In this paper the authors introduced a new generalized differintegral operator for meromorphic univalent functions in U* = {z : z ∈ C, 0 &lt; |z| &lt; 1}. The objective of this paper is to establish certain results concerning the Hadamard product of functions in the classes ∑^{∗,m}_{μ,λ} (α, β , γ, k) and ∑^h_{μ,λ} (α, β , γ, k)

Darus, Maslina

M. El-Ashwah, Rabha

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

LE MATEMATICHEVol. LXX (2015) – Fasc. I, pp. 179–189doi: 10.4418/2015.70.1.13HADAMARD PRODUCT CONCERNINGCERTAIN MEROMORPHIC FUNCTIONSRABHA M. EL-ASHWAH - MASLINA DARUSIn this paper the authors introduced a new generalized differintegraloperator for meromorphic univalent functions in U∗ = {z : z ∈ C, 0 <|z| < 1}. The objective of this paper is to establish certain results con-cerning the Hadamard product of functions in the classes ∑∗,mµ,λ (α,β ,γ,k)and ∑hµ,λ (α,β ,γ,k).1. IntroductionThroughout this paper, let the functions f of the form :ϕ(z) = c1z+∞∑n=2cnzn (c1 > 0; cn ≥ 0), (1)andψ(z) = d1z+∞∑n=2dnzn (d1 > 0; dn ≥ 0) (2)be regular and univalent in the unit disc U = {z : |z|< 1} and letf (z) =a0z+∞∑n=1anzn (a0 > 0; an ≥ 0), (3)Entrato in redazione: 25 maggio 2014AMS 2010 Subject Classification: 30C45.Keywords: Analytic, meromorphic, positive coefficients, Hadamard product.This work was partially supported by MOHE’s grant: FRGSTOPDOWN/2013/ST06/UKM/01/1.180 RABHA M. EL-ASHWAH - MASLINA DARUSfi(z) =a0,iz+∞∑n=1an,izn (a0,i > 0; an,i ≥ 0), (4)g(z) =b0z+∞∑n=1bnzn (b0 > 0; bn ≥ 0), (5)andg j(z) =b0, jz+∞∑n=1bn, jzn (b0, j > 0; bn, j ≥ 0), (6)be regular and univalent in the punctured unit disc U∗ = {z : 0 < |z| < 1} =U\{0}.For a function f (z) defined by (3) (with a0 = 1), El-Ashwah defined theintegral operator ℑβ ,mλ (α,µ) (see [10, with p = 1]) as follows:ℑβ ,mλ (α,µ) f (z) =1z+∞∑n=1[α+βα+β +(µ+λ )(n+1)]manzn(α,β ,µ,λ ≥ 0, α+β 6= 0, µ+λ 6= 0; m ∈ N0; z ∈U∗). (7)We note that(i) ℑ0,mλ (α,0) f (z) = Im(λ ,α) f (z)(α > 0,λ ≥ 0,m ∈ N0) (see El-Ashwah [11,with p = 1]);(ii) ℑ0,γ0 (1,1) f (z) = Pγ f (z)(γ > 0) (see Aqlan et al. [4, with p = 1]);(iii) ℑ0,m0 (α,µ) f (z) =Lm(α,µ) f (z)(α > 0,µ ≥ 0,m∈N0) (see Bulboaca et al.[6]);(iv) ℑ0,γ0 (1,µ) f (z) = Pγµ f (z)(γ > 0,µ > 0) (see Lashin [18]).Also we note that(i) ℑ0,m1 (α,1) f (z) = ℑm(α) f (z) =1z+∞∑n=1(αα+2(k+1))manzn(α > 0,m ∈N0):(ii) ℑ0,mp,λ (1,0) f (z) = ℑmp,λ f (z) =1zp+∞∑n=1(11+λ (k+1))manzn−p(λ ≥ 0,m ∈N0).Now we will extend the definition of the operator ℑβ ,mλ (α,µ) f (z) for m ∈Z= {0,±1,±2, . . .} and z ∈U∗ as follows:ℑβ ,m−1λ (α,µ) f (z) =(µ+λα+β)z−α+βµ+λ(z(α+βµ+λ)+1ℑβ ,mλ (α,µ) f (z))′,ℑβ ,m−2λ (α,µ) f (z) =(µ+λα+β)z−α+βµ+λ(z(α+βµ+λ)+1ℑβ ,m−1λ (α,µ) f (z))′,HADAMARD PRODUCT CONCERNING . . . 181ℑβ ,1λ (α,µ) f (z) =(µ+λα+β)z−α+βµ+λ(z(α+βµ+λ)+1ℑβ ,2λ (α,µ) f (z))′,ℑβ ,0λ (α,µ) f (z) =(µ+λα+β)z−α+βµ+λ(z(α+βµ+λ)+1ℑβ ,1λ (α,µ) f (z))′= f (z)ℑβ ,−1λ (α,µ) f (z) =(µ+λα+β)z−α+βµ+λ(z(α+βµ+λ)+1ℑβ ,0λ (α,µ) f (z))′,ℑβ ,−2λ (α,µ) f (z) =(µ+λα+β)z−α+βµ+λ(z(α+βµ+λ)+1ℑβ ,−1λ (α,µ) f (z))′,ℑβ ,−m+1λ (α,µ) f (z) =(µ+λα+β)z−α+βµ+λ(z(α+βµ+λ)+1ℑβ ,−m+2λ (α,µ) f (z))′,ℑβ ,−mλ (α,µ) f (z) =(µ+λα+β)z−α+βµ+λ(z(α+βµ+λ)+1ℑβ ,−m+1λ (α,µ) f (z))′.We see that for f ∈ Σ, we haveℑβ ,−mλ (α,µ) f (z) =1z+∞∑n=1[α+βα+β +(µ+λ )(n+1)]−manzn(α,β ,µ,λ ≥ 0, α+β 6= 0, µ+λ 6= 0; m ∈ N0; z ∈U∗).We note that(i)ℑ0,−m0 (α,µ) f (z)= Jm(α,µ) f (z)(α > 0,µ ≥ 0,m∈N0) (see El-Ashwah [13,with p = 1]);(ii) ℑ0,−m1 (1,µ) f (z) = I(m,µ) f (z)(µ ≥ 0,m ∈ N0) (see Cho et al. [7, 8]);(iii) ℑ0,−m1 (α,1) f (z)) = Dmα f (z)(α > 0,m ∈ N0) (see Al-Oboudi and Al-Zkeri[1]);(iv) ℑ0,−m1 (1,1) f (z) = Im f (z)(m ∈ N0) (see Uralegaddi