Differential Sandwich Theorem for Multivalent Meromorphic Functions associated with the Liu-Srivastava Operator

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1 KYUNGPOOK Math. J , DOI /KMJ Differential Sandwich Theorem for Multivalent Meromorhic Functions associated with the Liu-Srivastava Oerator Rosihan M. Ali, R. Chandrashekar and See Keong Lee School of Mathematical Sciences, Universiti Sains Malaysia, USM Penang, Malaysia rosihan@cs.usm.my, chandrasc82@hotmail.com and sklee@cs.usm.my A. Swaminathan Deartment of Mathematics, Indian Institute of Technology, Roorkee , Uttarkhand, India swamifma@iitr.ernet.in V. Ravichandran Deartment of Mathematics, University of Delhi, Delhi , India vravi@maths.du.ac.in Abstract. Differential subordination and suerordination results are obtained for multivalent meromorhic functions associated with the Liu-Srivastava linear oerator in the unctured unit disk. These results are derived by investigating aroriate classes of admissible functions. Sandwich-tye results are also obtained. Dedicated to Professor H. M. Srivastava on the occasion of his 70th birthday. 1. Introduction Let HU be the class of functions analytic in U := {z C : z < 1} and H[a, n] be the subclass of HU consisting of functions of the form fz = a + a n z n + a n+1 z n+1 +, with H H[1, 1]. Let Σ denote the class of all meromorhic -valent functions of the form 1.1 fz = 1 z + k=1 a k z k 1 * Corresonding Author. Received July 19, 2010; acceted January 20, Mathematics Subject Classification: Primary 30C80, Secondary 30C45. Key words and hrases: Hyergeometric function, subordination, suerordination, Liu- Srivastava linear oerator, convolution. 217

2 218 R. M. Ali, R. Chandrashekar, S. K. Lee, A. Swaminathan and V. Ravichandran that are analytic in the unctured oen unit disk U = {z C : 0 < z < 1}, and let Σ 1 := Σ. Let f and F be members of HU. A function f is said to be subordinate to F, or F is said to be suerordinate to f, written fz F z, if there exists a function w analytic in U with w0 = 0 and wz < 1 z U satisfying fz = F wz. If F is univalent, fz F z if and only if f0 = F 0 and fu F U. For two functions f, g Σ, where f is given by 1.1 and gz = 1 z + k=1 b kz k, the Hadamard roduct or convolution of f and g is defined by the series f gz := 1 z + k=1 a k b k z k =: g fz. For α j C j = 1, 2,..., l and β j C \ {0, 1, 2,...} j = 1, 2,..., m, the generalized hyergeometric function l F m α 1,..., α l ; β 1,..., β m ; z is defined by the infinite series α 1 k... α l k z k lf m α 1,..., α l ; β 1,..., β m ; z := β 1 k... β m k k! k=0 l m + 1; l, m N 0 := {0, 1, 2,...}, where a n is the Pochhammer symbol defined by a n := Γa + n Γa = Corresonding to the function { 1, n = 0; aa + 1a a + n 1, n N := {1, 2, 3...}. h α 1,..., α l ; β 1,..., β m ; z := z lf m α 1,..., α l ; β 1,..., β m ; z, the Liu-Srivastava oerator [17, 18] Hl,m α 1,..., α l ; β 1,..., β m : Σ Σ is defined by the Hadamard roduct 1.2 α 1,..., α l ; β 1,..., β m fz := h α 1,..., α l ; β 1,..., β m ; z fz For convenience, 1.2 is written as H l,m [β 1 ]fz := = 1 z + k=1 l,m H α 1,..., α l ; β 1,..., β m fz. α 1 k+... α l k+ a k z k β 1 k+... β m k+ k +!. Various authors have investigated the Liu-Srivastava oerator where 1.2 is framed using the notation H l,m [α 1 ]fz. Their works exloited the recurrence relation involving the arameter α 1 in the numerator satisfying α 1 H l,m [α 1 + 1]fz = z[h l,m [α 1 ]fz] + α 1 + H l,m [α 1 ]fz.

