MATEMATIQKI VESNIK originalni nauqni rad research paper 63, 4 (2011), 235–246 December 2011 THE QUASI-HADAMARD PRODUCTS OF UNIFORMLY CONVEX FUNCTIONS DEFINED BY DZIOK-SRIVASTAVA OPERATOR M. K. Aouf, A. Shamandy, A. O. Mostafa and E. A. Adwan Abstract. The purpose of this paper is to obtain many interesting results about the quasiHadamard products of uniformly convex functions defined by Dziok-Srivastava operator belonging to the class Tq,s ([α1 ]; α, β). 1. Introduction Let T denote the class of functions of the form ∞ P an z n (an ≥ 0) , f (z) = z − (1.1) n=2 which are analytic in the open unit disc U = {z : |z| < 1}. Let C(γ) and T ∗ (γ) denote the subclasses of T which are, respectively, convex and starlike functions of order γ, 0 ≤ γ < 1. For convenience, we write C(0) = C and T ∗ (0) = T ∗ (see [9]). A function f ∈ T is said to be in U ST (β, γ), the class of β-uniformly starlike functions of order γ, −1 ≤ γ < 1, if it satisfies the condition ¯ 0 ¯ ¾ ½ 0 ¯ zf (z) ¯ zf (z) − γ > β ¯¯ − 1¯¯ , (β ≥ 0) . (1.2) Re f (z) f (z) 0 Replacing f (z) in (1.2) by zf (z) we have the condition ¯ 00 ¯ ¾ ½ ¯ zf (z) ¯ zf 00 (z) ¯ ¯ −γ >β¯ 0 Re 1 + 0 ¯ (β ≥ 0) , ¯ f (z) ¯ f (z) required for the function f to be in the subclass U CT (β, γ) of β-uniformly convex functions of order γ (see [2]). Let fj (z) ∈ T (j = 1, . . . , t) be given by fj (z) = z − ∞ P n=2 an,j z n (an,j > 0; j = 1, 2, . . . , t). (1.3) 2010 AMS Subject Classification: 30C45. Keywords and phrases: Analytic functions; Dziok-Srivastava operator; uniformly convex functions; quasi-Hadamard product. 235 236 M. K. Aouf, A. Shamandy, A. O. Mostafa, E. A. Adwan Then the quasi-Hadamard product (or convolution) of these functions is defined by (f1 ∗ f2 ∗ · · · ∗ ft )(z) = z − ∞ ³Q t P n=2 j=1 ´ an,j z n . (1.4) For positive real parameters α1 , . . . , αq and β1 , . . . , βs , βi ∈ C \ Z − ; Z − = {0, −1, −2, . . . }; i = 1, 2, . . . , s, the Dziok-Srivastava operator (see [3] and [4]) Hq,s (α1 ) : T → T is given by Hq,s (α1 )f (z) = z q Fs (α1 , . . . αq ; β1 , . . . βs ) ∗ f (z) = z − ∞ P n=2 Ψn a n z n , (1.5) where (α1 )n−1 · · · (αq )n−1 , (β1 )n−1 · · · (βs )n−1 (n − 1)! ½ 1, n=0 Γ(θ + n) (θ)n = = Γ(θ) θ(θ + 1) · · · (θ + n − 1), n ∈ N. Ψn = (1.6) and q Fs (α1 , . . . , αq ; β1 , . . . , βs ; z) (q ≤ s + 1; s, q ∈ N0 = N ∪ {0}, N = {1, 2, . . . }; z ∈ U ) is the generalized hypergeometric function. q Fs (α1 , . . . , αq ; β1 , . . . , βs ; z) = ∞ (α ) . . . (α ) P 1 n q n n z . n=0 (β1 )n . . . (βs )n n! For −1 ≤ γ < 1, β ≥ 0, and for all z ∈ U , Aouf and Murugusundaramoorthy [1] defined the subclass Tq,s ([α1 ]; γ, β) of functions of T which satisfy: ¯ ¯ à ! 