s = 1,α1 = a(a > 0) ,α2 = 1 and β1 = c(c > 0) in (6), theclass SH2,1([a,1;c] ;γ) reduces to the class LH(a,c;γ)={f ∈ SH : ℜ{L(a,c)h(z)+L(a,c)g(z)z}< γ, 1 < γ ≤ 2,a,c > 0,z ∈U} ,where L(a,c) is the modified Carlson-Shaffer operator (see [3]), defined as fol-lows:L(a,c) f (z) = L(a,c)h(z)+L(a,c)g(z);2) Putting q = 2,s = 1,α1 = λ + 1(λ > −1) and α2 = β1 = 1 in (6), the classSH2,1([λ +1] ;γ) reduces to the class W H(λ ;γ)={f ∈ SH : ℜ{Dλh(z)+Dλg(z)z}< γ, 1 < γ ≤ 2,λ >−1,z ∈U} ,where Dλ is the modified Ruscheweyh derivative operator (see [13]), defined asfollows:Dλ f (z) = Dλh(z)+Dλg(z);3) Putting q = 2,s = 1,α1 = v+1(v >−1) ,α2 = 1 and β1 = v+2 in (6), theclass SH2,1([v+1,1;v+2] ;γ) reduces to the class ζH(v;γ)={f ∈ SH : ℜ{Jvh(z)+ Jvg(z)z}< γ, 1 < γ ≤ 2,v >−1,z ∈U} ,168 M. K. AOUF - A. O. MOSTAFA - A. SHAMANDY - E. A. ADWANwhere Jv is the modified generalized Bernardi-Libera-Livingston operator (see[10]), defined as follows:Jv f (z) = Jvh(z)+ Jvg(z);4) Putting q = 2,s = 1,α1 = 2,α2 = 1 and β1 = 2−µ (µ 6= 2,3, . . .) in (6), theclass SH2,1([2,1;2−µ] ;γ) reduces to the class FH(µ;γ)={f ∈ SH : ℜ{Ωµz h(z)+Ωµz g(z)z}< γ, 1 < γ ≤ 2,µ 6= 2,3, . . . ,z ∈U} ,where Ωµz is the modified Srivastava-Owa fractional derivative operator (see[12]), defined as follows:Ωµz f (z) =Ωµz h(z)+Ωµz g(z);5) Putting q= 2,s= 1,α1 = µ (µ > 0) ,α2 = 1 and β1 = λ +1(λ >−1) in (6),the class SH2,1([µ,1;λ +1] ;γ) reduces to the class £H(µ,λ ;γ) ={f ∈ SH : ℜ{Iµ,λh(z)+ Iµ,λg(z)z}< γ, 1 < γ ≤ 2,µ > 0,λ >−1,z ∈U} ,where Iλ ,µ is the modified Choi-Saigo-Srivastava operator (see [5]), defined asfollows:Iµ,λ f (z) = Iµ,λh(z)+ Iµ,λg(z);6) Putting q = 2,s = 1,α1 = 2,α2 = 1 and β1 = k + 1(k >−1) in (6), theclass SH2,1([2,1;k+1] ;γ) reduces to the class AH(k;γ)={f ∈ SH : ℜ{Ikh(z)+ Ikg(z)z}< γ, 1 < γ ≤ 2,k >−1,z ∈U} ,where Ik is the modified Noor integral operator (see [11]), defined as follows:Ik f (z) = Ikh(z)+ Ikg(z);7) Putting q= 2,s= 1,α1 = c(c > 0) ,α2 = λ +1(λ >−1) and β1 = a(a > 0)in (6), the class SH2,1([c,λ +1;a] ;γ) reduces to the class zH(c,a,λ ;γ)={f ∈ SH : ℜ{Iλ (a,c)h(z)+ Iλ (a,c)g(z)z}< γ, 1 < γ ≤ 2,c > 0,λ>−1,a > 0,z ∈U} ,SUBCLASS OF HARMONIC UNIVALENT FUNCTIONS 169where Iλ (a,c) is the modified Cho-Kwon-Srivastava operator (see [4]), definedas follows:Iλ (a,c) f (z) = Iλ (a,c)h(z)+ Iλ (a,c)g(z).2. Coefficient estimatesUnless otherwise mentioned, we shall assume in the reminder of this paper that,the parameters α1, . . . ,αq and β1, . . . ,βs are positive real numbers, 1< γ ≤ 2,z∈U and Ψn (α1) is defined