3 Differential Sandwich Theorem for the Liu-Rrivastava Oerator 219 The arameter β 1 in the denominator also satisfies a recurrence relation. Denoting 1.2 by [β 1 ]fz, it can be shown that H l,m 1.3 β Hl,m 1 [β 1 ]fz = z[ l,m H [β 1 + 1]fz] + β 1 + l,m H [β 1 + 1]fz. The analytic analogue of the Liu-Srivastava oerator known as the Dziok-Srivastava oerator with resect to the arameter β 1, was first investigated by Srivastava et al. [24], and more recently by Ali et al. [3]. However, the Liu-Srivastava oerator for the arameter β 1 given by 1.3 for functions f satisfying 1.1 seems yet to be investigated. Secial cases of the Liu-Srivastava linear oerator include the meromorhic 2,1 analogue of the Carlson-Shaffer linear oerator L a, c := H 1, a; c[14, 16, 27], the oerator D n+1 := L n+, 1, which is analogous to the Ruscheweyh derivative oerator [26], and the oerator J c, := c z c+ z 0 t c+ 1 ftdt = L c, c + 1 c > 0 studied by Uralegaddi and Somanatha [25]. It is clear that the Liu-Srivastava oerator investigated in [11, 22, 23] is the meromorhic analogue of the Dziok- Srivastava [12] linear oerator. To state our main results, the following definitions and theorems will be required. Denote by Q the set of all functions q that are analytic and injective on U \Eq, where Eq = {ζ U : lim qz = }, z ζ and are such that q ζ 0 for ζ U \ Eq. Further let the subclass of Q for which q0 = a be denoted by Qa and Q1 Q 1. Definition 1.1[19, Definition 2.3a,. 27]. Let Ω be a set in C, q Q and n be a ositive integer. The class of admissible functions Ψ n [Ω, q] consists of those functions ψ : C 3 U C satisfying the admissibility condition ψr, s, t; z Ω whenever r = qζ, s = kζq ζ, and Re t s + 1 k Re ζq ζ q ζ + 1, z U, ζ U \ Eq and k n. We write Ψ 1 [Ω, q] as Ψ[Ω, q]. In articular when qz = M Mz+a M+az, with M > 0 and a < M, qu = U M := {w : w < M}, q0 = a, Eq = and q Qa. In this case, we set Ψ n [Ω, q] := Ψ n [Ω, M, a], and in the secial case when the set Ω = U M, the class is simly denoted by Ψ n [M, a]. Definition 1.2.[20, Definition 3,. 817]. Let Ω be a set in C, q H[a, n] with q z 0. The class of admissible functions Ψ n[ω, q] consists of those functions

4 220 R. M. Ali, R. Chandrashekar, S. K. Lee, A. Swaminathan and V. Ravichandran ψ : C 3 U C satisfying the admissibility condition ψr, s, t; ζ Ω whenever r = qz, s = zq z m, and t Re s zq m Re z q z + 1, z U, ζ U and m n 1. In articular, we write Ψ 1[Ω, q] as Ψ [Ω, q]. For the above two classes of admissible functions, Miller and Mocanu [19, 20] roved the following theorems. Theorem 1.1[19, Theorem 2.3b,. 28]. Let ψ Ψ n [Ω, q] with q0 = a. If the analytic function z = a + a n z n + a n+1 z n+1 + satisfies z qz. ψz, z z, z 2 z; z Ω, Theorem 1.2[20, Theorem 1,. 818]. Let ψ Ψ n[ω, q] with q0 = a. If Qa and ψz, z z, z 2 z; z is univalent in U, imlies qz z. Ω {ψz, z z, z 2 z; z : z U} In the resent investigation, the differential subordination results of Miller and Mocanu [19, Theorem 2.3b,. 28] are extended for functions associated with l,m the Liu-Srivastava linear oerator H [β 1 ]. A similar roblem was first studied by Aghalary et al. [1, 2], and related results may be found in the works of [4, 5, 6, 7, 8, 9, 10, 13, 15]. Additionally, the corresonding differential suerordination roblem is also investigated, and several sandwich-tye results are obtained. Analogous results for analytic functions in the class associated with the Dziok- Srivastava oerator can be found in [3]. 2. Subordination results involving the Liu-Srivastava linear oerator The following class of admissible functions will be required in the first result. Definition 2.1. Let Ω be a set in C and q Q 1 H. The class of admissible functions Φ H [Ω, q] consists of those functions φ : C 3 U C satisfying the admissibility condition φu, v, w; z Ω whenever u = qζ, v = kζq ζ + β 1 + 1qζ β 1 C \ {0, 1, 2,...}, β β1 w u ζq ζ Re 2β k Re v u q ζ + 1,

