DIFFERENTIAL SUBORDINATION AND SUPERORDINATION OF ANALYTIC FUNCTIONS DEFINED BY THE MULTIPLIER TRANSFORMATION

M athematical I nequalities & A pplications Volume 12, Number 1 2009), 123 139 DIFFERENTIAL SUBORDINATION AND SUPERORDINATION OF ANALYTIC FUNCTIONS DEFINED BY THE MULTIPLIER TRANSFORMATION ROSIHAN M. ALI, V.RAVICHANDRAN AND N. SEENIVASAGAN communicated by R. Mohapatra) Abstract. Differential subordination and superordination results are obtained for analytic functions in the open unit disk which are associated with the multiplier transformation. These results are obtained by investigating appropriate classes of admissible functions. Sandwich-type results are also obtained. 1. Introduction Let H U) be the class of functions analytic in U := {z C : z < 1} and H [a, n] be the subclass of H U) consisting of functions of the form f z) = a + a n z n + a n+1 z n+1 +, with H 0 H [0, 1] and H H [1, 1].LetA p denote the class of all analytic functions of the form f z) =z p + k=p+1 a k z k z U) 1.1) and let A 1 := A. Let f and F be members of H U). The function f z) is said to be subordinate to Fz),orFz) is said to be superordinate to f z), if there exists a function wz) analytic in U with w0) =0 and wz) < 1 z U), such that f z) = Fwz)). Insuchacasewewrite f z) Fz). If F is univalent, f z) Fz) if and only if f 0) =F0) and f U) FU). For two functions f z) given by 1.1) and gz) =z p + k=p+1 b kz k, the Hadamard product or convolution) of f and g is defined by f g)z) := z p + k=p+1 a k b k z k =: g f )z). 1.2) Mathematics subject classification 2000): 30C80, 30C45. Keywords and phrases: Subordination, superordination, multiplier transformation, convolution. The authors acknowledge support from the FRGS and Science Fund research grants. This work was completed while the second and third authors were at USM. c D l,zagreb Paper MIA-12-11 123

124 ROSIHAN M. ALI, V.RAVICHANDRAN AND N. SEENIVASAGAN Motivated by the multiplier transformation on A, we define the operator I p n, λ ) on A p by the following infinite series I p n, λ )f z) := z p + k=p+1 ) n k + λ a k z k λ > p). 1.3) p + λ The operator I p n, λ ) is closely related to the Sǎlǎgean derivative operators [11]. The operator I n λ := I 1n, λ ) was studied recently by Cho and Srivastava [6] and Cho and Kim [7]. The operator I n := I 1 n, 1) was studied by Uralegaddi and Somanatha [13]. To prove our results, we need the following definitions and theorems. Denote by Q the set of all functions qz) that are analytic and injective on U\Eq) where Eq) ={ζ U : lim qz) = }, z ζ and are such that q ζ) 0forζ U \ Eq). Further let the subclass of Q for which q0) =a be denoted by Qa), Q0) Q 0 and Q1) Q 1. DEFINITION 1.1. [9, Definition 2.3a, p. 27] Let Ω beasetin C, q Q and n be a positive integer. The class of admissible functions Ψ n [Ω, q] consists of those functions ψ : C 3 U C that satisfy the admissibility condition ψr, s, t; z) Ω whenever r = qζ), s = kζq ζ),and { t } { ζq } R s + 1 ζ) k R q ζ) + 1, z U, ζ U \ Eq) and k n. We write Ψ 1 [Ω, q] as Ψ[Ω, q]. In particular when qz) =M Mz+a M+az, with M > 0 and a < M,qU) = U M := {w : w < M}, q0) =a, Eq) = and q Q. In this case, we set Ψ n [Ω, M, a] := Ψ n [Ω, q], and in the special case when the set Ω = U M, the class is simply denoted by Ψ n [M, a]. DEFINITION 1.2. [10, Definition 3, p. 817] Let Ω be a set in C, qz) H [a, n] with q z) 0. The class of admissible functions Ψ n [Ω, q] consists of those functions ψ : C 3 U C that satisfy the admissibility condition ψr, s, t; ζ) Ω whenever r = qz), s = zq z) m,and { t } R s + 1 1 { zq } m R z) q z) + 1, z U, ζ U and m n 1. In particular, we write Ψ 1 [Ω, q] as Ψ [Ω, q]. THEOREM 1.1. [9, Theorem 2.3b, p. 28] Let ψ Ψ n [Ω, q] with q0) =a. If the analytic function pz) =a + a n z n + a n+1 z n+1 + satisfies pz) qz). ψpz), zp z), z 2 p z); z) Ω,