and Somanatha [24]).On the other hand for α+β = µ+λ = m = 1, we haveℑβ ,−1λ (α,µ) f (z) =(z2 f (z))′z=1z+∞∑n=1(n+2)anzn.In general we will writeℑβ ,mλ (α,µ) f (z) =1z+∞∑n=1[α+β +(µ+λ )(n+1)α+β]manzn(α,β ,µ,λ ≥ 0, α+β 6= 0, µ+λ 6= 0; m ∈ Z; z ∈U∗). (8)182 RABHA M. EL-ASHWAH - MASLINA DARUSWith the help of the differintegral operator ℑβ ,mλ (α,µ) f (z), we define theclass ∑∗,mµ,λ (α,β ,γ,k) as follows.Denote by ∑∗,mµ,λ (α,β ,γ,k) the class of functions f (z) given by (3) whichsatisfy the condition−Rez(ℑβ ,mλ (α,µ) f (z))′ℑβ ,mλ (α,µ) f (z)+ γ> k∣∣∣∣∣∣∣z(ℑβ ,mλ (α,µ) f (z))′ℑβ ,mλ (α,µ) f (z)+1∣∣∣∣∣∣∣(0≤ γ < 1; k≥ 0; α,β ,µ,λ ≥ 0, α+β 6= 0, µ+λ 6= 0; m ∈ Z; z ∈U∗). (9)Taking m = k = 0, the class ∑∗,mµ,λ (α,β ,γ,k) with a0 = 1 will reduce to the class∑S∗(γ) (see Mogra et al. [21, with β = 1]), and El-Ashwah and Aouf ([12, withβ = 1 and k = 0]).Also, we observe that the class∑∗,mµ,λ (α,β ,γ,k) reduces to several interestingmany other classes for different choices of α,β ,µ,λ ,k and m.Using similar arguments as given in [5] (see also Aouf et al. [3, withλ = 0]), we can easily prove the following results for functions in the class∑∗,mµ,λ (α,β ,γ,k).A function f (z)∈∑∗,mµ,λ (α,β ,γ,k)(α,β ,µ,λ ≥ 0,α+β 6= 0,µ+λ 6= 0;m∈Z) if and only if∞∑n=1[α+β +(µ+λ )(n+1)α+β]m[n(k+1)+(k+ γ)]an ≤ (1− γ)a0, (10)where 0≤ γ < 1,k ≥ 0,α,β ,µ,λ ≥ 0,α+β 6= 0,µ+λ 6= 0 and m ∈ N0.In this paper we introduce the following class of meromorphic univalentfunctions in U∗.A function f (z) ∈ ∑hµ,λ (α,β ,γ,k) if and only if∞∑n=1[α+β +(µ+λ )(n+1)α+β]h[n(k+1)+(k+ γ)]an ≤ (1− γ)a0, (11)where 0≤ γ < 1,k ≥ 0,α,β ,µ,λ ≥ 0,α+β 6= 0,µ+λ 6= 0 and h ∈ N0.Further, ∑hµ,λ (α,β ,γ,k)⊂ ∑ϕµ,λ (α,β ,γ,k) if h > ϕ , the containment beingproper. Moreover, for any positive integer h > h−1 > · · ·> m+1 > m, we havethe following inclusion relation∑hµ,λ (α,β ,γ,k)⊂ ∑h−1µ,λ (α,β ,γ,k)⊂ ·· · ⊂ ∑m+1µ,λ (α,β ,γ,k)⊂ ∑∗,mµ,λ (α,β ,γ,k).We also note that, for every real number h, the class ∑hµ,λ (α,β ,γ,k) isnonempty as the functions of the formf (z) =a0z+∞∑n=1[α+β +(µ+λ )(n+1)α+β]−h{ (1− γ)n(k+1)+(k+ γ)}a0λnzn,(12)HADAMARD PRODUCT CONCERNING . . . 183where a0 > 0, λn ≥ 0 and∞∑n=1λn ≤ 1, satisfy the inequality (11).The quasi-Hadamard product of two or more functions has recently beendefined and used by Owa [22], Kumar [15–17], Mogra [19, 20], Aouf and Dar-wish [2], Darwish [9], Hossen [14] and Sekine [23]. Accordingly, the quasi-Hadamard product of two functions ϕ(z) and ψ(z) given by (1) and (2) is de-fined by(ϕ ∗ψ)(z) = c1d1z+∞∑k=2cndnzn.Let us define the Hadamard product of two meromorphic univalent functionsf (z) and g(z) by( f ∗g)(z) = a0b0z+∞∑n=1anbnzn. (13)Objective of this paper is to establish certain results concerning the Hadamardproduct of functions for the classes ∑∗,mµ,λ (α,β ,γ,k) and ∑m+1µ,λ (α,β ,γ,k) analo-gous to the results due to Mogra [19, 20].2. The Main TheoremsUnless otherwise mentioned we shall assume throughout the paper that 0≤ γ <1,k ≥ 0,α,β ,µ,λ ≥ 0,α+β 6= 0,µ+λ 6= 0 and m ∈ N0.Theorem 2.1. Let the functions fi(z) belong to the class ∑m+1µ,λ (α,β ,γ,k) forevery i = 1,2, . . . ,r. Then the Hadamard product f1 ∗ f2 ∗ · · · ∗ fr(z) belongs tothe class ∑r(m+2)−1µ,λ (α,β ,γ,k).Proof. It is sufficient to show that∞∑n=1{[α+β +(µ+λ )(n+1)α+β]r(m+2)−1[n(k+1)+(k+ γ)]r∏i=1an,i}≤ (1− γ)[r∏i=1a0,i]. (14)Since fi(z) ∈ ∑m+1µ,λ (α,β ,γ,k), we have∞∑n=1[α+β +(µ+λ )(n+1)α+β]m+1[n(k+1)+(k+ γ)]an,i ≤ (1− γ)a0,i (15)for every i = 1,2, . . . ,r. Therefore,[α+β +(µ+λ )(n+1)α+β]m+1[n(k+1)+(k+ γ)]an,i ≤ (1− γ)a0,i184 RABHA M. EL-ASHWAH - MASLINA DARUSoran,i ≤{[α+βα+β +(µ+λ )(n+1)]m+1 (1− γ)n(k+1)+(k+ γ)}a0,i,for every i = 1,2, . . . ,r impliesan,i ≤[α+β +(µ+λ )(n+1)α+β]−(m+2)a0,i, (16)for every i = 1,2, . . . ,r.Using (16) for i = 1,2, . . . ,r−1, and (15) for i = r, we obtain∞∑n=1{[α+β +(µ+λ )(n+1)α+β]r(m+2)−1[n(k+1)+(k+ γ)]r∏i=1an,i}≤∞∑n=1[α+β +(µ+λ )(n+1)α+β]r(m+2)−1[n(k+1)+(k+ γ)]an,r·[α+β +(µ+λ )(n+1)α+β]−(m+2)(r−1) r−1∏i=1a0,i=[r−1∏i=1a0,i]∞∑n=1{[α+β +(µ+λ )(n+1)α+β]m+1[n(k+1)+(k+ γ)]an,r}≤ (1− γ)[r∏i=1a0,i].Hence f1 ∗ f2 ∗ · · · ∗ fr(z) ∈ ∑r(m+2)−1µ,λ (α,β ,γ,k). This completes the proof ofthe theorem.Theorem 2.2. Let the functions fi(z) belong to the class ∑∗,mµ,λ (α,β ,γ,k) forevery i = 1,2, . . . ,r. Then the Hadamard product f1 ∗ f2 ∗ · · · ∗ fr(z) belongs tothe class ∑r(m+1)−1µ,λ (α,β ,γ,k).Proof. Since fi(z) ∈ ∑∗,mµ,λ (α,β ,γ,k), we have∞∑n=1{[α+β +(µ+λ )(n+1)α+β]m[n(k+1)+(k+ γ)]an,i}≤ (1− γ)a0,i (17)for every i = 1,2, . . . ,r. Thereforean,i ≤{[α+βα+β +(µ+λ )(n+1)]m (1− γ)n(k+1)+(k+ γ)}a0,i,HADAMARD PRODUCT CONCERNING . . . 185and hencean,i ≤[α+β +(µ+λ )(n+1)α+β]−(m+1)a0,i, (18)for every i = 1,2, . . . ,r.Using (18) for i = 1,2, . . . ,r−1, and (17) for i = r, we get∞∑n=1{[α+β +(µ+λ )(n+1)α+β]r(m+1)−1[n(k+1)+(k+ γ)]r∏i=1an,i}≤∞∑n=1[α+β +(µ+λ )(n+1)α+β]r(m+1)−1[n(k+1)+(k+ γ)]an,r·[[α+β +(µ+λ )(n+1)α+β]−(m+1)(r−1) r−1∏i=1a0,i]=[r−1∏i=1a0,i]∞∑n=1{[α+β +(µ+λ )(n+1)α+β]m[n(k+1)+(k+ γ)]an,r}≤ (1− γ)[r∏i=1a0,i].Hence f1 ∗ f2 ∗ · · · ∗ fr(z) ∈ ∑r(m+1)−1µ,λ (α,β ,γ,k). This completes the proof ofthe theorem.Theorem 2.3. Let the functions fi(z) belong to the class∑m+1µ,λ (α,β ,γ,k) for ev-ery i = 1,2, . . . ,r and let the functions g j(z) belong to the class ∑∗,mµ,λ (α,β ,γ,k)for every j = 1,2, . . . ,q. Then the Hadamard product f1 ∗ f2 ∗ · · · ∗ fr ∗g1 ∗g2 ∗· · · ∗gq(z) belongs to the class ∑r(m+2)+q(m+1)−1µ,λ (α,β ,γ,k).Proof. We denote the Hadamard product f1 ∗ f2 ∗ · · · ∗ fr ∗g1 ∗g2 ∗ · · · ∗gq(z) bythe function H(z), for the sake of convenience. Clearly,H(z) =[r∏i=1a0,i.q∏j=1b0, j]z−1+∞∑n=1[r∏i=1an,i.q∏j=1bn, j]zn.To prove the theorem, we need to show that∞∑n=1{[α+β +(µ+λ )(n+1)α+β]r(m+2)+q(m+1)−1·[n(k+1)+(k+ γ)][r∏i=1an,i.q∏j=1bn, j]}≤ (1− γ)[r∏i=1a0,i.q∏j=1b0, j].186 RABHA M. EL-ASHWAH - MASLINA DARUSSince fi(z) ∈ ∑m+1µ,λ (α,β ,γ,k), the inequalities (15) and (16) hold for every i =1,2, . . . ,r. Further, since g j(z) ∈ ∑∗,mµ,λ (α,β ,γ,k), we have∞∑n=1{[α+β +(µ+λ )(n+1)α+β]m[n(k+1)+(k+ γ)]bn, j}≤ (1− γ)b0, j,(19)for every j = 1,2, . . . ,q. Whence we obtainbn, j ≤[α+β +(µ+λ )(n+1)α+β]−(m+1)b0, j, (20)for every j = 1,2, . . . ,q.Using (16) for i = 1,2, . . . ,r, (20) for j = 1,2, . . .q− 1 and (19) for j = q,we get∞∑n=1{[α+β +(µ+λ )(n+1)α+β]r(m+2)+q(m+1)−1[n(k+1)+(k+ γ)][r∏i=1an,i.q∏j=1bn, j]}≤∞∑n=1{[α+β +(µ+λ )(n+1)α+β]r(m+2)+q(m+1)−1[n(k+1)+(k+ γ)][[α+β +(µ+λ )(n+1)α+β]−r(m+2) r∏i=1a0,i.q∏j=1bn, j]}≤∞∑n=1[α+β +(µ+λ )(n+1)α+β]r(m+2)+q(m+1)−1[n(k+1)+(k+ γ)]·[[α+β+(µ+λ )(n+1)α+β]−r(m+2).[α+β+(µ+λ )(n+1)α+β]−(m+1)(q−1).r∏i=1a0,i.q−1∏j=1b0, j]bn,q=[r∏i=1a0,i.q−1∏j=1b0, j]∞∑n=1{[α+β +(µ+λ )(n+1)α+β]m[n(k+1)+(k+ γ)]bn,q}≤ (1− γ)[r∏i=1a0,i.q∏j=1b0, j].Hence H(z) ∈ ∑r(m+2)+q(m+1)−1µ,λ (α,β ,γ,k).We note that the required estimate can also be obtained by using (16) fori = 1,2, . . . ,r−1; (20) for j = 1,2, . . . ,qand (15) for i = r. This completes theproof of the theorem.Remark 2.4. By specializing the parameters α,β ,µ,λ ,k and m we can obtaincorresponding results for various subclasses associated with various operators.Note that other work related to differential operators can be seen in the fol-lowing references (e.g. see [25, 30]).HADAMARD PRODUCT CONCERNING . . . 