by (5).Theorem 2.1. Let f = h+g be such that h(z) and g(z) given by (2). Further-more, let∞∑n=2Ψn (α1) |an|+∞∑n=1Ψn (α1) |bn| ≤ γ−1. (7)Then f (z) is sense-preserving, harmonic univalent in U and f (z)∈ SHq,s([α1];γ).Proof. If z1 6= z2, then∣∣∣∣ f (z1)− f (z2)h(z1)−h(z2)∣∣∣∣≥ 1− ∣∣∣∣g(z1)−g(z2)h(z1)−h(z2)∣∣∣∣= 1−∣∣∣∣∣∣∣∣∞∑n=1bn (zn1− zn2)(z1− z2)+∞∑n=2an (zn1− zn2)∣∣∣∣∣∣∣∣> 1−∞∑n=1n |bn|1−∞∑n=2n |an|≥ 1−Ψn(α1)γ−1 |bn|Ψn(α1)γ−1 |an|≥ 0,which proves univalence. Note that f (z) is sense-preserving in U . This is be-cause ∣∣∣h′(z)∣∣∣≥ 1− ∞∑n=2n |an| |z|n−1> 1−∞∑n=2n |an| ≥∞∑n=2Ψn(α1)γ−1 |an|≥∞∑n=1Ψn(α1)γ−1 |bn| ≥∞∑n=1n |bn|>∞∑n=1n |bn|∣∣zn−1∣∣≥ ∣∣∣g′ (z)∣∣∣ .170 M. K. AOUF - A. O. MOSTAFA - A. SHAMANDY - E. A. ADWANNow we will show that f (z) ∈ SHq,s([α1] ;γ). We only need to show that if (7)holds then the condition (6) is satisfied. Using the fact that Re{w} < γ if andonly if |1−w|< |w− (2γ−1)|, it suffices to show that∣∣∣∣∣∣∣∣Hq,s(α1)h(z)+Hq,s(α1)g(z)z−1Hq,s(α1)h(z)+Hq,s(α1)g(z)z− (2γ−1)∣∣∣∣∣∣∣∣< 1.We have∣∣∣∣∣∣Hq,s(α1)h(z)+Hq,s(α1)g(z)z −1Hq,s(α1)h(z)+Hq,s(α1)g(z)z − (2γ−1)∣∣∣∣∣∣=∣∣∣∣∣∣∞∑n=2Ψn(α1)anzn−1+∞∑n=1Ψn(α1)bnzn−12(γ−1)−∞∑n=2Ψn(α1)anzn−1+∞∑n=1Ψn(α1)bnzn−1∣∣∣∣∣∣≤∞∑n=2Ψn (α1) |an| |z|n−1+∞∑n=1Ψn (α1) |bn| |z|n−12(γ−1)−∞∑n=2Ψn (α1) |an| |z|n−1−∞∑n=1Ψn (α1) |bn| |z|n−1<∞∑n=2Ψn (α1) |an|+∞∑n=1Ψn (α1) |bn|2(γ−1)−∞∑n=2Ψn (α1) |an|−∞∑n=1Ψn (α1) |bn|,which is bounded above by 1 by using (7). This completes the proof of Theorem2.1.The harmonic univalent functions of the formf (z) = z+∞∑n=2γ−1Ψn(α1)xnzn+∞∑n=1γ−1Ψn(α1)ynzn, (8)where∞∑n=2|xn|+∞∑n=1|yn| = 1, show that the coefficient bound given by (7) issharp. It is worthy to note that the function of the form (8) belongs to theclass SHq,s([α1] ;γ) for all∞∑n=2|xn|+∞∑n=1|yn| ≤ 1 because coefficient inequality(7) holds.Theorem 2.2. A function f (z) of the form (3) is in the class SHq,s([α1] ;γ) if andonly if∞∑n=2Ψn (α1) |an|+∞∑n=1Ψn (α1) |bn| ≤ γ−1. (9)SUBCLASS OF HARMONIC UNIVALENT FUNCTIONS 171Proof. Since SHq,s([α1] ;γ)⊂ SHq,s([α1] ;γ), we only need to prove the “only if”part of this theorem. To this end , for functions f (z) of the form (3), we noticethat the conditionℜ{Hq,s(α1)h(z)+Hq,s(α1)g(z)z}< γ,i.e.