SUBORDINATION AND SUPERORDINATION FOR ANALYTIC FUNCTIONS 125 THEOREM 1.2. [10, Theorem 1, p. 818] Let ψ Ψ n[ω, q] with q0) =a. If pz) Qa) and ψpz), zp z), z 2 p z); z) is univalent in U, Ω {ψpz), zp z), z 2 p z); z) : z U} qz) pz). In the present investigation, the differential subordination result of Miller and Mocanu [9, Theorem 2.3b, p. 28] is extended for functions associated with the multiplier transformation I p n, λ ), and we obtain certain other related results. A similar problem for analytic functions defined by Dizok-Srivastava linear operator was considered by Ali et al. [4] see also [1], [2], [3], [5]). Additionally, the corresponding differential superordination problem is investigated, and several sandwich-type results are obtained. 2. Subordination Results involving the Multiplier Transformation DEFINITION 2.1. Let Ω beasetin C and qz) Q 0 H [0, p]. The class of admissible functions Φ I [Ω, q] consists of those functions : C 3 U C that satisfy the admissibility condition u, v, w; z) Ω whenever u = qζ), v = kζq ζ)+λqζ), { ) 2 w λ 2 } { u ζq } R )v λ u 2λ ζ) k R q ζ) + 1, z U, ζ U \ Eq) and k p. THEOREM 2.1. Let Φ I [Ω, q].iffz) A p satisfies { I p n, λ )f z), I p n + 1, λ )f z), I p n + 2, λ )f z); z) : z U} Ω, 2.1) I p n, λ )f z) qz). Proof. Define the analytic function pz) in U by In view of the relation pz) := I p n, λ )f z). 2.2) p + λ )I p n + 1, λ )f z) =z[i p n, λ )f z)] + λ I p n, λ )f z), 2.3) from 2.2),weget I p n + 1, λ )f z) = zp z)+λpz). 2.4) Further computations show that I p n + 2, λ )f z) = z2 p z)+2λ + 1)zp z)+λ 2 pz) ) 2. 2.5)

126 ROSIHAN M. ALI, V.RAVICHANDRAN AND N. SEENIVASAGAN Define the transformations from C 3 to C by u = r, v = s + λ r, w = t +2λ + 1)s + λ 2 r ) 2. 2.6) Let ψr, s, t; z) =u, v, w; z) = r, s + λ r, t +2λ + 1)s + λ 2 ) r ) 2 ; z. 2.7) The proof shall make use of Theorem 1.1. Using equations 2.2), 2.4) and 2.5), from 2.7), we obtain ψpz), zp z), z 2 p z); z) = I p n, λ )f z), I p n + 1, λ )f z), I p n + 2, λ )f z); z). 2.8) Hence 2.1) becomes ψpz), zp z), z 2 p z); z) Ω. The proof is completed if it can be shown that the admissibility condition for Φ I [Ω, q] is equivalent to the admissibility condition for ψ as given in Definition 1.1. Note that t s + 1 = )2 w λ 2 u )v λ u 2λ, and hence ψ Ψ p [Ω, q]. By Theorem 1.1, pz) qz) or I p n, λ )f z) qz). If Ω C is a simply connected domain, Ω = hu) for some conformal mapping hz) of U onto Ω. In this case the class Φ I [hu), q] is written as Φ I [h, q]. The following result is an immediate consequence of Theorem 2.1. THEOREM 2.2. Let Φ I [h, q].iffz) A p satisfies I p n, λ )f z), I p n + 1, λ )f z), I p n + 2, λ )f z); z) hz), 2.9) I p n, λ )f z) qz). Our next result is an extension of Theorem 2.2 to the case where the behavior of qz) on U is not known. COROLLARY 2.1. Let Ω C and let qz) be univalent in U, q0) =0. Let Φ I [Ω, q ρ ] for some ρ 0, 1) where q ρ z) =qρz).iffz) A p and I p n, λ )f z), I p n + 1, λ )f z), I p n + 2, λ )f z); z) Ω, I p n, λ )f z) qz).

