Making use of a convolution structure, we introduce a new class of
analytic functions $mathbb{T}^{p}_{g}(lambda,alpha, beta, )$
defined in the open unit disc and investigate its various
characteristics. Further we obtained distortion bounds, extreme
points and radii of close-to-convexity, starlikeness and convexity
for functions belonging to the class
$mathbb{T}^{p}_{g}(lambda,alpha, beta).

Gangadharan Murugusundaramoorthy

Santosh Bose Joshi

English

Hrčak - Portal of scientific journals of Croatia

MATHEMATICAL COMMUNICATIONS 311Math. Commun., Vol. 15, No. 2, pp. 311-322 (2010)Certain classes of multivalent functions with negativecoefficients associated with a convolution structureGangadharanMurugusundaramoorthy1,∗and SantoshBose Joshi21 School of Advanced Sciences, VIT University, Vellore-632 014, India2 Department of Mathematics, Walchand College of Engineering, Sangli, IndiaReceived March 6, 2009; accepted November 13, 2009Abstract. Making use of a convolution structure, we introduce a new class of analyticfunctions Tpg(λ, α, β, ) defined in the open unit disc and investigate its various characteris-tics. Further we obtained distortion bounds, extreme points and radii of close-to-convexity,starlikeness and convexity for functions belonging to the class Tpg(λ, α, β).AMS subject classifications: 30C45Key words: Analytic, starlikeness, convexity, Hadamard product (convolution)1. IntroductionLet A(p) denote a class of functions of the formf(z) = zp +∞∑n=p+1anzn, p ∈ N = {1, 2, 3, . . . }, (1)which are analytic in the open disc U = {z : z ∈ C; |z| < 1}. For functions f ∈ A(p)given by (1) and g ∈ A(p) given byg(z) = zp +∞∑n=p+1bnzn, p ∈ N = {1, 2, 3, . . . }we define the Hadamard product (or convolution) of f and g byf(z) ∗ g(z) = (f ∗ g)(z) = zp +∞∑n=p+1anbnzn, z ∈ U. (2)In terms of the Hadamard product (or convolution), we choose g as a fixedfunction in A(p) such that (f ∗g)(z) exists for any f ∈ A(p), and for various choicesof g we get different linear operators which have been studied in the recent past.To illustrate some of these cases which arise from the convolution structure (2), weconsider the following examples.∗Corresponding author. Email addresses: gmsmoorthy@yahoo.com (G.Murugusundaramoorthy),Joshisb@hotmail.com (S. B. Joshi)http://www.mathos.hr/mc c©2010 Department of Mathematics, University of Osijek312 G.Murugusundaramoorthy and S.B. Joshi1. Forg(z) = zp +∞∑n=p+1(α1)n−p . . . (αr)n−p(β1)n−p . . . (βs)n−pzn(n− p)! , (3)we get the Dziok–Srivastava operatorΛ(α1, α2, · · · , αr; β1, β2, · · · , βs; z)f(z) ≡ Hpr,sf(z) := (f ∗ g)(z),introduced by Dziok and Srivastava [6]; where α1, α2, · · ·αr, β1, β2, · · · . . . , βsare complex parameters, βj /∈ {0,−1,−2, · · · } for j = 1, 2, · · · , s, r ≤ s +1, r, s ∈ N ∪ {0} . Here (a)ν denotes a well-known Pochhammer symbol (orshifted factorial).Remark 1. When α1 = a, β1 = c s = 1and r = 2, Dizok-Srivastava opera-tor reduces to Carlson-Shaffer operator [3], further when r = 1, s = 0; α1 =ν + 1, α2 = 1; β1 = 1, then the above Dziok-Srivastava operator yields theRuscheweyh derivative operator introduced by Patel and Cho [10].2. Forg(z) = φp(a, c, z) := zp +∞∑n=p+1(a)n−1(c)n−1zn (c 6= 0,−1,−2, · · · ), (4)we get the multivalent Carlson-Shaffer operator Lp(a, c)f(z) := (f ∗g)(z). TheoperatorL(a, c)f(z) ≡ L1(a, c)f(z) ≡ zF (a, 1; c; z) ∗ f(z)was introduced by Carlson-Shaffer [3] where F (a, b; c; z) is the Gaussian hy-pergeometric function.3. For g(z) = zp(1−z)ν+p (ν ≥ −p), we obtain the multivalent Ruscheweyh operatordefined byDν+p−1f(z) := (f ∗ g)(z) = zp +∞∑n=p+1(ν + n− 1n− p)anzn. (5)The operator Dν+p−1f was introduced by Patel and Cho [10]. In particular,Dνf : A → A for p = 1 and ν ≥ −1 was introduced by Ruschweyh [15]. Notethat for n ∈ N0,Dnf(z) =z(zn−1f(z))(n)n!.4. Forg(z) = zp +∞∑n=p+1(np)mzn ( m ≥ 0) , (6)we get the multivalent Sălăgean operator Dmp f(z) introduced by Shenan et al.[14]. In particular, the differential operator Dm ≡ Dm1 was initially studied bySălăgean [12].Certain classes of multivalent functions 3135. Forg(z) = zp +∞∑n=p+1n(n + lp + l)mzn ( l ≥ 0; m ∈ Z) , (7)we obtain the multiplier transformation Ip(l, m) := (f ∗ g)(z) introduced byRavichandran et al. [11]. In particular, I(l, m) ≡ I1(l,m) was studied by Choand Kim [4] and Cho and Srivastava [5].6. Forg(z) = zp +∞∑n=p+1(n + lp + l)mzn ( l ≥ 0; m ∈ Z) , (8)we get multiplier transformation Jp(l, m) := (f ∗g)(z). In particular J (l, m) ≡J1(l, m) introduced by Cho and Srivastava [4].Remark 2. We note that for l = 0 the above operator reduces to the multivalentSălăgean operator Dmp f(z) introduced by Shenan et al. [14].Motivated by the earlier works of [6, 8, 10, 14, 11, 16] we introduced a new sub-class of multivalent functions with negative coefficients and discuss some interestingproperties of this generalized function class.For 0 ≤ λ ≤ 1, 0 ≤ α < 1 and 0 < β ≤ 1, we let Pg(λ, α, β) be the subclass ofA(p) consisting of functions of the form (1) and satisfying the inequality∣∣∣∣Jg,λ(z)− 1Jg,λ(z) + (1− 2α)∣∣∣∣ < β, (9)whereJg,λ(z) = (1− λ) (f ∗ g)zp+λ(f ∗ g)′pzp−1, (10)(f ∗ g)(z) is given by (2) and g is fixed function for all z ∈ U. We further letTpg(λ, α, β) = Pg(λ, α, β) ∩ T (p), whereT (p) :={f ∈ A(p) : f(z) = z −∞∑n=p+1|an|zn, z ∈ U}. (11)We deem it proper to mention below some of the function classes which emergefrom the function class Tpg(λ, α, β) defined above. Indeed, we observe that if wespecialize the function g(z) by means of (3) to (8), and denote the correspondingreducible classes of functions of Tpg(λ, α, β), as