5 Differential Sandwich Theorem for the Liu-Rrivastava Oerator 221 z U, ζ U \ Eq and k 1. Choosing qz = 1 + M z, M > 0, Definition 2.1 easily gives the following definition. Definition 2.2. Let Ω be a set in C and M > 0. The class of admissible functions Φ H [Ω, M] consists of those functions φ : C 3 U C such that 2.1 φ 1 + Me iθ, 1 + k + β β Me iθ, 1 + L + β k + β 1 Me iθ ; z Ω β 1 β whenever z U, θ R, ReLe iθ k 1kM for all real θ, β 1 C \ {0, 1, 2,...} and k 1. In the secial case Ω = qu = {ω : ω 1 < M}, the class Φ H [Ω, M] is simly denoted by Φ H [M]. Theorem 2.1. Let φ Φ H [Ω, q]. If f Σ satisfies 2.2 { φ [β 1 + 2]fz, z Hl,m [β 1 + 1]fz, z Hl,m [β 1 ]fz; z z Hl,m } : z U Ω, z Hl,m [β 1 + 2]fz qz, z U. Proof. Define the analytic function in U by 2.3 z := z Hl,m [β 1 + 2]fz. In view of the relation 1.3, it follows from 2.3 that 2.4 z Hl,m [β 1 + 1]fz = 1 β [β 1 + 1z + z z]. Further comutations show that 2.5 z 1 Hl,m [β 1 ]fz = β 1 β [z2 z + 2β 1 + 1z z] + z. Define the transformations from C 3 to C by 2.6 u = r, v = s + β 1 + 1r β Let ψr, s, t; z = φu, v, w; z 2.7 = φ r, s + β 1 + 1r β 1 + 1, w = t + 2β 1 + 1s + β 1 β 1 + 1r. β 1 β 1 + 1, t + 2β 1 + 1s + β 1 β 1 + 1r ; z. β 1 β 1 + 1

6 222 R. M. Ali, R. Chandrashekar, S. K. Lee, A. Swaminathan and V. Ravichandran From 2.3, 2.4 and 2.5, Equation 2.7 yields 2.8 ψz, z z, z 2 z; z = φ [β 1 + 2]fz, z Hl,m [β 1 + 1]fz, z Hl,m [β 1 ]fz; z. z Hl,m Hence 2.2 becomes ψz, z z, z 2 z; z Ω. To comlete the roof, it is left to show that the admissibility condition for φ Φ H [Ω, q] is equivalent to the admissibility condition for ψ as given in Definition 1.1. Note that t s + 1 = β 1w u 2β 1 + 1, v u and hence ψ Ψ n [Ω, q]. By Theorem 1.1, z qz or z Hl,m [β 1 + 2]fz qz. If Ω C is a simly connected domain, Ω = hu for some conformal maing h of U onto Ω. In this case the class Φ H [hu, q] is written as Φ H [h, q]. The following result is an immediate consequence of Theorem 2.1. Theorem 2.2. Let φ Φ H [h, q] with q0 = 1. If f Σ and β 1 C \ {0, 1, 2,...} satisfies 2.9 φ [β 1 + 2]fz, z Hl,m [β 1 + 1]fz, z Hl,m [β 1 ]fz; z hz, z Hl,m z Hl,m [β 1 + 2]fz qz. The next result is an extension of Theorem 2.1 to the case where the behavior of q on U is not known. Corollary 2.1. Let Ω C, q be univalent in U and q0 = 1. Let φ Φ H [Ω, q ρ ] for some ρ 0, 1 where q ρ z = qρz. If f Σ and φ z Hl,m [β 1 + 2]fz, z Hl,m [β 1 + 1]fz, z Hl,m [β 1 ]fz; z Ω, z Hl,m [β 1 + 2]fz qz. Proof. Theorem 2.1 yields z Hl,m [β 1 + 2]fz q ρ z. The result now follows from the fact that q ρ z qz. Theorem 2.3. Let h and q be univalent in U, with q0 = 1, and set q ρ z = qρz and h ρ z = hρz. Let φ : C 3 U C satisfy one of the following conditions:

7 Differential Sandwich Theorem for the Liu-Rrivastava Oerator φ Φ H [h, q ρ ] for some ρ 0, 1, or 2. there exists ρ 0 0, 1 such that φ Φ H [h ρ, q ρ ] for all ρ ρ 0, 1. If f Σ satisfies 2.9, z Hl,m [β 1 + 2]fz qz. Proof. The result is similar to the roof of Theorem 2.3d [19,. 30] and is omitted. The next theorem yields the best dominant of the differential subordination 2.9. Theorem 2.4. Let h be univalent in U, and φ : C 3 U C. Suose that the differential equation 2.10 φ qz, zq z, z 2 q z; z = hz has a solution q with q0 = 1 and satisfy one of the following conditions: 1. q Q 1 and φ Φ H [h, q], 2. q is univalent in U and φ Φ H [h, q ρ ] for some ρ 0, 1, or 3. q is univalent in U and there exists ρ 0 0, 1 such that φ Φ H [h ρ, q ρ ] for all ρ ρ 0, 1. If f Σ satisfies 2.9, and q is the best dominant. z Hl,m [β 1 + 2]fz qz, Proof. Following the same arguments in [19, Theorem 2.3e,. 31], the function q is a dominant from Theorems 2.2 and 2.3. Since q satisfies 2.10, it is also a solution of 2.9 and therefore q will be dominated by all dominants. Hence q is the best dominant. Corollary 2.2. Let φ Φ H [Ω, M]. If f Σ satisfies φ [β 1 + 2]fz, z Hl,m [β 1 + 1]fz, z Hl,m [β 1 ]fz; z Ω, z Hl,m z Hl,m [β 1 + 2]fz 1 < M. Corollary 2.3. Let φ Φ H [M]. If f Σ satisfies φ [β 1 + 2]fz, z Hl,m [β 1 + 1]fz, z Hl,m [β 1 ]fz; z 1 < M, z Hl,m z Hl,m [β 1 + 2]fz 1 < M.

8 224 R. M. Ali, R. Chandrashekar, S. K. Lee, A. Swaminathan and V. Ravichandran The following examle is easily obtained by taking φu, v, w; z = v in Corollary 2.3. Examle 2.1. If Re β 1 k and f Σ satisfies z Hl,m [β 1 + 1]fz 1 < M, for l = 2, 3,.... z Hl,m [β 1 + l]fz 1 < M Corollary 2.4. Let M > 0. If f Σ and z Hl,m [β 1 + 1]fz z Hl,m [β 1 + 2]fz < M β 1 + 1, z Hl,m [β 1 + 2]fz 1 < M. Proof. Let φu, v, w; z = v u and Ω = hu where hz = M β 1+1 z, M > 0. It suffices to show that φ Φ H [Ω, M], that is, the admissible condition 2.1 is satisfied. This follows since φ 1 + Me iθ, 1 + k + β Me iθ, 1 + L + β k + β 1 Me iθ ; z β β 1 β = km β M β 1 + 1, z U, θ R, β 1 C \ {0, 1, 2,...} and k 1. From Corollary 2.2, the required result is obtained. Definition 2.3. Let Ω be a set in C and q Q 1 H. The class of admissible functions Φ H,1 [Ω, q] consists of those functions φ : C 3 U C satisfying the admissibility condition φu, v, w; z Ω whenever u = qζ, v = β 1 + 1qζ β kζq ζ qζ, β 1 C \ {0, 1, 2,...}, qζ 0, [ β + 1u β + 1 Re vβ + 2 β + 1u vu v z U, ζ U \ Eq and k 1. β ] w 1 β + 1 ζq ζ 1 k Re v q ζ + 1,