SUBORDINATION AND SUPERORDINATION FOR ANALYTIC FUNCTIONS 127 Proof. Theorem 2.1 yields I p n, λ )f z) q ρ z). The result is now deduced from q ρ z) qz). THEOREM 2.3. Let hz) and qz) be univalent in U, with q0) =0 and set q ρ z) =qρz) and h ρ z) =hρz).let: C 3 U C satisfy one of the following conditions: 1) Φ I [h, q ρ ],forsome ρ 0, 1),or 2) there exists ρ 0 0, 1) such that Φ I [h ρ, q ρ ], for all ρ ρ 0, 1). If f z) A p satisfies 2.9), I p n, λ )f z) qz). Proof. The proof is similar to the proof of [9, Theorem 2.3d, p. 30] and is therefore omitted. The next theorem yields the best dominant of the differential subordination 2.9). THEOREM 2.4. Let hz) be univalent in U. Let : C 3 U C. Suppose that the differential equation qz), zq z)+λqz), z2 q z)+2λ + 1)zq z)+λ 2 ) qz) ) 2 ; z = hz) 2.10) has a solution qz) with q0) =0 and satisfy one of the following conditions: 1) qz) Q 0 and Φ I [h, q], 2) qz) is univalent in U and Φ I [h, q ρ ],forsome ρ 0, 1),or 3) qz) is univalent in U and there exists ρ 0 0, 1) such that Φ I [h ρ, q ρ ], for all ρ ρ 0, 1). If f z) A p satisfies 2.9), and qz) is the best dominant. I p n, λ )f z) qz), Proof. Following the same arguments in [9, Theorem 2.3e, p. 31], we deduce that qz) is a dominant from Theorems 2.2 and 2.3. Since qz) satisfies 2.10) it is also a solution of 2.9) and therefore qz) will be dominated by all dominants. Hence qz) is the best dominant. In the particular case qz) =Mz, M > 0, and in view of the Definition 2.1, the class of admissible functions Φ I [Ω, q], denoted by Φ I [Ω, M], is described below. DEFINITION 2.2. Let Ω be a set in C and M > 0. The class of admissible functions Φ I [Ω, M] consists of those functions : C 3 U C such that Me iθ, k + λ Meiθ, L +2λ + 1)k + λ 2 )Me iθ ) ) 2 ; z Ω 2.11) whenever z U, θ R, RLe iθ ) k 1)kM for all real θ and k p.