187AcknowledgementsWe thank the referee for the comments given towards the manuscript.REFERENCES[1] F. M. Al-Oboudi - H. A. Al-Zkeri, Applications of Briot-Bouquet differential sub-ordination to certain classes of meromorphic functions, Arab. J. Math. Sci. 12 (1)(2005), 1–14.[2] M. K. Aouf - H. E. Darwish, Hadamard product of certain meromorphic univalentfunctions with positive coefficients, South. Asian Bull. Math. 30 (2006), 23–28.[3] M. K.Aouf - N. Magesh - J. Jothibasu - S. Murthy, On certain subclasses of mero-morphic functions with positive coefficients, Stud. Univ. Babe¸s-Bolyai Math. 58(1) (2013), 31–42.[4] E. Aqlan - J. M. Jahangiri - S. R. Kulkarni, Certain integral operators applied tomeromorphic p-valent functions, J. Nat. Geom. 24 (2003), 111–120.[5] W. G. Atshan - S. R. Kulkarni, Subclass of meromorphic functions with positivecoefficients defined by Ruscheweyh derivative, I, J. Rajasthan Acad. Phys. Sci. 6(2) (2007), 129–140.[6] T. Bulboaca - M. K. Aouf - R. M. El-Ashwah Convolution properties for sub-classes of meromorphic univalent functions of complex order, Filomat 26 (1)(2012), 153–163.[7] N. E. Cho - O. S. Kwon - H. M Srivastava, Inclusion and argument properties forcertain subclasses of meromorphic functions associated with a family of multipliertransformations, J. Math. Anal. Appl. 300 (2004), 505–520.[8] N. E. Cho - O. S. Kwon - H. M. Srivastava, Inclusion relationships for certainsubclasses of meromorphic functions associated with a family of multiplier trans-formations, Integral Transforms Special Functions 16 (18) (2005), 647–659.[9] H. E. Darwish, The quasi-Hadamard product of certain starlike and convex func-tions, Appl. Math. Letters 20 (2007), 692–695.[10] R. M. El-Ashwah, Differential inequalities for meromorphic p-valent functionsAssociated with Generalized Integral Operator, Le Matematiche 68 (2) (2013),131–139.[11] R. M. El-Ashwah, Properties of certain class of p-valent meromorphic functionsassociated with new integral operator, Acta Univ. Apulensis 29 (2012), 255–264.[12] R. M. El-Ashwah - M. K. Aouf, Hadamard product of certain meromorphic star-like and convex functions, Comput. Math. Appl. 57 (2009), 1102–1106.[13] R. M. El-Ashwah, A note on certain meromorphic p-valent functions, Appl. Math.Letters 22 (2009), 1756–1759.188 RABHA M. EL-ASHWAH - MASLINA DARUS[14] H. M. Hossen, Quasi-Hadamard product of certain p-valent functions, Demon-stratio Math. 33 (2) (2000), 277–281.[15] V. Kumar, Hadamard product of certain starlike functions, J. Math. Anal. Appl.110 (1985), 425–128.[16] V. Kumar, Hadamard product of certain starlike functions II, J. Math. Anal. Appl.113 (1986), 230–234.[17] V. Kumar, Quasi-Hadamard product of certain univalent function, J. Math. Anal.Appl. 126 (1987), 70–77.[18] A. Y. Lashin, On certain subclass of meromorphic functions associated with cer-tain integral operators, Comput. Math Appl. 59 (1) (2010), 524–531.[19] M. L. Mogra, Hadamard product of certain meromorphic univalent functions, J.Math. Anal. Appl. 157 (1991), 10–16.[20] M. L. Mogra, Hadamard product of certain meromorphic starlike and convexfunctions, Tamkang J. Math. 25 (2) (1994), 157–162.[21] M. L. Mogra - T. Reddy - O. P. Juneja, Meromorphic univalent functions withpositive coefficients, Bull. Aust. Math. Soc. 32 (1985), 161–176.[22] S. Owa, On the Hadamard product of univalent functions, Tamkang J. Math. 14(1983), 15–21.[23] T. Sekine, On quasi-Hadamard products of p-valent functions with negative coef-ficients, in: H. M. Srivastava and S. Owa (Eds.), Univalent Functions, FractionalCalculus, and Their Applications, Halsted Press (Ellis Horwood Limited, Chices-ter), Brisbane and Toronto, 1989, 317–328.