9 Differential Sandwich Theorem for the Liu-Rrivastava Oerator 225 In the articular case qz = 1 + Mz, M > 0, Definition 2.3 yields the following definition. Definition 2.4. Let Ω be a set in C and M > 0. The class of admissible functions Φ H,1 [Ω, M] consists of those functions φ : C 3 U C satisfying φ 1 + Me iθ β Me iθ, β Me iθ k + 1, β Me iθ [β Me iθ k + 1] [β Me iθ k + 1][β 1 2Me iθ k + 1] 1 + Me iθ L + 2kMe iθ ; z Ω whenever z U, θ R, ReLe iθ k 1kM for all real θ, β 1 C \ {0, 1, 2,...} and k 1. In the secial case Ω = qu = {ω : ω 1 < M}, the class Φ H,1 [Ω, M] is simly denoted by Φ H,1 [M]. Theorem 2.5. Let φ Φ H,1 [Ω, q]. If f Σ satisfies { } Hl,m [β 1 + 3]fz 2.11 φ H l,m [β 1 + 2]fz, [β 1 +2]fz H l,m [β 1 + 1]fz, [β 1 + 1]fz H l,m ; z : z U Ω, [β 1 ]fz H l,m [β 1 + 3]fz H l,m [β 1 + 2]fz qz. Proof. Define the analytic function in U by 2.12 z := H l,m [β 1 + 3]fz H l,m [β 1 + 2]fz. This imlies z z z which from 1.3 yields l,m z[ H [β 1 + 3]fz] l,m z[ H [β 1 + 2]fz] := H l,m [β 1 + 3]fz H l,m, [β 1 + 2]fz 2.13 H l,m [β 1 + 2]fz H l,m [β 1 + 1]fz = β 1 + 1z β z z z. Further comutations show that 2.14 H l,m [β 1 + 1]fz H l,m = [β 1 ]fz β 1 2 z z z z β 1 z z z [z2 z+2z z] β 1 2 z z z 1.

10 226 R. M. Ali, R. Chandrashekar, S. K. Lee, A. Swaminathan and V. Ravichandran Define the transformations from C 3 to C by 2.15 u = r, v = β 1 + 1r β s r, w = β 1 β 1+2 s r r s r t+2s β s r Let 2.16 ψr, s, t; z : = φu, v, w; z β 1 + 1r = φ r, β s r, β 1 β 1+2 s r r s r t+2s β 1; z. 1+2 s r The roof shall make use of Theorem 1.1. Using equations 2.12, 2.13 and 2.14 in 2.16 yield 2.17 ψz, z z, z 2 z; z Hl,m [β 1 + 3]fz = φ H l,m [β 1 + 2]fz, [β 1 + 2]fz H l,m [β 1 + 1]fz, [β 1 + 1]fz H l,m ; z. [β 1 ]fz Hence 2.11 becomes ψz, z z, z 2 z; z Ω. To comlete the roof, the admissibility condition for φ Φ H,2 [Ω, q] is shown to be equivalent to the admissibility condition for ψ as given in Definition 1.1. Note that [ t s + 1 = Re β + 1u β + 1 β ] vβ + 2 β + 1u vu v w 1 β + 1 1, v and hence ψ Ψ[Ω, q]. By Theorem 1.1, z qz or H l,m [β 1 + 3]fz H l,m [β 1 + 2]fz qz. As in the revious cases, if Ω C is a simly connected domain, Ω = hu for some conformal maing h of U onto Ω. In this case the class Φ H,1 [hu, q] is written as Φ H,1 [h, q]. The following result is an immediate consequence of Theorem 2.5. Theorem 2.6. Let φ Φ H,1 [h, q]. If f Σ satisfies Hl,m [β 1 + 3]fz φ H l,m [β 1 + 2]fz, [β 1 + 2]fz H l,m [β 1 + 1]fz, [β 1 + 1]fz H l,m ; z hz, [β 1 ]fz H l,m [β 1 + 3]fz H l,m [β 1 + 2]fz qz.