listed below.When g(z) is as defined in (3), the class Tpg(λ, α, β) reduces to the subclassHT (λ, α, β) with (9) andJH,λ(z) = (1− λ)Hps,r[α1, β1]f(z)zp+ λ(Hps,r[α1, β1]f(z))′pzp−1.When g(z) is as defined in (4), the class Tpg(λ, α, β) reduces to the subclassLT (λ, α, β) with (9) andJL,λ(z) = (1− λ)Lp(a, c)f(z)zp+λ(Lp(a, c)f(z))′pzp−1.314 G.Murugusundaramoorthy and S.B. JoshiWhen g(z) is as defined in (5), the class Tpg(λ, α, β) reduces to the subclassRT ν(λ, α, β) with (9) andJν,λ(z) = (1− λ)Dν+p−1f(z)zp+ λ(Dν+p−1f(z))′pzp−1.When g(z) is as defined in (7), the class Tpg(λ, α, β) reduces to the subclassIT (λ, α, β) with (9) andJl,λ(z) = (1− λ)Ip(m, l)f(z)zp+ λ(Ip(m, l))′pzp−1.When g(z) is as defined in (8),the class Tpg(λ, α, β) reduces to the subclass JT (λ, α, β)with (9) andJn,λ(z) = (1− λ)Jp(l, m)f(z)zp+ λ(Jp(l, m))′pzp−1.When g(z) is as defined in (6), the class Tpg(λ, α, β) reduces to the subclassST (λ, α, β) with (9) andJm,λ(z) = (1− λ)Dmp f(z)zp+ λDm+1p f(z)pzp−1.The purpose of the present paper is to investigate the various properties and char-acteristics of functions belonging to the above defined subclass Tpg(λ, α, β) of mul-tivalent functions in the open unit disk U . Apart from deriving a set of coefficientbounds for this function class, we also establish distortion bounds and several in-clusion relationships involving the multivalent functions with negative coefficientsbelonging to this subclass.2. Coefficient boundsIn this section we obtain coefficient estimates and extreme points for the classTpg(λ, α, β).Theorem 1. Let the function f be defined by (11). Then f ∈ Tpg(λ, α, β) if andonly if∞∑n=p+1(p + nλ)(1 + β)an bn ≤ 2pβ(1− α) (12)Certain classes of multivalent functions 315Proof. Suppose f satisfies (12). Then for |z| = r < 1,|Jg,λ(z)− 1| − β |Jg,λ(z) + (1− 2α)| =∣∣∣∣∣−∞∑n=p+1(p + nλ)p(1 + β)anbnzn−p∣∣∣∣∣−β∣∣∣∣∣2(1− α)−∞∑n=p+1(p + nλ)panbnzn−p∣∣∣∣∣≤∞∑n=p+1(p + nλ)panbn − 2β(1− α)+∞∑n=p+1(p + nλ)pβanbn=∞∑n=p+1(p + nλ)p[1 + β]anbn − 2β(1− α)≤ 0, by (12).Hence, by maximum modulus theorem and (9), f ∈ Tpg(λ, α, β). To prove the con-verse, assume that∣∣∣∣Jg,λ(z)− 1Jg,λ(z) + (1− 2α)∣∣∣∣ =∣∣∣∣∣−∑∞n=p+1 (p+nλ)p anbnzn−p2(1− α)−∑∞n=p+1 (p+nλ)p anbnzn−p∣∣∣∣∣ ≤ β, z ∈ U.Thus,Re{ ∑∞n=p+1(p+nλ)p anbnzn−p2(1− α)−∑∞n=p+1 (p+nλ)p anbnzn−p}< β. (13)Choose values of z on the real axis so that Jg,λ(z) is real. Upon clearing the de-nominator in (13) and letting z → 1− through real values, we obtain the desiredinequality (12).Corollary 1. If f(z) of the form (11) is in Tpg(λ, α, β), thenan ≤ 2pβ(1− α)(p + nλ)[1 + β]bn , n ≥ p + 1, (14)with the equality only for functions of the formf(z) = zp − 2pβ(1− α)(p + nλ)[1 + β]bnzn, n ≥ p + 1. (15)Corresponding to the various subclasses which arise from the function classTpg(λ, α, β), by suitably choosing the function g(z) as mentioned in (3) to (8), wearrive at the following corollaries giving the coefficient bound inequalities for thesesubclasses of functions.316 G.Murugusundaramoorthy and S.B. JoshiCorollary 2. A function f ∈ HT (λ, α, β) if and only if∞∑n=p+1(p + nλ)[1 + β]an Γn ≤ 2pβ(1− α).whereΓn =1(n− p)!