128 ROSIHAN M. ALI, V.RAVICHANDRAN AND N. SEENIVASAGAN COROLLARY 2.2. Let Φ I [Ω, M].Iffz) A p satisfies I p n, λ )f z), I p n + 1, λ )f z), I p n + 2, λ )f z); z) Ω, I p n, λ )f z) < M. In the special case Ω = qu) ={ω : ω < M}, the class Φ I [Ω, M] is simply denoted by Φ I [M]. COROLLARY 2.3. Let Φ I [M].Iffz) A p satisfies I p n, λ )f z), I p n + 1, λ )f z), I p n + 2, λ )f z); z) < M, I p n, λ )f z) < M. REMARK 2.1. When Ω = U and M = 1, Corollary 2.2 reduces to [1, Theorem 2, p. 271]. WhenΩ = U, λ = a 1 a > 0), p = 1andM = 1, Corollary 2.2 reduces to [8, Theorem 2, p. 231]. WhenΩ = U, λ = 1, p = 1andM = 1, Corollary 2.2 reduces to [5, Theorem 1, p. 477]. COROLLARY 2.4. If M > 0 and f z) A p satisfies ) 2 I p n + 2, λ )f z) )I p n + 1, λ )f z) λ 2 I p n, λ )f z) < [2p 1)p 1)] M, I p n, λ )f z) < M. 2.12) Proof. This follows from Corollary 2.2 by taking u, v, w; z) =) 2 w )v λ 2 u and Ω = hu) where hz) =[2p 1)p 1)]Mz, M > 0. To use Corollary 2.2, we need to show that Φ I [Ω, M], that is, the admissible condition 2.11) is satisfied. This follows since Me iθ, k + λ Meiθ, L +2λ + 1)k + λ 2 )Me iθ ) 2 ; z) = L +2λ + 1)k + λ 2 )Me iθ k + λ )Me iθ λ 2 Me iθ = L +2k 1)λ Me iθ 2k 1)λ M + RLe iθ ) 2k 1)λ M + kk 1)M [2p 1)p 1)] M z U, θ R, RLe iθ ) kk 1)M and k p. Hence by Corollary 2.2, we deduce the required result. DEFINITION 2.3. Let Ω beasetin C and qz) Q 0 H 0. The class of admissible functions Φ I,1 [Ω, q] consists of those functions : C 3 U C that satisfy the admissibility condition u, v, w; z) Ω

SUBORDINATION AND SUPERORDINATION FOR ANALYTIC FUNCTIONS 129 whenever u = qζ), v = kζq ζ)+ 1)qζ), { ) 2 w 1) 2 } { u ζq } ζ) R 2 1) k R )v 1)u q ζ) + 1, z U, ζ U \ Eq) and k 1. THEOREM 2.5. Let Φ I,1 [Ω, q].iffz) A p satisfies { Ip n, λ )f z), I pn + 1, λ )f z), I ) } pn + 2, λ )f z) ; z : z U Ω, 2.13) I p n, λ )f z) qz). Proof. Define an analytic function pz) in U by pz) := I pn, λ )f z). 2.14) By making use of 2.3),weget, I p n + 1, λ )f z) Further computations show that I p n + 2, λ )f z) = zp z)+ 1)pz). 2.15) = z2 p z)+[2) 1]zp z)+ 1) 2 pz) ) 2. 2.16) Define the transformations from C 3 to C by s + 1)r u = r, v =, w = t +[2) 1]s + 1)2 r ) 2. 2.17) Let ψr, s, t; z) = u, v, w; z) 2.18) s + 1)r = r,, t +[2) 1]s + ) 1)2 r ) 2 ; z. The proof shall make use of Theorem 1.1. Using equations 2.14), 2.15) and 2.16), from 2.18), we obtain ψpz), zp z), z 2 p Ip n, λ )f z) z); z) =, I pn + 1, λ )f z), I ) pn + 2, λ )f z) ; z. 2.19) Hence 2.13) becomes ψpz), zp z), z 2 p z); z) Ω.