11 Differential Sandwich Theorem for the Liu-Rrivastava Oerator 227 Corollary 2.5. Let φ Φ H,1 [Ω, M]. If f Σ satisfies φ Hl,m [β 1 + 3]fz H l,m [β 1 + 2]fz, H l,m [β 1 + 2]fz H l,m [β 1 + 1]fz, [β 1 + 1]fz H l,m ; z Ω, [β 1 ]fz H l,m [β 1 + 3]fz H l,m [β 1 + 2]fz 1 < M. Corollary 2.6. Let φ Φ H,1 [M]. If f Σ satisfies Hl,m φ [β 1 + 3]fz H l,m [β 1 + 2]fz, [β 1 + 2]fz H l,m [β 1 + 1]fz, [β 1 + 1]fz H l,m ; z 1 [β 1 ]fz < M, H l,m [β 1 + 3]fz H l,m [β 1 + 2]fz 1 < M. 3. Suerordination of the Liu-Srivastava linear oerator The dual roblem of differential subordination, that is, differential suerordination of the Liu-Srivastava linear oerator is investigated in this section. For this urose, the following class of admissible functions will be required. Definition 3.1. Let Ω be a set in C and q H with zq z 0. The class of admissible functions Φ H [Ω, q] consists of those functions φ : C3 U C satisfying the admissibility condition φu, v, w; ζ Ω whenever u = qz, z U, ζ U and m 1. v = zq z + mβ 1 + 1qz mβ β 1 C \ {0, 1, 2,...}, β1 w u Re 2β zq v u m Re z q z + 1, Theorem 3.1. Let φ Φ H [Ω, q]. If f Σ, z Hl,m [β 1 + 2]fz Q 1 and φ z Hl,m [β 1 + 2]fz, z Hl,m [β 1 + 1]fz, z Hl,m [β 1 ]fz; z

12 228 R. M. Ali, R. Chandrashekar, S. K. Lee, A. Swaminathan and V. Ravichandran is univalent in U, { 3.1 Ω φ [β 1 +2]fz, z Hl,m [β 1 +1]fz, z Hl,m [β 1 ]fz;z z Hl,m } :z U imlies qz z Hl,m [β 1 + 2]fz. Proof. From 2.8 and 3.1, it follows that Ω { ψ z, z z, z 2 z; z : z U }. From 2.6, it is clear that the admissibility condition for φ Φ H [Ω, q] is equivalent to the admissibility condition for ψ as given in Definition 1.2. Hence ψ Ψ [Ω, q], and by Theorem 1.2, qz z or qz z Hl,m [β 1 + 2]fz. If Ω C is a simly connected domain, Ω = hu for some conformal maing h of U onto Ω. In this case the class Φ H [hu, q] is written as Φ H [h, q]. Proceeding similarly as in the revious section, the following result is an immediate consequence of Theorem 3.1. Theorem 3.2. Let q H, h be analytic in U and φ Φ H [h, q]. If f Σ, z Hl,m [β 1 + 2]fz Q 1 and φ z Hl,m [β 1 + 2]fz, z Hl,m [β 1 + 1]fz, z Hl,m [β 1 ]fz; z is univalent in U, 3.2 hz φ [β 1 + 2]fz, z Hl,m [β 1 + 1]fz, z Hl,m [β 1 ]fz; z z Hl,m imlies qz z Hl,m [β 1 + 2]fz. Theorem 3.1 and 3.2 can only be used to obtain subordinants of differential suerordination of the form 3.1 or 3.2. The following theorem roves the existence of the best subordinant of 3.2 for certain φ. Theorem 3.3. Let h be analytic in U and φ : C 3 U C. Suose that the differential equation φqz, zq z, z 2 q z; z = hz has a solution q Q 1. If φ Φ H [h, q], f Σ, z Hl,m [β 1 + 2]fz Q 1 and φ z Hl,m [β 1 + 2]fz, z Hl,m [β 1 + 1]fz, z Hl,m [β 1 ]fz; z