ℜ{1+∞∑n=2Ψn (α1) |an|zn−1+∞∑n=1Ψn (α1) |bn|zn−1}< γ.Letting z→ 1− along the real axis, we obtain the inequality (9). This completesthe proof of Theorem 2.Remark 2.3. Putting q= 2,s= 1, α1 = 2 and α2 = β1 = 1 in Theorems 2.1 and2.2, we obtain the result obtained by Dixit and Porwal [7, Theorem 2.1].3. Distortion theoremTheorem 3.1. Let the function f (z) given in (3) belong to the class SHq,s([α1];γ).Then for |z|= r < 1, we have(1−|b1|)r− γ−1−|b1|Ψ2 (α1) r2 ≤ | f (z)| ≤ (1+ |b1|)r+ γ−1−|b1|Ψ2 (α1) r2 (10)for |b1| ≤ γ−1. The results are sharp with equality for the functions f (z) definedbyf (z) = z+b1z+γ−1−b1Ψ2 (α1)z2 (11)andf (z) = z−b1z− γ−1−b1Ψ2 (α1) z2. (12)Proof. We only prove the right-hand inequality. The proof for the left-handinequality is similar and will be omitted. Sincef (z) = z+∞∑n=2Ψn (α1)anzn+∞∑n=1Ψn (α1)bnzn,then| f (z)| ≤ (1+ |b1|)r+∞∑n=2(|an|+ |bn|)rn ≤ (1+ |b1|)r+∞∑n=2(|an|+ |bn|)r2 == (1+ |b1|)r+ γ−1Ψ2 (α1)∞∑n=2Ψ2 (α1)γ−1 (|an|+ |bn|)r2172 M. K. AOUF - A. O. MOSTAFA - A. SHAMANDY - E. A. ADWAN≤ (1+ |b1|)r+ γ−1Ψ2 (α1)∞∑n=2Ψn (α1)γ−1 (|an|+ |bn|)r2≤ (1+ |b1|)r+ γ−1−|b1|Ψ2 (α1) r2.The functions f (z) given by (11) and (12), respectively, for |b1| ≤ γ − 1 showthat the bounds given in Theorem 3.1 are sharp.4. Extreme pointsTheorem 4.1. Let f (z) be given by (3). Then f (z) ∈ SHq,s([α1] ;γ) if and only iff (z) =∞∑n=1(µnhn(z)+ηngn(z)) , (13)where h1(z) = z,hn(z) = z+γ−1Ψn (α1)zn (n = 2,3, . . .) (14)andgn(z) = z+γ−1Ψn (α1)zn (n = 1,2, . . .) , (15)µn ≥ 0,ηn ≥ 0,∞∑n=1(µn+ηn) = 1. In particular, the extreme points of the classSHq,s([α1] ;γ) are {hn} and {gn}, respectively.Proof. Suppose thatf (z) =∞∑n=1(µnhn(z)+ηngn(z)) = z+∞∑n=2γ−1Ψn (α1)µnzn +∞∑n=1γ−1Ψn (α1)ηnzn.Then∞∑n=2Ψn (α1)γ−1(γ−1Ψn (α1)µn)+∞∑n=1Ψn (α1)γ−1(γ−1Ψn (α1)ηn)=∞∑n=2µn+∞∑n=1ηn = 1−µ1 ≤ 1and so f (z) ∈ SHq,s([α1] ;γ).SUBCLASS OF HARMONIC UNIVALENT FUNCTIONS 173Conversely, if f (z) ∈ SHq,s([α1] ;γ), then|an| ≤ γ−1Ψn (α1) (n≥ 2)and|bn| ≤ γ−1Ψn (α1) (n≥ 1) .Settingµn =Ψn (α1)γ−1 |an| (n = 2,3, . . .)andηn =Ψn (α1)γ−1 |bn| (n = 1,2, . . .) .Since 0≤ µn ≤ 1 (n = 2,3, . . .) and 0≤ ηn ≤ 1 (n = 1,2, . . .), µ1 = 1−∞∑n=2µn+∞∑n=1ηn ≥ 0, then, we can see that f (z) can be expressed in the form (13). Thiscompletes the proof.5. Convolution and convex combinationFor our next theorem, we need to define the convolution of two harmonic func-tions.For harmonic functions of the form:f (z) = z+∞∑n=2|an|zn +∞∑n=1|bn|zn (16)andF (z) = z+∞∑n=2|An|zn +∞∑n=1|Bn|zn, (17)the convolution of f and F is given by( f ∗F)(z) = f (z)∗F (z) = z+∞∑n=2|anAn|zn +∞∑n=1|bnBn|zn. (18)Using this definition, the next theorem shows that the class SHq,s([α1] ;γ) isclosed under convolution.Theorem 5.1. For 1 < γ ≤ λ ≤ 2,let f (z) ∈ SHq,s([α1] ;λ ), where f (z) is givenby (16) and F ∈ SHq,s([α1] ;γ), where F (z) is given by (17). Then f ∗ F ∈SHq,s([α1] ;λ ) ⊂ SHq,s([α1] ;γ).174 M. K. AOUF - A. O. MOSTAFA - A. SHAMANDY - E. A. ADWANProof. We wish to show that the coefficients of f ∗F satisfy the required con-dition given in Theorem 2.2. For f ∈ SHq,s([α1] ;λ ) we note that |an| ≤ 1 and|bn| ≤ 1. Now, for the convolution function f ∗F we obtain∞∑n=2Ψn (α1)λ −1 |anAn|zn+∞∑n=1Ψn (α1)λ −1 |bnBn|zn≤∞∑n=2Ψn (α1)λ −1 |An|zn+∞∑n=1Ψn (α1)λ −1 |Bn|zn≤∞∑n=2Ψn (α1)γ−1 |An|zn+∞∑n=1Ψn (α1)γ−1 |Bn|zn≤ 1.Therefore f ∗F ∈ SHq,s([α1] ;λ ) ⊂ SHq,s([α1] ;γ).Now we show that the class SHq,s([α1] ;γ) is closed under convex combina-tions of its members.Theorem 5.2. The class SHq,s([α1] ;γ) is closed under convex combination.Proof. For i = 1,2,3, . . . , let fi ∈ SHq,s([α1] ;γ), where fi is given byfi = z+∞∑n=2|ani |zn +∞∑n=1|bni |zn.Then by using Theorem 2.2, we have∞∑n=2Ψn (α1)γ−1 |ani |zn +∞∑n=1Ψn (α1)γ−1 |bni |zn ≤ 1. (19)For∞∑n=1ti = 1, 0≤ ti ≤ 1, the convex combination of fi may be written as∞∑i=1ti fi (z) = z+∞∑n=2(∞∑i=1ti |ani |)zn +∞∑n=1(∞∑i=1ti |bni |)zn. (20)Then by (19), we have∞∑n=2Ψn (α1)γ−1(∞∑i=1ti |ani |)+∞∑n=1Ψn (α1)γ−1(∞∑i=1ti |bni |)=∞∑i=1ti(∞∑n=2Ψn (α1)γ−1 |ani | +∞∑n=1Ψn (α1)γ−1 |bni |)≤∞∑i=1ti = 1.SUBCLASS OF HARMONIC UNIVALENT FUNCTIONS 175This is the condition required by (7) and so∞∑i=1ti fi (z) ∈ SHq,s([α1] ;γ).6. A family of integral operatorsTheorem 6.1. Let the function f (z) defined by (1) be in the class SHq,s([α1] ;γ)and let c be a real number such that c >−1. Then the function F (z) defined byF (z) =c+1zcz∫0tc−1h(t)dt+c+1zcz∫0tc−1g(t)dt (c >−1) (21)also belongs to the class SHq,s([α1] ;γ).Proof. Let the function f (z) be defined by (1). Then from the representation(21) of F (z), it follows thatF (z) = z+∞∑n=2dnzn+∞∑n=1ζnzn,wheredn =(c+1c+n)|an| and ζn =(c+1c+n)|bn| .Therefore, we have∞∑n=2Ψn (α1)dn+∞∑n=1Ψn (α1)ζn=∞∑n=2Ψn (α1)(c+1c+n)|an|+∞∑n=1Ψn (α1)(c+1c+n)|bn|≤∞∑n=2Ψn (α1) |an|+∞∑n=1Ψn (α1) |bn| ≤ (1−α) ,since f (z) ∈ SHq,s([α1] ;γ). Hence, by Theorem 2.2, F (z) ∈ SHq,s([α1] ;γ). Thiscompletes the proof.Remark 6.2. Specializing the parameters q;s;α1, . . .αq and β1, . . . ,βs in theabove results, we obtain the corresponding results for the corresponding classesdefined in the introduction.176 M. K. AOUF - A. O. MOSTAFA - A. SHAMANDY - E. A. ADWANREFERENCES[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. Pureand Appl. Math. 8 (2) (2007), 1–8.[3] B. C. Carlson - D. B. Shaffer, Starlike and prestarlike hypergeometric functions,J. Math. Anal. Appl. 15 (1984), 737–745.