[24] B. A. Uralegaddi - C. Somanatha, New criteria for meromorphic starlike univalentfunctions, Bull. Austral. Math. Soc. 43 (1991), 137–140.[25] S. F. Ramadan - M. Darus, On inclusion properties of generalized differential op-erator involving Hadamard product, Inter. J. Applied Mathematics and Statistics33 (3) (2013), 18–27.[26] E. A. Eljamal - M. Darus, Inclusion properties for certain subclasses of p-valentfunctions associated with new generalized derivative operator, VladikavkavMathematical Journal 15 (2) (2013), 27–34.[27] E. El-Yagubi - M. Darus, Subclasses of analytic functions defined by new gen-eralized derivative operator, Journal of Quality and Management Analysis 9 (1)(2013), 47–56.[28] S. F. Ramadan - M. Darus, Certain subordination results for a class of analyticfunctions defined by the generalized differential operator, Far East J. Math. Sci.54 (1) (2011), 37–45.[29] M. H. Al-Abbadi - M. Darus, Differential Subordination Defined by New Gener-alised Derivative Operator for Analytic Functions, International Journal of Math-ematics and Mathematical Sciences 2010, Article ID 369078 (2010), 15 pages.[30] R. W. Ibrahim - M. Darus, On univalent function defined by a generalized differ-ential operator, Journal of Applied Analysis (Lodz) 16 (2) (2010), 305–313.HADAMARD PRODUCT CONCERNING . . . 189RABHA M. EL-ASHWAHDepartment of MathematicsFaculty of Science (Damietta Branch)Mansoura UniversityNew Damietta 34517, Egypte-mail: r elashwah@yahoo.comMASLINA DARUSSchool of Mathematical SciencesFaculty of Science and TechnologyUniversiti Kebangsaan MalaysiaBangi 43600, Selangor D. Ehsan, Malaysiae-mail: maslina@ukm.edu.my

Hadamard Product Concerning Certain Meromorphic Functions

In this paper the authors introduced a new generalized differintegral operator for meromorphic univalent functions in U* = {z : z ∈ C, 0 &lt; |z| &lt; 1}. The objective of this paper is to establish certain results concerning the Hadamard product of functions in the classes ∑^{∗,m}_{μ,λ} (α, β , γ, k) and ∑^h_{μ,λ} (α, β , γ, k).</p

Rabha M. El-Ashwah

Maslina Darus

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LE MATEMATICHEVol. LXX (2015) – Fasc. I, pp. 179–189doi: 10.4418/2015.70.1.13HADAMARD PRODUCT CONCERNINGCERTAIN MEROMORPHIC FUNCTIONSRABHA M. EL-ASHWAH - MASLINA DARUSIn this paper the authors introduced a new generalized differintegraloperator for meromorphic univalent functions in U∗ = {z : z ∈ C, 0 <|z| < 1}. The objective of this paper is to establish certain results con-cerning the Hadamard product of functions in the classes ∑∗,mµ,λ (α,β ,γ,k)and ∑hµ,λ (α,β ,γ,k).1. IntroductionThroughout this paper, let the functions f of the form :ϕ(z) = c1z+∞∑n=2cnzn (c1 > 0; cn ≥ 0), (1)andψ(z) = d1z+∞∑n=2dnzn (d1 > 0; dn ≥ 0) (2)be regular and univalent in the unit disc U = {z : |z|< 1} and letf (z) =a0z+∞∑n=1anzn (a0 > 0; an ≥ 0), (3)Entrato in redazione: 25 maggio 2014AMS 2010 Subject Classification: 30C45.Keywords: Analytic, meromorphic, positive coefficients, Hadamard product.This work was partially supported by MOHE’s grant: FRGSTOPDOWN/2013/ST06/UKM/01/1.180 RABHA M. EL-ASHWAH - MASLINA DARUSfi(z) =a0,iz+∞∑n=1an,izn (a0,i > 0; an,i ≥ 0), (4)g(z) =b0z+∞∑n=1bnzn (b0 > 0; bn ≥ 0), (5)andg j(z) =b0, jz+∞∑n=1bn, jzn (b0, j > 0; bn, j ≥ 0), (6)be regular and univalent in the punctured unit disc U∗ = {z : 0 < |z| < 1} =U\{0}.For a function f (z) defined by (3) (with a0 = 1), El-Ashwah defined theintegral operator ℑβ ,mλ(α,µ) (see [10, with p = 1]) as follows:ℑβ ,mλ(α,µ) f (z) =1z+∞∑n=1[α +βα +β +(µ +λ )(n+1)]manzn(α,β ,µ,λ ≥ 0, α +β 6= 0, µ +λ 6= 0; m ∈ N0; z ∈U∗). (7)We note that(i) ℑ0,mλ(α,0) f (z) = Im(λ ,α) f (z)(α > 0,λ ≥ 0,m ∈ N0) (see El-Ashwah [11,with p = 1]);(ii) ℑ0,γ0 (1,1) f (z) = Pγ f (z)(γ > 0) (see Aqlan et al. [4, with p = 1]);(iii) ℑ0,m0 (α,µ) f (z) =Lm(α,µ) f (z)(α > 0,µ ≥ 0,m∈N0) (see Bulboaca et al.