130 ROSIHAN M. ALI, V.RAVICHANDRAN AND N. SEENIVASAGAN The proof is completed if it can be shown that the admissibility condition for Φ I,1 [Ω, q] is equivalent to the admissibility condition for ψ as given in Definition 1.1. Note that t s + 1 = )2 w 1) 2 u 2 1), )v 1)u and hence ψ Ψ[Ω, q]. By Theorem 1.1, pz) qz) or I p n, λ )f z) qz). If Ω C is a simply connected domain, Ω = hu), for some conformal mapping hz) of U onto Ω. In this case the class Φ I,1 [hu), q] is written as Φ I,1 [h, q]. In the particular case qz) =Mz, M > 0, the class of admissible functions Φ I,1 [Ω, q], denoted by Φ I,1 [Ω, M]. The following result is an immediate consequence of Theorem 2.5. THEOREM 2.6. Let Φ I,1 [h, q].iffz) A p satisfies Ip n, λ )f z), I pn + 1, λ )f z), I ) pn + 2, λ )f z) ; z hz), 2.20) I p n, λ )f z) qz). DEFINITION 2.4. Let Ω be a set in C and M > 0. The class of admissible functions Φ I,1 [Ω, M] consists of those functions : C 3 U C such that Me iθ, k + 1 Me iθ, L +[2) 1)k + 1)2 ]Me iθ ) ) 2 ; z Ω 2.21) whenever z U, θ R, RLe iθ ) k 1)kM for all real θ and k 1. COROLLARY 2.5. Let Φ I,1 [Ω, M].Iffz) A p satisfies Ip n, λ )f z), I pn + 1, λ )f z), I ) pn + 2, λ )f z) ; z Ω, I p n, λ )f z) < M. In the special case Ω = qu) ={ω : ω < M}, the class Φ I,1 [Ω, M] is simply denoted by Φ I,1 [M]. COROLLARY 2.6. Let Φ I,1 [M].Iffz) A p satisfies Ip n, λ )f z), I pn + 1, λ )f z), I ) pn + 2, λ )f z) ; z < M, I p n, λ )f z) < M.

SUBORDINATION AND SUPERORDINATION FOR ANALYTIC FUNCTIONS 131 REMARK 2.2. When Ω = U, λ = a 1 a > 0), p = 1andM = 1, Corollary 2.5 reduces to [8, Theorem 2, p. 231]. When Ω = U, λ = 1, p = 1andM = 1, Corollary 2.5 reduces to [5, Theorem1, p. 477]. COROLLARY 2.7. If f z) A p,, I p n + 1, λ )f z) < M I p n, λ )f z) < M. This follows from Corollary 2.6 by taking u, v, w; z) =v. COROLLARY 2.8. If M > 0 and f z) A p satisfies λ + I pn + 2, λ )f z) p)2 +) I pn + 1, λ )f z) 1) 2 I pn, λ )f z) < [3) 1]M, I p n, λ )f z) < M. 2.22) Proof. This follows from Corollary 2.5 by taking u, v, w; z) =λ +p) 2 w+λ + p)v 1) 2 u and Ω = hu) where hz) =3) 1)Mz, M > 0. Touse Corollary 2.5, we need to show that Φ I,1 [Ω, M], that is, the admissible condition 2.21) is satisfied. This follows since Me iθ, k + 1 Me iθ, L +[2) 1)k + 1)2 ]Me iθ ) 2 ; z) = L+[2) 1)k+λ +p 1) 2 ]Me iθ +k+λ +p 1)Me iθ λ +p 1) 2 Me iθ = L +[2k + 1)) 1]Me iθ [2k + 1)) 1]M + RLe iθ ) [2k + 1)) 1]M + kk 1)M 3) 1)M z U, θ R, RLe iθ ) kk 1)M and k 1. Hence by Corollary 2.5, we deduce the required result. DEFINITION 2.5. Let Ω beasetin C and qz) Q 1 H. The class of admissible functions Φ I,2 [Ω, q] consists of those functions : C 3 U C that satisfy the admissibility condition whenever u, v, w; z) Ω )qζ)+ kζq ζ) qζ) u = qζ), v = 1 { )vw v) R )2u v) v u z U, ζ U \ Eq) and k 1. ) } k R qζ) 0), { ζq } ζ) q ζ) + 1,