0 ¯ z(H (α )f (z))0 ¯ z(Hq,s (α1 )f (z)) ¯ ¯ q,s 1 Re −γ >β¯ − 1¯ , z ∈ U. (1.7) ¯ Hq,s (α1 )f (z) ¯ Hq,s (α1 )f (z) They also proved [1] that the necessary and sufficient condition for functions f (z) of the form (1.1) to be in the class Tq,s ([α1 ]; γ, β) is that: ∞ P n=2 [n(1 + β) − (γ + β)]Ψn an ≤ 1 − γ. (1.8) We note that for suitable choices of q, s, γ and β, we obtain the following subclasses studied by various authors. (1) For q = 2 and s = α1 = α2 = β1 = 1 in (1.7), the class T2,1 ([1]; γ, β) reduces to the class Sp T (γ, β) (−1 ≤ γ < 1, β ≥ 0) and the class Sp T (γ, 1) which for β = 1 reduces to the class Sp T (γ) (see [2]). (2) For q = 2, s = 1, α1 = a (a > 0) , α2 = 1 and β1 = c (c > 0) in (1.7), the class T2,1 ([a]; γ, β) reduces to the class Sp T (a, c; γ, β) (−1 ≤ γ < 1, β ≥ 0) (see [5]). (3) For q = 2, s = 1, α1 = λ + 1 (λ > −1), α2 = 1 and β1 = 1 in (1.7), the class T2,1 (λ + 1, 1; 1; γ, β) reduces to the class Sp T (λ; γ, β) (−1 ≤ γ < 1, β ≥ 0) (see [8]). Quasi-Hadamard products of uniformly convex functions 237 (4) For q = 2, s = 1, α1 = v + 1 (v > −1), α2 = 1 and β1 = v + 2 in (1.7), the class T2,1 (v + 1, 1; v + 2; γ, β) reduces to the class Sp T (v; γ, β) (−1 ≤ γ < 1, β ≥ 0) (see [1]). (5) For q = 2, s = 1, α1 = 2, α2 = 1 and β1 = 2 − µ (µ 6= 2, 3, . . . ) in (1.7), the class T2,1 (2, 1; 2 − µ; γ, β) reduces to the class Sp T (µ; γ, β) (−1 ≤ γ < 1, β ≥ 0) (see [1]). 2. Main results Unless otherwise mentioned, we shall assume in the remainder of this paper that the parameters α1 , . . . , αq and β1 , . . . , βs are positive real numbers, −1 ≤ γ < 1, β ≥ 0, z ∈ U , Ψn is defined by (1.6), Ψn ≥ 1 and j = 1, 2, . . . , t. Theorem 1. Let the functions fj (z) defined by (1.3) be in the class Tq,s ([α1 ]; γj , β). Then we have (f1 ∗ · · · ∗ ft )(z) ∈ Tq,s ([α1 ]; δ, β), where Qt (1 + β) j=1 (1 − γj ) δ = 1 − Qt . Qt t−1 − j=1 (1 − γj ) j=1 (2 + β − γj )Ψ2 (2.1) The result is sharp for the functions fj (z) given by fj (z) = z − (1 − γj ) z2. (2 + β − γj )Ψ2 (2.2) Proof. Employing the technique used earlier by Schild and Silverman [7] and Owa [6], we prove Theorem 1 by using mathematical induction on t. For t = 2, (1.8) gives ∞ [n(1 + β) − (γ + β)]Ψ P j n an,j ≤ 1 (j = 1, 2). (2.3) (1 − γj ) n=2 By the Cauchy-Schwarz inequality, we have s ∞ 2 P Q [n(1+β)−(γj +β)]Ψn √ an,2 an,2 ≤ 1. (2.4) (1−γj ) n=2 j=1 To prove the case when t = 2, we need to find the largest δ(−1 ≤ δ < 1) such that ∞ [n(1 + β) − (δ + β)]Ψ P n an,1 an,2 ≤ 1, (1 − δ) n=2 thus, it suffices to show that [n(1 + β) − (δ + β)]Ψn an,1 an,2 ≤ (1 − δ) or, equivalently, to √ an,1 an,2 ≤ (1 − δ) qQ 2 qQ j=1 [n(1 (2.5) + β) − (γj + β)]Ψn √ qQ an,1 an,2 2 (1 − γ ) j j=1 2 j=1 [n(1 + β) − (γj + β)]Ψn qQ . 