[6]);(iv) ℑ0,γ0 (1,µ) f (z) = Pγµ f (z)(γ > 0,µ > 0) (see Lashin [18]).Also we note that(i) ℑ0,m1 (α,1) f (z) = ℑm(α) f (z) =1z+∞∑n=1(αα +2(k+1))manzn(α > 0,m ∈N0):(ii) ℑ0,mp,λ (1,0) f (z) = ℑmp,λ f (z) =1zp+∞∑n=1(11+λ (k+1))manzn−p(λ ≥ 0,m ∈N0).Now we will extend the definition of the operator ℑβ ,mλ(α,µ) f (z) for m ∈Z= {0,±1,±2, . . .} and z ∈U∗ as follows:ℑβ ,m−1λ(α,µ) f (z) =(µ+λα+β)z−α+βµ+λ(z(α+βµ+λ)+1ℑβ ,mλ(α,µ) f (z))′,ℑβ ,m−2λ(α,µ) f (z) =(µ+λα+β)z−α+βµ+λ(z(α+βµ+λ)+1ℑβ ,m−1λ(α,µ) f (z))′,HADAMARD PRODUCT CONCERNING . . . 181ℑβ ,1λ(α,µ) f (z) =(µ+λα+β)z−α+βµ+λ(z(α+βµ+λ)+1ℑβ ,2λ(α,µ) f (z))′,ℑβ ,0λ(α,µ) f (z) =(µ+λα+β)z−α+βµ+λ(z(α+βµ+λ)+1ℑβ ,1λ(α,µ) f (z))′= f (z)ℑβ ,−1λ(α,µ) f (z) =(µ+λα+β)z−α+βµ+λ(z(α+βµ+λ)+1ℑβ ,0λ(α,µ) f (z))′,ℑβ ,−2λ(α,µ) f (z) =(µ+λα+β)z−α+βµ+λ(z(α+βµ+λ)+1ℑβ ,−1λ(α,µ) f (z))′,ℑβ ,−m+1λ(α,µ) f (z) =(µ+λα+β)z−α+βµ+λ(z(α+βµ+λ)+1ℑβ ,−m+2λ(α,µ) f (z))′,ℑβ ,−mλ(α,µ) f (z) =(µ+λα+β)z−α+βµ+λ(z(α+βµ+λ)+1ℑβ ,−m+1λ(α,µ) f (z))′.We see that for f ∈ Σ, we haveℑβ ,−mλ(α,µ) f (z) =1z+∞∑n=1[α +βα +β +(µ +λ )(n+1)]−manzn(α,β ,µ,λ ≥ 0, α +β 6= 0, µ +λ 6= 0; m ∈ N0; z ∈U∗).We note that(i) ℑ0,−m0 (α,µ) f (z)= Jm(α,µ) f (z)(α > 0,µ ≥ 0,m∈N0) (see El-Ashwah [13,with p = 1]);(ii) ℑ0,−m1 (1,µ) f (z) = I(m,µ) f (z)(µ ≥ 0,m ∈ N0) (see Cho et al. [7, 8]);(iii) ℑ0,−m1 (α,1) f (z)) = Dmα f (z)(α > 0,m ∈ N0) (see Al-Oboudi and Al-Zkeri[1]);(iv) ℑ0,−m1 (1,1) f (z) = Im f (z)(m ∈ N0) (see Uralegaddi and Somanatha [24]).On the other hand for α +β = µ +λ = m = 1, we haveℑβ ,−1λ(α,µ) f (z) =(z2 f (z))′z=1z+∞∑n=1(n+2)anzn.In general we will writeℑβ ,mλ(α,µ) f (z) =1z+∞∑n=1[α +β +(µ +λ )(n+1)α +β]manzn(α,β ,µ,λ ≥ 0, α +β 6= 0, µ +λ 6= 0; m ∈ Z; z ∈U∗). (8)182 RABHA M. EL-ASHWAH - MASLINA DARUSWith the help of the differintegral operator ℑβ ,mλ(α,µ) f (z), we define theclass ∑∗,mµ,λ (α,β ,γ,k) as follows.Denote by ∑∗,mµ,λ (α,β ,γ,k) the class of functions f (z) given by (3) whichsatisfy the condition−Rez(ℑβ ,mλ(α,µ) f (z))′ℑβ ,mλ(α,µ) f (z)+ γ> k∣∣∣∣∣∣∣z(ℑβ ,mλ(α,µ) f (z))′ℑβ ,mλ(α,µ) f (z)+1∣∣∣∣∣∣∣(0≤ γ < 1; k≥ 0; α,β ,µ,λ ≥ 0, α +β 6= 0, µ +λ 6= 0; m ∈ Z; z ∈U∗). (9)Taking m = k = 0, the class ∑∗,mµ,λ (α,β ,γ,k) with a0 = 1 will reduce to the class∑S∗(γ) (see Mogra et al. [21, with β = 1]), and El-Ashwah and Aouf ([12, withβ = 1 and k = 0]).Also, we observe that the class ∑∗,mµ,λ (α,β ,γ,k) reduces to several interestingmany other classes for different choices of α,β ,µ,λ ,k and m.Using similar arguments as given in [5] (see also Aouf et al. [3, withλ = 0]), we can easily prove the following results for functions in the class∑∗,mµ,λ (α,β ,γ,k).A function f (z)∈∑∗,mµ,λ (α,β ,γ,k)(α,β ,µ,λ ≥ 0,α+β 6= 0,µ+λ 6= 0;m∈Z) if and only if∞∑n=1[α +β +(µ +λ )(n+1)α +β]m[n(k+1)+(k+ γ)]an ≤ (1− γ)a0, (10)where 0≤ γ < 1,k ≥ 0,α,β ,µ,λ ≥ 0,α +β 6= 0,µ +λ 6= 0 and m ∈ N0.In this paper we introduce the following class of meromorphic univalentfunctions in U∗.A function f (z) ∈ ∑hµ,λ (α,β ,γ,k) if and only if∞∑n=1[α +β +(µ +λ )(n+1)α +β]h[n(k+1)+(k+ γ)]an ≤ (1− γ)a0, (11)where 0≤ γ < 1,k ≥ 0,α,β ,µ,λ ≥ 0,α +β 6= 0,µ +λ 6= 0 and h ∈ N0.Further, ∑hµ,λ (α,β ,γ,k)⊂ ∑ϕµ,λ (α,β ,γ,k) if h > ϕ , the containment beingproper. Moreover, for any positive integer h > h−1 > · · ·> m+1 > m, we havethe following inclusion relation∑hµ,λ (α,β ,γ,k)⊂ ∑h−1µ,λ (α,β ,γ,k)⊂ ·· · ⊂ ∑m+1µ,λ (α,β ,γ,k)⊂ ∑∗,mµ,λ (α,β ,γ,k).We also note that, for every real number h, the class ∑hµ,λ (α,β ,γ,k) isnonempty as the functions of the formf (z) =a0z+∞∑n=1[α +β +(µ +λ )(n+1)α +β]−h{ (1− γ)n(k+1)+(k+ γ)}a0λnzn,(12)HADAMARD PRODUCT CONCERNING . . . 