132 ROSIHAN M. ALI, V.RAVICHANDRAN AND N. SEENIVASAGAN THEOREM 2.7. Let Φ I,2 [Ω, q] and I p n, λ )f z) 0.Iffz) A p satisfies { Ip n + 1, λ )f z), I pn + 2, λ )f z) I p n, λ )f z) I p n + 1, λ )f z), I ) } pn + 3, λ )f z) I p n + 2, λ )f z) ; z : z U Ω, 2.23) I p n + 1, λ )f z) qz). I p n, λ )f z) Proof. Define an analytic function pz) in U by By making use of 2.3) and 2.24),weget pz) := I pn + 1, λ )f z). 2.24) I p n, λ )f z) I p n + 2, λ )f z) I p n + 1, λ )f z) = 1 Further computations show that I p n+3, λ )f z) I p n+2, λ )f z) = pz)+ 1 λ +p zp z) Define the transformations from C 3 to C by u = r, v = r+ 1 s ), w = r+ 1 r [ ] )pz)+ zp z). 2.25) pz) )zp zp z)+ z) pz) + pz) zp z) pz) )pz)+ zp z) pz) [ s r + )s + s r s r )2 + t r )r + s r ) 2 + z 2 p z) pz). 2.26) ]. 2.27) Let ψr, s, t; z) =u, v, w; z) 2.28) 1 [ = r, λ +p)r+ s ] [ 1, λ +p)r+ s λ +p r λ +p r +λ +p)s+ s r s r )2 + t ] ) r )r+ s ; z. r The proof shall make use of Theorem 1.1. Using equations 2.24), 2.25) and 2.26), from 2.28), we obtain ψpz), zp z), z 2 p Ip n+1, λ )f z) z); z) = I p n, λ )f z), I pn+2, λ )f z) I p n+1, λ )f z), I ) pn+3, λ )f z) I p n+2, λ )f z) ; z. 2.29) Hence 2.23) becomes ψpz), zp z), z 2 p z); z) Ω. The proof is completed if it can be shown that the admissibility condition for Φ I,2 [Ω, q] is equivalent to the admissibility condition for ψ as given in Definition 1.1. Note that t )vw v) + 1 = )2u v), s v u

SUBORDINATION AND SUPERORDINATION FOR ANALYTIC FUNCTIONS 133 and hence ψ Ψ[Ω, q]. By Theorem 1.1, pz) qz) or I p n + 1, λ )f z) qz). I p n, λ )f z) If Ω C is a simply connected domain, Ω = hu), for some conformal mapping hz) of U onto Ω. In this case the class Φ I,2 [hu), q] is written as Φ I,2 [h, q]. In the particular case qz) =1+Mz, M > 0, the class of admissible functions Φ I,2 [Ω, q] becomes the class Φ I,2 [Ω, M]. Proceeding similarly as in the previous section, the following result is an immediate consequence of Theorem 2.7. THEOREM 2.8. Let Φ I,2 [h, q].iffz) A p satisfies Ip n + 1, λ )f z), I pn + 2, λ )f z) I p n, λ )f z) I p n + 1, λ )f z), I ) pn + 3, λ )f z) I p n + 2, λ )f z) ; z hz), 2.30) I p n + 1, λ )f z) qz). I p n, λ )f z) DEFINITION 2.6. Let Ω be a set in C. The class of admissible functions Φ I,2 [Ω, M] consists of those functions : C 3 U C such that 1 + Me iθ, 1 + k +)1 + Meiθ ) )1 + Me iθ Me iθ, 1 + k +)1 + Meiθ ) ) )1 + Me iθ Me iθ ) M + e iθ )[Le iθ +[ + 1]kM +)km 2 e iθ ] k 2 M 2 ) + )M + e iθ )[)e iθ +2)+k)M +)M 2 e iθ ] ; z Ω z U, θ R, RLe iθ ) k 1)kM for all real θ and k 1. 2.31) COROLLARY 2.9. Let Φ I,2 [Ω, M].Iffz) A p satisfies Ip n + 1, λ )f z), I pn + 2, λ )f z) I p n, λ )f z) I p n + 1, λ )f z), I ) pn + 3, λ )f z) I p n + 2, λ )f z) ; z Ω, I p n + 1, λ )f z) 1 + Mz. I p n, λ )f z) When Ω = {ω : ω 1 < M} = qu), the class Φ I,2 [Ω, M] is denoted by Φ I,2 [M] COROLLARY 2.10. Let Φ I,2 [M].Iffz) A p satisfies Ip n + 1, λ )f z), I pn + 2, λ )f z) I p n, λ )f z) I p n + 1, λ )f z), I ) pn + 3, λ )f z) I p n + 2, λ )f z) ; z 1 < M, I p n + 1, λ )f z) 1 I p n, λ )f z) < M.