2 [n(1 + β) − (δ + β)]Ψn (1 − γ ) j j=1 238 M. K. Aouf, A. Shamandy, A. O. Mostafa, E. A. Adwan By noting that √ qQ 2 j=1 (1 an,1 an,2 ≤ qQ 2 j=1 [n(1 − γj ) , + β) − (γj + β)]Ψn consequently, we need only to prove that Q2 (1 − δ) j=1 (1 − γj ) ≤ Q2 [n(1 + β) − (δ + β)] [n(1 + β) − (γ + β)]Ψ j n j=1 which is equivalent to (n − 1)(1 + β) Q2 − γj ) . Q2 j=1 [n(1 + β) − (γj + β)]Ψn − j=1 (1 − γj ) δ ≤ 1 − Q2 j=1 (1 Since Q2 − γj ) , Q2 j=1 [n(1 + β) − (γj + β)]Ψn − j=1 (1 − γj ) B(n) = 1 − Q2 (n − 1)(1 + β) j=1 (1 is an increasing function of n (n > 2), then Q2 − γj ) . Q2 j=1 (1 − γj ) j=1 (2 + β − γj )Ψ2 − δ ≤ B(2) = 1 − Q2 (1 + β) j=1 (1 Therefore, the result is true for t = 2. Suppose that the result is true for any positive integer t = k. Then we have (f1 ∗ · · · ∗ fk ∗ fk+1 )(z) ∈ Tq,s ([α1 ]; λ, β), where λ=1− (1 + β)(1 − γk+1 )(1 − δ) , (2 + β − γk+1 )(2 + β − δ)Ψ2 − (1 − γk+1 )(1 − δ) and δ is given by (2.2). After simple calculations, we have Qk+1 (1 + β) j=1 (1 − γj ) . λ = 1 − Qk+1 Qk+1 k j=1 (1 − γj ) j=1 (2 + β − γj )Ψ2 − (2.6) This shows that the result is true for t = k + 1. Therefore, by mathematical induction, the result is true for any positive integer t (t ≥ 2). Taking the functions fj (z) given by (2.2), we have (f1 ∗ · · · ∗ ft )(z) = z − t Y (1 − γj ) z 2 = z − H2 z 2 , (2 + β − γ )Ψ j 2 j=1 which shows that t ∞ [n(1 + β) − (δ + β)]Ψ P (2 + β − δ]Ψ2 Q (1 − γj ) n H2 = · = 1; (1 − δ) (1 − δ) (2 + β − γj )Ψ2 n=2 j=1 (2.7) Quasi-Hadamard products of uniformly convex functions 239 Consequently, the result is sharp for functions fj (z) given by (2.2). This completes the proof of Theorem 1. Letting γj = γ in Theorem 1, we obtain the following corollary. Corollary 1. Let the functions fj (z) defined by (1.3) be in the class Tq,s ([α1 ]; γ, β). Then we have (f1 ∗ · · · ∗ ft )(z) ∈ Tq,s ([α1 ]; δ, β), δ =1− (1 + β)(1 − γ)t . (2 + β − γ)t Ψt−1 − (1 − γ)t 2 (2.8) The result is sharp for the functions fj (z) given by fj (z) = z − (1 − γ) z2. (2 + β − γ)Ψ2 (2.9) Putting t = 2 in Corollary 1, we obtain the following corollary. Corollary 2. Let the functions fj (z) (j = 1, 2) defined by (1.3) be in the class Tq,s ([α1 ]; γ, β). Then we have (f1 ∗ f2 )(z) ∈ Tq,s ([α1 ]; δ, β), where δ =1− (1 + β)(1 − γ)2 . (2 + β − γ)2 Ψ2 − (1 − γ)2 The result is sharp. Next, similarly by applying the method of proof of Theorem 1, we easily get the following result. Theorem 2. Let the functions fj (z) defined by (1.3) be in the class Tq,s ([α1 ]; γ, ζj ), ζj ≥ 0. Then we have (f1 ∗ · · · ∗ ft )(z) ∈ Tq,s ([α1 ]; γ, η), where Qt t−1 j=1 (2 + ζj − γ)Ψ2 η= + γ − 2. (2.10) (1 − γ)t−1 The result is sharp for the functions fj (z) given by fj (z) = z − (1 − γ) z2. (2 + ζj − γ)Ψ2 (2.11) Let ζj = β (j = 1, 2, . . . , t) in Theorem 2, we obtain the following corollary. Corollary 3. Let the functions fj (z) defined by (1.3) be in the class Tq,s ([α1 ]; γ, β). Then we have (f1 ∗ · · · ∗ ft )(z) ∈ Tq,s ([α1 ]; γ, η), where η= (2 + β − γ)t Ψt−1 2 + γ − 2. (1 − γ)t−1 The result is sharp for the functions fj (z) given by (2.9). 