183where a0 > 0, λn ≥ 0 and∞∑n=1λn ≤ 1, satisfy the inequality (11).The quasi-Hadamard product of two or more functions has recently beendefined and used by Owa [22], Kumar [15–17], Mogra [19, 20], Aouf and Dar-wish [2], Darwish [9], Hossen [14] and Sekine [23]. Accordingly, the quasi-Hadamard product of two functions ϕ(z) and ψ(z) given by (1) and (2) is de-fined by(ϕ ∗ψ)(z) = c1d1z+∞∑k=2cndnzn.Let us define the Hadamard product of two meromorphic univalent functionsf (z) and g(z) by( f ∗g)(z) = a0b0z+∞∑n=1anbnzn. (13)Objective of this paper is to establish certain results concerning the Hadamardproduct of functions for the classes ∑∗,mµ,λ (α,β ,γ,k) and ∑m+1µ,λ (α,β ,γ,k) analo-gous to the results due to Mogra [19, 20].2. The Main TheoremsUnless otherwise mentioned we shall assume throughout the paper that 0≤ γ <1,k ≥ 0,α,β ,µ,λ ≥ 0,α +β 6= 0,µ +λ 6= 0 and m ∈ N0.Theorem 2.1. Let the functions fi(z) belong to the class ∑m+1µ,λ (α,β ,γ,k) forevery i = 1,2, . . . ,r. Then the Hadamard product f1 ∗ f2 ∗ · · · ∗ fr(z) belongs tothe class ∑r(m+2)−1µ,λ (α,β ,γ,k).Proof. It is sufficient to show that∞∑n=1{[α +β +(µ +λ )(n+1)α +β]r(m+2)−1[n(k+1)+(k+ γ)]r∏i=1an,i}≤ (1− γ)[r∏i=1a0,i]. (14)Since fi(z) ∈ ∑m+1µ,λ (α,β ,γ,k), we have∞∑n=1[α +β +(µ +λ )(n+1)α +β]m+1[n(k+1)+(k+ γ)]an,i ≤ (1− γ)a0,i (15)for every i = 1,2, . . . ,r. Therefore,[α +β +(µ +λ )(n+1)α +β]m+1[n(k+1)+(k+ γ)]an,i ≤ (1− γ)a0,i184 RABHA M. EL-ASHWAH - MASLINA DARUSoran,i ≤{[α +βα +β +(µ +λ )(n+1)]m+1 (1− γ)n(k+1)+(k+ γ)}a0,i,for every i = 1,2, . . . ,r impliesan,i ≤[α +β +(µ +λ )(n+1)α +β]−(m+2)a0,i, (16)for every i = 1,2, . . . ,r.Using (16) for i = 1,2, . . . ,r−1, and (15) for i = r, we obtain∞∑n=1{[α +β +(µ +λ )(n+1)α +β]r(m+2)−1[n(k+1)+(k+ γ)]r∏i=1an,i}≤∞∑n=1[α +β +(µ +λ )(n+1)α +β]r(m+2)−1[n(k+1)+(k+ γ)]an,r·[α +β +(µ +λ )(n+1)α +β]−(m+2)(r−1) r−1∏i=1a0,i=[r−1∏i=1a0,i]∞∑n=1{[α +β +(µ +λ )(n+1)α +β]m+1[n(k+1)+(k+ γ)]an,r}≤ (1− γ)[r∏i=1a0,i].Hence f1 ∗ f2 ∗ · · · ∗ fr(z) ∈ ∑r(m+2)−1µ,λ (α,β ,γ,k). This completes the proof ofthe theorem.Theorem 2.2. Let the functions fi(z) belong to the class ∑∗,mµ,λ (α,β ,γ,k) forevery i = 1,2, . . . ,r. Then the Hadamard product f1 ∗ f2 ∗ · · · ∗ fr(z) belongs tothe class ∑r(m+1)−1µ,λ (α,β ,γ,k).Proof. Since fi(z) ∈ ∑∗,mµ,λ (α,β ,γ,k), we have∞∑n=1{[α +β +(µ +λ )(n+1)α +β]m[n(k+1)+(k+ γ)]an,i}≤ (1− γ)a0,i (17)for every i = 1,2, . . . ,r. Thereforean,i ≤{[α +βα +β +(µ +λ )(n+1)]m (1− γ)n(k+1)+(k+ γ)}a0,i,HADAMARD PRODUCT CONCERNING . . . 185and hencean,i ≤[α +β +(µ +λ )(n+1)α +β]−(m+1)a0,i, (18)for every i = 1,2, . . . ,r.Using (18) for i = 1,2, . . . ,r−1, and (17) for i = r, we get∞∑n=1{[α +β +(µ +λ )(n+1)α +β]r(m+1)−1[n(k+1)+(k+ γ)]r∏i=1an,i}≤∞∑n=1[α +β +(µ +λ )(n+1)α +β]r(m+1)−1[n(k+1)+(k+ γ)]an,r·[[α +β +(µ +λ )(n+1)α +β]−(m+1)(r−1) r−1∏i=1a0,i]=[r−1∏i=1a0,i]∞∑n=1{[α +β +(µ +λ )(n+1)α +β]m[n(k+1)+(k+ γ)]an,r}≤ (1− γ)[r∏i=1a0,i].Hence f1 ∗ f2 ∗ · · · ∗ fr(z) ∈ ∑r(m+1)−1µ,λ (α,β ,γ,k). This completes the proof ofthe theorem.Theorem 2.3. Let the functions fi(z) belong to the class ∑m+1µ,λ (α,β ,γ,k) for ev-ery i = 1,2, . . . ,r and let the functions g j(z) belong to the class ∑∗,mµ,λ (α,β ,γ,k)for every j = 1,2, . . . ,q. Then the Hadamard product f1 ∗ f2 ∗ · · · ∗ fr ∗g1 ∗g2 ∗· · · ∗gq(z) belongs to the class ∑r(m+2)+q(m+1)−1µ,λ (α,β ,γ,k).Proof. We