13 Differential Sandwich Theorem for the Liu-Rrivastava Oerator 229 is univalent in U, hz φ z Hl,m [β 1 + 2]fz, z Hl,m [β 1 + 1]fz, z Hl,m [β 1 ]fz; z imlies qz z Hl,m [β 1 + 2]fz, and q is the best subordinant. Proof. omitted. The roof is similar to the roof of Theorem 2.4 and is therefore Combining Theorems 2.2 and 3.2, we obtain the following sandwich-tye theorem. Corollary 3.1. Let h 1 and q 1 be analytic functions in U, h 2 be univalent function in U, q 2 Q 1 with q 1 0 = q 2 0 = 1 and φ Φ H [h 2, q 2 ] Φ H [h 1, q 1 ]. If f Σ, z Hl,m [β 1 + 2]fz H Q 1 and φ [β 1 + 2]fz, z Hl,m [β 1 + 1]fz, z Hl,m [β 1 ]fz; z z Hl,m is univalent in U, h 1 z φ [β 1 + 2]fz, z Hl,m [β 1 + 1]fz, z Hl,m [β 1 ]fz; z h 2 z z Hl,m imlies q 1 z z Hl,m [β 1 + 2]fz q 2 z. Definition 3.2. Let Ω be a set in C and q H with zq z 0. The class of admissible functions Φ H,1 [Ω, q] consists of those functions φ : C3 U C satisfying the admissibility condition φu, v, w; ζ Ω whenever mβ 1 + 1qz u = qz, v = mβ zq z mqz, β 1 C \ {0, 1, 2,...}, [ β +1u β+1 Re vβ +2 β +1u vu v β ] w 1 β+1 v 1 1 zq m Re z q z +1, z U, ζ U and m 1. Next the dual result of Theorem 2.5 for differential suerordination is given. Theorem 3.4. Let φ Φ H,1 [Ω, q]. If f Σ, φ Hl,m [β 1 + 3]fz H l,m [β 1 + 2]fz, H l,m [β 1 + 2]fz H l,m [β 1 + 1]fz, [β 1+3]fz H l,m Q [β 1 and 1+2]fz H l,m [β 1 + 1]fz H l,m ; z [β 1 ]fz