240 M. K. Aouf, A. Shamandy, A. O. Mostafa, E. A. Adwan Putting t = 2 in Corollary 3, we obtain the following corollary. Corollary 4. Let the functions fj (z) (j = 1, 2) defined by (1.3) be in the class Tq,s ([α1 ]; γ, β). Then we have (f1 ∗ f2 )(z) ∈ Tq,s ([α1 ]; γ, η), where η= (2 + β − γ)2 Ψ2 + γ − 2. (1 − γ) The result is sharp. Theorem 3. Let the functions fj (z) defined by (1.3) be in the class Tq,s ([α1 ]; γj , β). Then the function F (z) = z − ∞ ³P t P n=2 j=1 ´ n am n,j z (m > 1) (2.12) belongs to the class Tq,s ([α1 ]; δt , β), where δt = 1 − t(1 + β)(1 − γ)m (γ = min {γj }), 1≤j≤t (2 + β − γ)m Ψm−1 − t(1 − γ)m 2 (2.13) and (2 + β − γ)m Ψm−1 ≥ t(2 + β)(1 − γ)m . The result is sharp for the functions 2 fj (z) (j = 1, 2, . . . , t) given by (2.2). Proof. By virtue of (1.8), we have ∞ [n(1 + β) − (γ + β)]Ψ P j n an,j ≤ 1. (1 − γj ) n=2 By the Cauchy-Schwarz inequality, we have ³P ´m ∞ ³ [n(1 + β) − (γ + β)]Ψ ´m ∞ [n(1 + β) − (γ + β)]Ψ P j n j n am ≤ a ≤ 1. n,j n,j (1 − γj ) (1 − γj ) n=2 n=2 (2.14) It follows from (2.14) that à ! µ ¶m ∞ t P 1 P [n(1 + β) − (γj + β)]Ψn m an,j ≤ 1. t j=1 (1 − γj ) n=2 By setting γ = min1≤j≤t {γj }, the last inequality gives à µ ! ¶m t ∞ P P m 1 [n(1 + β) − (γ + β)]Ψn (α1 ) an,j ≤ 1. t (1 − γ) n=2 j=1 Therefore, to prove our result we need to fined the largest δt such that ∞ [n(1 + β) − (δ + β)]Ψ (α ) P t P t n 1 am n,j ≤ 1, (1 − δt ) n=2 j=1 241 Quasi-Hadamard products of uniformly convex functions that is, that [n(1 + β) − (δt + β)]Ψn 1 ≤ (1 − δt ) t µ [n(1 + β) − (γ + β)]Ψn (1 − γ) ¶m which leads to δt ≤ 1 − t(n − 1)(1 + β)(1 − γ)m m−1 [n(1 + β) − (γ + β)]m (Ψn ) − t(1 − γ)m . Now let R(n) = 1 − t(n − 1)(1 + β)(1 − γ)m [n(1 + β) − (γ + β)]m Ψnm−1 − t(1 − γ)m (n ≥ 2). Since R(n) is an increasing function of n (n > 2), then we have δt ≤ R(2) = 1 − t(1 + β)(1 − γ)m , (2 + β − γ)m Ψm−1 − t(1 − γ)m 2 and by noting that (2+β −γ)m Ψm−1 ≥ t(2+β)(1−γ)m , we can see that 0 ≤ δt < 1. 2 The result is sharp for the functions fj (z) (j = 1, 2, . . . , t) given by (2.2). This completes the proof of Theorem 3. Putting m = 2 and γj = γ (j = 1, . . . , t) in Theorem 3, we obtain the following corollary. Corollary 5. Let the functions fj (z) (j = 1, 2, . . . , t) defined by (1.3) be in the class Tq,s ([α1 ]; γ, β). Then the function G(z) = z − ∞ ³P t P n=2 j=1 ´ a2n,j z n , (2.15) belongs to the class Tq,s ([α1 ]; δt , β), where δt = 1 − t(1 + β)(1 − γ)2 (2 + β − γ)2 Ψ2 − t(1 − γ)2 (2.16) and (2 + β − γ)2 Ψ2 ≥ t(2 + β)(1 − γ)2 . The result is sharp for the functions fj (z) (j = 1, 2, . . . , t) given by (2.9). Similarly by applying the method of proof of Theorem 3, we