denote the Hadamard product f1 ∗ f2 ∗ · · · ∗ fr ∗g1 ∗g2 ∗ · · · ∗gq(z) bythe function H(z), for the sake of convenience. Clearly,H(z) =[r∏i=1a0,i.q∏j=1b0, j]z−1 +∞∑n=1[r∏i=1an,i.q∏j=1bn, j]zn.To prove the theorem, we need to show that∞∑n=1{[α +β +(µ +λ )(n+1)α +β]r(m+2)+q(m+1)−1·[n(k+1)+(k+ γ)][r∏i=1an,i.q∏j=1bn, j]}≤ (1− γ)[r∏i=1a0,i.q∏j=1b0, j].186 RABHA M. EL-ASHWAH - MASLINA DARUSSince fi(z) ∈ ∑m+1µ,λ (α,β ,γ,k), the inequalities (15) and (16) hold for every i =1,2, . . . ,r. Further, since g j(z) ∈ ∑∗,mµ,λ (α,β ,γ,k), we have∞∑n=1{[α +β +(µ +λ )(n+1)α +β]m[n(k+1)+(k+ γ)]bn, j}≤ (1− γ)b0, j,(19)for every j = 1,2, . . . ,q. Whence we obtainbn, j ≤[α +β +(µ +λ )(n+1)α +β]−(m+1)b0, j, (20)for every j = 1,2, . . . ,q.Using (16) for i = 1,2, . . . ,r, (20) for j = 1,2, . . .q− 1 and (19) for j = q,we get∞∑n=1{[α +β +(µ +λ )(n+1)α +β]r(m+2)+q(m+1)−1[n(k+1)+(k+ γ)][r∏i=1an,i.q∏j=1bn, j]}≤∞∑n=1{[α +β +(µ +λ )(n+1)α +β]r(m+2)+q(m+1)−1[n(k+1)+(k+ γ)][[α +β +(µ +λ )(n+1)α +β]−r(m+2) r∏i=1a0,i.q∏j=1bn, j]}≤∞∑n=1[α +β +(µ +λ )(n+1)α +β]r(m+2)+q(m+1)−1[n(k+1)+(k+ γ)]·[[α+β+(µ+λ )(n+1)α+β]−r(m+2).[α+β+(µ+λ )(n+1)α+β]−(m+1)(q−1).r∏i=1a0,i.q−1∏j=1b0, j]bn,q=[r∏i=1a0,i.q−1∏j=1b0, j]∞∑n=1{[α +β +(µ +λ )(n+1)α +β]m[n(k+1)+(k+ γ)]bn,q}≤ (1− γ)[r∏i=1a0,i.q∏j=1b0, j].Hence H(z) ∈ ∑r(m+2)+q(m+1)−1µ,λ (α,β ,γ,k).We note that the required estimate can also be obtained by using (16) fori = 1,2, . . . ,r−1; (20) for j = 1,2, . . . ,qand (15) for i = r. This completes theproof of the theorem.Remark 2.4. By specializing the parameters α,β ,µ,λ ,k and m we can obtaincorresponding results for various subclasses associated with various operators.Note that other work related to differential operators can be seen in the fol-lowing references (e.g. see [25, 30]).HADAMARD PRODUCT CONCERNING . . . 187AcknowledgementsWe thank the referee for the comments given towards the manuscript.REFERENCES[1] F. M. Al-Oboudi - H. A. Al-Zkeri, Applications of Briot-Bouquet differential sub-ordination to certain classes of meromorphic functions, Arab. J. Math. Sci. 12 (1)(2005), 1–14.[2] M. K. Aouf - H. E. Darwish, Hadamard product of certain meromorphic univalentfunctions with positive coefficients, South. Asian Bull. Math. 30 (2006), 23–28.[3] M. K.Aouf - N. Magesh - J. Jothibasu - S. Murthy, On certain subclasses of mero-morphic functions with positive coefficients, Stud. Univ. Babȩs-Bolyai Math. 58(1) (2013), 31–42.[4] E. Aqlan - J. M. Jahangiri - S. R. 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Darus, Certain subordination results for a class of analyticfunctions defined by the generalized differential operator, Far East J. Math. Sci.54 (1) (2011), 37–45.[29] M. H. Al-Abbadi - M. Darus, Differential Subordination Defined by New Gener-alised Derivative Operator for Analytic Functions, International Journal of Math-ematics and Mathematical Sciences 2010, Article ID 369078 (2010), 15 pages.[30] R. W. Ibrahim - M. Darus, On univalent function defined by a generalized differ-ential operator, Journal of Applied Analysis (Lodz) 16 (2) (2010), 305–313.HADAMARD PRODUCT CONCERNING . . . 189RABHA M. EL-ASHWAHDepartment of MathematicsFaculty of Science (Damietta Branch)Mansoura UniversityNew Damietta 34517, Egypte-mail: r elashwah@yahoo.comMASLINA DARUSSchool of Mathematical SciencesFaculty of Science and TechnologyUniversiti Kebangsaan MalaysiaBangi 43600, Selangor D. Ehsan, Malaysiae-mail: maslina@ukm.edu.my

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Hadamard Product Concerning Certain Meromorphic Functions

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