(α1)n−p . . . (αl)n−p(β1)n−p . . . (βm)n−p(16)Remark 3. For specific choices of parameters α1, β1, s, r as stated in Remark 1,Corollary 2 would yield the coefficient bound inequalities for the subclasses of func-tions LT (λ, α, β) and RT ν(λ, α, β)).Corollary 3. A function f ∈ IT (λ, α, β) if and only if∞∑n=p+1(p + nλ)[1 + β]n(n + lp + l)man ≤ 2pβ(1− α).Corollary 4. A function f ∈ J T (λ, α, β) if and only if∞∑n=p+1(p + nλ)[1 + β](n + lp + l)man ≤ 2pβ(1− α).Remark 4. When l = 0, Corollary 4 would give the coefficient bound inequality forthe subclass of functionsST (λ, α, β).Theorem 2 (Extreme points). Letfp(z) = zpandfn(z) = zp − 2pβ(1− α)(p + nλ)[1 + β]bn zn, n ≥ p + 1, (17)for 0 ≤ α < 1, 0 < β ≤ 1, 0 ≤ λ ≤ 1. Then f(z) is in the class Tpg(λ, α, β) if andonly if it can be expressed in the formf(z) = µpzp +∞∑n=p+1µnfn(z), (18)where µn ≥ 0 and∑∞n=p µn = 1.Proof. Suppose f(z) can be written as in (18). Thenf(z) = µpzp −∞∑n=p+1µn[zp − 2pβ(1− α)(p + nλ)[1 + β]bnzn]= zp −∞∑n=p+1µn2pβ(1− α)(p + nλ)[1 + β]bnzn.Certain classes of multivalent functions 317Now,∞∑n=p+1(p + nλ)[1 + β]bn2pβ(1− α) µn2pβ(1− α)(p + nλ)[1 + β]bn=∞∑n=p+1µn = 1− µ1 ≤ 1.Thus f ∈ Tpg(λ, α, β). Conversely, let us have f ∈ Tpg(λ, α, β). Then by using (14),we setµn =(p + nλ)[1 + β]bn2pβ(1− α) an, n ≥ p + 1and µp = 1 −∑∞n=p+1 µn. Then we have f(z) =∑∞n=p+1 µnfn(z) and hence thiscompletes the proof of Theorem 2.Corollary 5. Letfp(z) = zpandfn(z) = zp − 2pβ(1− α)Γn(p + nλ)[1 + β]zn, n ≥ p + 1, (19)for 0 ≤ α < 1, 0 < β ≤ 1, λ ≥ 0. Then f(z) is in the class HT (λ, α, β) if and onlyif it can be expressed in the formf(z) = µpzp +∞∑n=p+1µnfn(z),whereµn ≥ 0,∞∑n=pµn = 1and Γn is given by (16).Remark 5. For specific choices of parameters r, s, α1, β1 (as mentioned in the Re-marks 1 and 2 ), Corollary 5 we can prove analogous results for the subclasses offunctions LT (λ, α, β) and RT ν(λ, α, β). Further on lines similar to the above the-orem one can easily prove the extreme points results for the classes IT (λ, α, β) andJ T (λ, α, β).3. Distortion boundsIn this section we obtain distortion bounds for the class Tpg(λ, α, β).Theorem 3. If f ∈ Tpg(λ, α, β), thenrp − 2pβ(1− α)(p + pλ + λ)[1 + β]bp+1rp+1 ≤ |f(z)| ≤ rp + 2pβ(1− α)(p + pλ + λ)[1 + β]bp+1rp+1(20)318 G.Murugusundaramoorthy and S.B. Joshiholds if the sequence {σn} is non-increasing for n > p andprp−1 − 2p(p + 1)β(1− α)(p + pλ + λ)[1 + β]bp+1rp ≤ |f ′(z)| ≤ prp−1 + 2p(p + 1)β(1− α)(p + pλ + λ)[1 + β]bp+1rp(21)holds