easily get the following result. Theorem 4. Let the functions fj (z) defined by (1.3) be in the class Tq,s ([α1 ]; γ, ζj ), ζj ≥ 0. Then the function F (z) defined by (2.12) belongs to the class Tq,s ([α1 ]; γ, ηt ), where ηt = (2 + β − γ)m Ψ2m−1 +γ−2 t(1 − γ)m−1 (β = min {ζj }), 1≤j≤t 242 M. K. Aouf, A. Shamandy, A. O. Mostafa, E. A. Adwan and (2 + β − γ)m Ψm−1 ≥ t(2 − γ)(1 − γ)m−1 . The result is sharp for the functions 2 fj (z) (j = 1, 2, . . . , t) given by (2.11). Putting m = 2 and ζj = β (j = 1, 2, . . . ., t) in Theorem 4, we obtain the following corollary. Corollary 6. Let the functions fj (z) (j = 1, 2, . . . , t) defined by (1.3) be in the class Tq,s ([α1 ]; γ, β). Then the function G(z) defined by (2.15) belongs to the class Tq,s ([α1 ]; γ, ηt ), where (2 + β − γ)2 Ψ2 +γ−2 t(1 − γ) ηt = and (2 + β − γ)2 Ψ2 ≥ t(2 − γ)(1 − γ). The result is sharp for the functions fj (z) given by (2.9). Theorem 5. Let the functions fj (z) (j = 1, 2, . . . , t) defined by (1.3) be in the class Tq,s ([α1 ]; γj , β) (j = 1, 2, . . . , t) and let the functions gm (z) defined by gm (z) = z − ∞ P bn,m z n (bn,m > 0; m = 1, 2, . . . , s), n=2 (2.17) be in the class Tq,s ([α1 ]; γm , β) (m = 1, 2, . . . , s), then (f1 ∗ f2 ∗ · · · ∗ ft ∗ g1 ∗ g2 ∗ · · · ∗ gs )(z) ∈ Tq,s ([α1 ]; Ω, β), where (1 + β) Ω=1− t Q j=1 t Q (2 + β − γj ) j=1 s Q (2 + β − m=1 (1 − γj ) s Q (1 − γm ) m=1 γm )Ψ2t+s−1 − t Q (1 − γj ) j=1 s Q m=1 . (1 − γm ) (2.18) The result is sharp for the functions fj (z) given by (2.2) and the functions gm (z) given by (1 − γm ) z 2 (m = 1, 2, . . . , s). (2.19) gm (z) = z − (2 + β − γm )Ψ2 Proof. From Theorem 1 we note that, if f (z) ∈ Tq,s ([α1 ]; δ, β) and g(z) ∈ Tq,s ([α1 ]; µ, β), then (f ∗ g)(z) ∈ Tq,s ([α1 ]; Ω, β), where Ω=1− (1 + β)(1 − δ)(1 − µ) . (2 + β − δ)(2 + β − µ)Ψ2 − (1 − δ)(1 − µ) (2.20) Since Theorem 1 leads to (f1 ∗ f2 ∗ · · · ∗ ft )(z) ∈ Tq,s ([α1 ]; δ, β), where δ is defined by (2.1) and (g1 ∗ g2 ∗ · · · ∗ gs )(z) ∈ Tq,s ([α1 ]; µ, β) with Qs (1 + β) m=1 (1 − γm ) µ = 1 − Qs . (2.21) Qs s−1 − m=1 (1 − γm ) m=1 (2 + β − γm )Ψ2 Quasi-Hadamard products of uniformly convex functions 243 Then, we have (f1 ∗ f2 ∗ · · · ∗ ft ∗ g1 ∗ g2 ∗ · · · ∗ gs )(z) ∈ Tq,s ([α1 ]; Ω, β), where Ω is given by (2.18), this completes the proof of Theorem 5. Letting γj = γ (j = 1, 2, . . . , t) and γm = γ (m = 1, 2, . . . , s) in Theorem 5, we obtain the following corollary. Corollary 7. Let the functions fj (z) (j = 1, 2, . . . , t) defined by (1.3) and let the functions gm (z) defined by (2.17) be in the class Tq,s ([α1 ]; γ, β). Then we have (f1 ∗ f2 ∗ · · · ∗ ft ∗ g1 ∗ g2 ∗ . . . . ∗ gs )(z) ∈ Tq,s ([α1 ]; Ω, β), where Ω=1− (1 + β)(1 − γ)t+s . (2 + β − γ)t+s Ψt+s−1 − (1 − γ)t+s 2 The result is sharp for the functions fj (z) given