[4] N. E. Cho - O. S. Kwon - H. M. Srivastava, Inclusion relationships and argumentproperties for certain subclasses of multivalent functions associated with a familyof linear operators, J. Math. Anal. Appl. 292 (2004), 470–483.[5] J. H. Choi - M. Saigo - H. M. Srivastava, Some inclusion properties of a certainfamily of integral operators, J. Math. Anal. Appl. 276 (2002), 432–445.[6] J. Clunie - T. Shell-Small, Harmonic univalent functions, Ann. Acad. Sci. Fenn.Ser. A. I. Math. 9 (1984), 3–25.[7] K. K. Dixit - S. Porwal, A subclass of harmonic univalent functions with positivecoefficients, Tamkang J. Math. 41(3) (2010), 261–269.[8] J. Dziok - H. M. Srivastava, Classes of analytic functions with the generalizedhypergeometric function, Appl. Math. Comput. 103 (1999), 1–13.[9] J. Dziok - H. M. Srivastava, Certain subclasses of analytic functions associatedwith the generalized hypergeometric function, Integral Transform. Spec. Funct.14 (2003), 7–18.[10] R. J. Libera, Some classes of regular univalent functions, Proc. Amer. Math. Soc.16 (1969), 755–758.[11] K. I. Noor, On new classes of integral operators, J. Natur. Geom. 16 (1999), 71–80.[12] S. Owa - H. M. Srivastava, Univalent and starlike generalized hypergeometricfunctions, Canad. J. Math. 39 (1987), 1057–1077.[13] S. Ruscheweyh, New criteria for univalent functions, Proc. Amer. Math. Soc. 49(1975), 109–115.M. K. AOUFDepartment of MathematicsFaculty of Science,Mansoura UniversityMansoura 35516, Egypte-mail: mkaouf127@yahoo.comSUBCLASS OF HARMONIC UNIVALENT FUNCTIONS 177A. O. MOSTAFADepartment of MathematicsFaculty of Science,Mansoura UniversityMansoura 35516, Egypte-mail: adelaeg254@yahoo.comA. SHAMANDYDepartment of MathematicsFaculty of Science,Mansoura UniversityMansoura 35516, Egypte-mail: shamandy16@hotmail.comE. A. ADWANDepartment of MathematicsFaculty of Science,Mansoura UniversityMansoura 35516, Egypte-mail: eman.a2009@yahoo.com

Subclass of harmonic univalent functions defined by Dziok-Srivastava operator

In this paper we introduce a new class of harmonic univalent functions defined by the Dziok-Srivastava operator. Coefficient estimates, extreme points, distortion bounds and convex combination for functions belonging to this class are obtained and also for a class preserving the integral operator.<br /

M. K. Aouf

A. O. Mostafa

A. Shamandy

E. A. Adwan

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Subclass of harmonic univalent functions defined by Dziok-Srivastava operator

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