if the sequence {σn/n} is non-increasing for n > p, where σn = (p+nλ)bn, (n >p).The bounds in (20) and (21) are sharp, since the equalities are attained by thefunctionf(z) = zp − 2pβ(1− α)(p + pλ + λ)[1 + β]bp+1zp+1 |z| = ±r. (22)Proof. In view of Theorem 1, we have∞∑n=p+1an ≤ 2pβ(1− α)(p + pλ + λ)[1 + β]bp+1 (23)Using (11) and (23), we obtain|z|p − |z|p+1∞∑n=p+1an ≤ |f(z)| ≤ |z|p + |z|p+1∞∑n=p+1anrp − rp+1 2pβ(1− α)(p + pλ + λ)[1 + β]bp+1≤ |f(z)| ≤ rp + rp+1 2pβ(1− α)(p + pλ + λ)[1 + β]bp+1.(24)Hence (20) follows from (24).Further, since∞∑n=p+1nan ≤ 2p(p + 1)β(1− α)(p + pλ + λ)[1 + β]bp+1 .Hence (21) follows fromprp−1 − (p + 1)rp∞∑n=p+1an ≤ |f ′(z)| ≤ prp−1 + (p + 1)rp∞∑n=p+1an.Corollary 6. If f ∈ HT (λ, α, β), thenrp − 2pβ(1− α)(p + pλ + λ)[1 + β]Γp+1rp+1 ≤ |f(z)| ≤ rp + 2pβ(1− α)(p + pλ + λ)[1 + β]Γp+1rp+1(25)holds if the sequence {σn} is non-increasing for n > p andprp−1 − 2p(p + 1)β(1− α)(p + pλ + λ)[1 + β]Γp+1rp ≤ |f ′(z)| ≤ prp−1 + 2p(p + 1)β(1− α)(p + pλ + λ)[1 + β]Γp+1rp(26)Certain classes of multivalent functions 319holds if the sequence {σn/n} is non-increasing for n > p, where σn = (p + nλ)Γn,(n > p).The bounds in (25) and (26) are sharp forf(z) = zp − 2pβ(1− α)(p + pλ + λ)[1 + β]Γp+1zp+1 (27)where Γp+1 is given by (16).Remark 6. For specific choices of parameters r, s, α1, β1 (as mentioned in Remarks1 and 2 ), Corollary 6 we can deduce analogous results for the subclasses of functionsLT (λ, α, β) and RT ν(λ, α, β). Further on lines similar to the distortion theorem onecan easily prove the distortion bounds for functions in the classes IT (λ, α, β) andJ T (λ, α, β).Corollary 7. If f ∈ ST (λ, α, β), thenrp − pβ(1− α)2m−1(p + pλ + λ)[1 + β]rp+1 ≤ |f(z)| ≤ rp + pβ(1− α)2m−1(p + pλ + λ)[1 + β]rp+1(28)prp−1 − pp + 1β(1− α)2m−1(p + pλ + λ)[1 + β]rp ≤ |f ′(z)| ≤ rp + p(p + 1)β(1− α)2m−1(p + pλ + λ)[1 + β]rp.(29)The bounds in (28) and (29) are sharp forf(z) = zp − pβ(1− α)2m−1(p + pλ + λ)[1 + β]zp+1. (30)4. Radius of starlikeness and convexityThe radii of close-to-convexity, starlikeness and convexity for the class Tpg(λ, α, β)are given in this section.Theorem 4. Let the function f(z) defined by (11) belong to the class Tpg(λ, α, β).Then f(z) p−valently is close-to-convex of order δ (0 ≤ δ < p) in the disc |z| < r1,wherer1 := infn≥p+1[(p− δ)(p + nλ)[1 + β] bn2pnβ(1− α)] 1n−p. (31)Proof. Given f ∈ T and f is close-to-convex of order δ, we have∣∣∣∣f ′(z)zp−1− p∣∣∣∣ < p− δ. (32)For the left-hand side of (32) we have|f′(z)zp−1− p| ≤∞∑n=p+1nan|z|n−p.320 G.Murugusundaramoorthy and S.B. JoshiThe last expression is less than p− δ if∞∑n=p+1np− δ an|z|n−p < 1.Using the fact, that f ∈ Tpg(λ, α, β) if and only if∞∑n=p+1(p + nλ)[1 + β]anbn2pβ(1− α) ≤ 1.We can say (32) is true ifnp− δ |z|n−p ≤ (p + nλ)[1 + β]bn2pβ(1− α) .Or, equivalently,|z|n−p =[(p− δ)(p + nλ)[1 + β] bn2pnβ(1− α)]which completes the proof.Theorem 5. Let f ∈ Tpg(λ, α, β). Then1. f is p−valently starlike of order δ(0 ≤ δ < p) in the disc |z| < r2; that is,Re{zf ′(z)f(z)}> δ, (|z| < r2 ; 0 ≤ δ < p), wherer2 = infn≥p+1{(p− δ)(p + nλ)[1 + β] bn2pβ(1− α)(k + p− δ)} 1n.2. f is p − valently convex of order δ (0 ≤ δ < p) in the disc |z| < r3, that isRe{1 + zf′′(z)f ′(z)}> δ, (|z| < r3; 0 ≤ δ < p), wherer3 = infn≥p+1{(p− δ)(p + nλ)[1 + β] bn2nβ(1− α)(n− δ)} 1n.Each of these results are sharp for the extremal function f(z) given by (17).Proof. Given f ∈ T and f is p−valently starlike of order δ, we have∣∣∣∣zf ′(z)f(z)− p∣∣∣∣ < p− δ. (33)For the left-hand side of (33) we have∣∣∣∣zf ′(z)f(z)− p∣∣∣∣ ≤∑∞n=p+1(n− p)an |z|n1−∑∞n=p+1 an |z|n.Certain classes of multivalent functions 321The last expression is less than p− δ if∞∑n=p+1n− δp− δ an |z|n < 1.Using the fact, that f ∈ Tpg(λ, α, β) if and only if∞∑n=p+1(p + nλ)[1 + β]anbn2pβ(1− α) < 1.We can say (33) is true ifn− δp− δ |z|n <(p + nλ)[1 + β] bn2pβ(1− α) .Or, equivalently,|z|n < (p− δ)(p + nλ)[1 + β] bn2pβ(1− α)(n− δ)which yields the starlikeness of the family.(ii) Using the fact that f is convex if and only if zf ′ is starlike, we canprove (2), on lines similar the proof of (1).Concluding remarks: Corresponding to the various subclasses which arise fromthe function class Tpg(λ, α, β), by suitably choosing the function g(z) as mentioned in(3) to (8), and for specific choices of parameters α1, β1, r, s we can deduce analogousresults for the subclasses of functions introduced in this paper. Furthermore, bysuitably choosing the values of g, α, β, λ, and p = 1 the class Tpg(λ, α, β) and theabove subclasses reduce to the various subclasses introduced and studied in theliterature (see [1, 2, 7, 9, 13, 17]).AcknowledgementThe authors express their sincerest thanks to the referee for useful comments.References[1] O.Altintas, A subclass of analytic functions with negative coefficients, Bull. Sci-Engg. Hacettepe Univ. 19(1990), 15–24.[2] G. J. Bao, Y. Ling, On some classes of analytic functions with negative coefficients,J. Harbin Inst. Tech. 23(1991), 100–103.[3] B.C.Carlson, D.B. Shaffer, Starlike and prestarlike hypergeometric functions,SIAM J. Math. Anal. 15(1984), 737–745.[4] N.E.Cho, T.H.Kim, Multiplier transformations and strongly close-to-convex func-tions, Bull. Korean Math. Soc. 40(2003), 399–410.[5] N.E.Cho, H.M. Srivastava, Argument estimates of certain analytic functions de-fined by a class of multiplier transformations, Math. Comput. 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Certain classes of multivalent functions with negative coefficients associated with a convolution structure

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