by (2.9) and the functions gm (z) given by (1 − γ) gm (z) = z − z 2 (m = 1, 2, . . . , s). (2 + β − γ)Ψ2 Letting t = s = 2 in Corollary 7, we obtain the following corollary. Corollary 8. Let the functions fj (z) (j = 1, 2) defined by (1.3) and let the functions gm (z) (m = 1, 2) defined by (2.17) be in the class Tq,s ([α1 ]; γ, β). Then we have (f1 ∗ f2 ∗ g1 ∗ g2 )(z) ∈ Tq,s ([α1 ]; Ω, β), where Ω=1− (1 + β)(1 − γ)4 . (2 + β − γ)4 Ψ32 − (1 − γ)4 The result is sharp. Putting q = 2, s = 1 and α1 = α2 = β1 = 1 in Theorem 5, we obtain the following corollary. Corollary 9. Let fj (z) (j = 1, 2, . . . , t) defined by (1.3) be in the class Sp T (γj , β) and let the functions gm (z) (m = 1, 2, . . . , s) defined by (2.17) be in the class Sp T (γm , β) (m = 1, 2, . . . , s), then (f1 ∗ f2 ∗ · · · ∗ ft ∗ g1 ∗ g2 ∗ · · · ∗ gs )(z) ∈ Sp T (τ, β), where Qs Qt (1 + β) j=1 (1 − γj ) m=1 (1 − γm ) . τ = 1 − Qt Qs Qt Qs m=1 (2 + β − γm ) − j=1 (1 − γj ) m=1 (1 − γm ) j=1 (2 + β − γj ) The result is sharp for the functions fj (z) given by fj (z) = z − (1 − γj ) z2 (2 + β − γj ) (j = 1, 2, . . . , t) and the functions gm (z) given by gm (z) = z − (1 − γm ) z2 (2 + β − γm ) (m = 1, 2, . . . , s). 244 M. K. Aouf, A. Shamandy, A. O. Mostafa, E. A. Adwan Putting q = 2, s = 1, α1 = a(a > 0), α2 = 1 and β1 = c (c > 0) in Theorem 5, we obtain the following corollary. Corollary 10. Let fj (z) (j = 1, 2, . . . , t) defined by (1.3) be in the class Sp T (a, c; γj , β) and let the functions gm (z) (m = 1, 2, . . . , s) defined by (2.17) be in the class Sp T (a, c; γm , β) (m = 1, 2, . . . , s), then (f1 ∗ f2 ∗ · · · ∗ ft ∗ g1 ∗ g2 ∗ · · · ∗ gs )(z) ∈ Sp T (a, c; ζ, β), where ct+s−1 (1 + β) t Q j=1 ζ = 1− at+s−1 t Q j=1 (2 + β − γj ) s Q m=1 (1 − γj ) (2 + β − γm ) − s Q (1 − γm ) m=1 ct+s−1 t Q (1 − γj ) j=1 . s Q m=1 (1 − γm ) The result is sharp for the functions fj (z) given by fj (z) = z − (1 − γj )c z2 (2 + β − γj )a (j = 1, 2, . . . , t) and the functions gm (z) given by gm (z) = z − (1 − γm )c z 2 (m = 1, 2, . . . , s). (2 + β − γm )a Putting q = 2, s = 1, α1 = λ + 1(λ > −1), α2 = 1 and β1 = 1 in Theorem 5, we obtain the following corollary. Corollary 11. Let fj (z) (j = 1, 2, . . . , t) defined by (1.3) be in the class Sp T (λ; γj , β) and let the functions gm (z) (m = 1, 2, . . . , s) defined by (2.17) be in the class Sp T (λ; γm , β) (m = 1, 2, . . . , s), then (f1 ∗ f2 ∗ · · · ∗ ft ∗ g1 ∗ g2 ∗ · · · ∗ gs )(z) ∈ Sp T (λ; ν, β), where t Q (1 + β) ν =1− (λ + 1)t+s−1 t Q j=1 (1 − γj ) j=1 s Q (2 + β − γj ) s Q (1 − γm ) m=1 (2 + β − γm ) − m=1 t Q (1 − γj ) j=1 The result is sharp for the functions fj (z) given by fj (z) = z − (1 − γj ) z2 (2 + β − γj )(λ + 1) (j = 1, 2, . . . , t) s Q . (1 − γm ) m=1 245 Quasi-Hadamard products of uniformly