14 230 R. M. Ali, R. Chandrashekar, S. K. Lee, A. Swaminathan and V. Ravichandran is univalent in U, { Hl,m [β 1 + 3]fz 3.3 Ω φ H l,m [β 1 + 2]fz, [β 1 + 2]fz H l,m [β 1 + 1]fz, imlies qz Proof. From 2.17 and 3.3, it follows that H l,m [β 1 + 3]fz H l,m [β 1 + 2]fz. } [β 1 + 1]fz H l,m ; z :z U [β 1 ]fz Ω { φ z, z z, z 2 z; z : z U }. From 2.15, the admissibility condition for φ Φ H,2 [Ω, q] is equivalent to the admissibility condition for ψ as given in Definition 1.2. Hence ψ Ψ [Ω, q], and by Theorem 1.2, qz z or qz H l,m [β 1 + 3]fz H l,m [β 1 + 2]fz. If Ω C is a simly connected domain, Ω = hu for some conformal maing h of U onto Ω. In this case the class Φ H,1 [hu, q] is written as Φ H,1 [h, q]. Proceeding similarly as in the revious section, the following result is an immediate consequence of Theorem 3.4. Theorem 3.5. Let q H, h be analytic in U and φ Φ H,1 [h, q]. If f Σ, [β 1+3]fz H Q l,m [β 1 and φ Hl,m [β 1+3]fz 1+2]fz H, [β 1+2]fz l,m [β 1+2]fz H, [β 1+1]fz ; z is univalent in U, l,m [β 1+1]fz H l,m [β 1]fz Hl,m [β 1 + 3]fz hz φ H l,m [β 1 + 2]fz, [β 1 + 2]fz H l,m [β 1 + 1]fz, [β 1 + 1]fz H l,m ; z [β 1 ]fz imlies qz H l,m [β 1 + 3]fz H l,m [β 1 + 2]fz. Combining Theorems 2.6 and 3.5, the following sandwich-tye theorem is obtained. Corollary 3.2. Let h 1 and q 1 be analytic functions in U, h 2 be univalent function in U, q 2 Q 1 with q 1 0 = q 2 0 = 1 and φ Φ H,1 [h 2, q 2 ] Φ H,1 [h 1, q 1 ]. If f Σ, [β 1+3]fz [β 1+2]fz H Q 1 and φ Hl,m [β 1 + 3]fz H l,m [β 1 + 2]fz, H l,m [β 1 + 2]fz H l,m [β 1 + 1]fz, [β 1 + 1]fz H l,m ; z [β 1 ]fz

15 Differential Sandwich Theorem for the Liu-Rrivastava Oerator 231 is univalent in U, h 1 z φ Hl,m [β 1 + 3]fz H l,m [β 1 + 2]fz, H l,m [β 1 + 2]fz H l,m [β 1 + 1]fz, [β 1 + 1]fz H l,m ; z h 2 z [β 1 ]fz imlies q 1 z H l,m [β 1 + 3]fz H l,m [β 1 + 2]fz q 2z. Acknowledgements This work was suorted in art by the FRGS grant from Universiti Sains Malaysia. The second author gratefully acknowledges suort from a USM Fellowshi. This work was comleted during the visit of the fourth and fifth authors to Universiti Sains Malaysia. References [1] R. Aghalary, R. M. Ali, S. B. Joshi and V. Ravichandran, Inequalities for analytic functions defined by certain linear oerator, Internat. J. Math. Sci., 42005, [2] R. Aghalary, S. B. Joshi, R. N. Mohaatra and V. Ravichandran, Subordinations for analytic functions defined by the Dziok-Srivastava linear oerator, Al. Math. Comut., , [3] R. M. Ali, R. Chandrashekar, S. K. Lee, A. Swaminathan and V. Ravichandran, Differential sandwich theorem for multivalent analytic functions associated with the Dziok-Srivastava oerator, Tamsui Oxf. J. Math. Sc., to aear. [4] R. M. Ali and V. Ravichandran, Differential subordination for meromorhic functions defined by a linear oerator, J. Anal. Al , [5] R. M. Ali, V. Ravichandran and N. Seenivasagan, Differential subordination and suerordination of analytic functions defined by the Dziok-Srivastava linear oerator, Journal of Franklin Institute, to aear. [6] R. M. Ali, V. Ravichandran and N. Seenivasagan, Differential subordination and suerordination of analytic functions defined by the multilier transformation, Math. Inequal. Al., , [7] R. M. Ali, V. Ravichandran and N. Seenivasagan, Differential subordination and suerordination for meromorhic functions defined by certain multilier transformation, Bull. Malaysian Math. Sci. Soc., , [8] R. M. Ali, V. Ravichandran and N. Seenivasagan, Subordination and suerordination on Schwarzian derivatives, J. Inequal. Al., 2008, Article ID , [9] R. M. Ali, V. Ravichandran and N. Seenivasagan, Subordination and suerordination of the Liu-Srivastava linear oerator on meromorhic functions, Bull. Malaysian Math. Sci. Soc., ,

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