convex functions and the functions gm (z) given by (1 − γm ) z2 (2 + β − γm )(λ + 1) gm (z) = z − (m = 1, 2, . . . , s). Putting q = 2, s = 1, α1 = v + 1(v > −1), α2 = 1 and β1 = v + 2 in Theorem 5, we obtain the following corollary. Corollary 12. Let fj (z) (j = 1, 2, . . . , t) defined by (1.3) be in the class Sp T (v; γj , β) and let the functions gm (z) (m = 1, 2, . . . , s) defined by (2.17) be in the class Sp T (v; γm , β) (m = 1, 2, . . . , s), then (f1 ∗ f2 ∗ · · · ∗ ft ∗ g1 ∗ g2 ∗ · · · ∗ gs )(z) ∈ Sp T (v; σ, β), where σ = 1− t Q (v + 2)t+s−1 (1 + β) j=1 (v + 1)t+s−1 t Q j=1 (2 + β − γj ) s Q j=1 (1 − γj ) s Q (1 − γm ) j=1 (2 + β − γm ) − (v + 2)t+s−1 t Q j=1 (1 − γj ) s Q . (1 − γm ) j=1 The result is sharp for the functions fj (z) given by fj (z) = z − (1 − γj )(v + 2) z2 (2 + β − γj )(v + 1) (j = 1, 2, . . . , t) and the functions gm (z) given by (1 − γm )(v + 2) z2 (2 + β − γm )(v + 1) gm (z) = z − (m = 1, 2, . . . , s). Putting q = 2, s = 1, α1 = 2, α2 = 1 and β1 = 2 − µ (µ 6= 2, 3, . . . ) in Theorem 5, we obtain the following corollary. Corollary 13. Let fj (z) (j = 1, 2, . . . , t) defined by (1.3) be in the class Sp T (µ; γj , β) and let the functions gm (z) (m = 1, 2, . . . , s) defined by (2.17) be in the class Sp T (µ; γm , β) (m = 1, 2, . . . , s), then (f1 ∗ f2 ∗ · · · ∗ ft ∗ g1 ∗ g2 ∗ · · · ∗ gs )(z) ∈ Sp T (µ; κ, β), where κ = 1− (2 − µ)t+s−1 (1 + β) t Q j=1 2t+s−1 t Q (2 + β − γj ) j=1 s Q (1 − γj ) (2 + β − γm ) − (2 − m=1 s Q (1 − γm ) m=1 µ)t+s−1 t Q j=1 (1 − γj ) s Q . (1 − γm ) m=1 246 M. K. Aouf, A. Shamandy, A. O. Mostafa, E. A. Adwan The result is sharp for the functions fj (z) given by fj (z) = z − (2 − µ)(1 − γj ) 2 z 2(2 + β − γj ) (j = 1, 2, . . . , t) and the functions gm (z) given by gm (z) = z − (2 − µ)(1 − γm ) 2 z 2(2 + β − γm ) (m = 1, 2, . . . , s). Acknowledgements. The authors would like to thank the referees of the paper for their helpful suggestions. REFERENCES [1] M.K. Aouf, G. Murugusundaramoorthy, On a subclass of uniformly convex functions defined by the Dziok-Srivastava operator, Austral. J. 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Schild, H. Silverman, Convolutions of univalent functions with negative coefficients, Ann. Univ. Mariae Curie-Sklodowska Sect. A 29 (1975), 99–107. [8] S. Shams, S.R. Kulkarni, On class of univalent functions defined by Ruscheweyh derivatives, Kyungpook Math. J. 43 (2003), 579–585. [9] H. Silverman, Univalent functions with negative coefficients, Proc. Amer. Math. Soc. 51 (1975), 109–116. (received 17.05.2010; in revised form 24.09.2010) Department of Mathematics, Faculty of Science, Mansoura University, Mansoura 35516, Egypt E-mail: mkaouf127@yahoo.com, shamandy16@hotmail.com, adelaeg254@yahoo.com, eman.a2009@yahoo.com