134 ROSIHAN M. ALI, V.RAVICHANDRAN AND N. SEENIVASAGAN COROLLARY 2.11. If M > 0 and f z) A p satisfies I p n + 2, λ )f z) I p n + 1, λ )f z) I pn + 1, λ )f z) I p n, λ )f z) < M )1 + M), I p n + 1, λ )f z) 1 I p n, λ )f z) < M. This follows from Corollary 2.9 by taking u, v, w; z) =v u and Ω = hu) M where hz) = λ+p)1+m) z. 3. Superordination of the Multiplier Transformation The dual problem of differential subordination, that is, differential superordination of the multiplier transformation is investigated in this section. For this purpose the class of admissible functions is given in the following definition. DEFINITION 3.1. Let Ω beasetin C and qz) H [0, p] with zq z) 0. The class of admissible functions Φ I[Ω, q] consists of those functions : C 3 U C that satisfy the admissibility condition u, v, w; ζ) Ω whenever u = qz), v = zq z)/m)+λqz), { ) 2 w λ 2 } u R )v λ u 2λ 1 { zq } m R z) q z) + 1, z U, ζ U and m p THEOREM 3.1. Let Φ I [Ω, q].iffz) A p, I p n, λ )f z) Q 0 and is univalent in U, I p n, λ )f z), I p n + 1, λ )f z), I p n + 2, λ )f z); z) Ω { I p n, λ )f z), I p n + 1, λ )f z), I p n + 2, λ )f z); z) : z U} 3.1) qz) I p n, λ )f z). Proof. From 2.8) and 3.1),wehave Ω { ψ pz), zp z), z 2 p z); z ) : z U }. From 2.6), we see that the admissibility condition for Φ I [Ω, q] is equivalent to the admissibility condition for ψ as given in Definition 1.2. Hence ψ Ψ p [Ω, q], and by Theorem 1.2, qz) pz) or qz) I p n, λ )f z).

SUBORDINATION AND SUPERORDINATION FOR ANALYTIC FUNCTIONS 135 If Ω C is a simply connected domain, Ω = hu) for some conformal mapping hz) of U onto Ω. In this case the class Φ I[hU), q] is written as Φ I[h, q]. Proceeding similarly as in the previous section, the following result is an immediate consequence of Theorem 3.1. THEOREM 3.2. Let qz) H [0, p], hz) is analytic on U and Φ I[h, q].if f z) A p,i p n, λ )f z) Q 0 and I p n, λ )f z), I p n + 1, λ )f z), I p n + 2, λ )f z); z) is univalent in U, hz) I p n, λ )f z), I p n + 1, λ )f z), I p n + 2, λ )f z); z) 3.2) qz) I p n, λ )f z). Theorem 3.1 and 3.2 can only be used to obtain subordinants of differential superordination of the form 3.1) or 3.2). The following theorem proves the existence of the best subordinant of 3.2) for certain. THEOREM 3.3. Let hz) be analytic in U and : C 3 U C. Suppose that the differential equation qz), zq z)+λqz), z2 q z)+2λ + 1)zq z)+λ 2 ) qz) ) 2 ; z = hz) 3.3) has a solution qz) Q 0.If Φ I [h, q],fz) A p,i p n, λ )f z) Q 0 and I p n, λ )f z), I p n + 1, λ )f z), I p n + 2, λ )f z); z) is univalent in U, hz) I p n, λ )f z), I p n + 1, λ )f z), I p n + 2, λ )f z); z) qz) I p n, λ )f z) and qz) is the best subordinant. Proof. The proof is similar to the proof of Theorem 2.4 and is therefore omitted. Combining Theorems 2.2 and 3.2, we obtain the following sandwich theorem. COROLLARY 3.1. Let h 1 z) and q 1 z) be analytic functions in U, h 2 z) be univalent function in U, q 2 z) Q 0 with q 1 0) =q 2 0) =0 and Φ I [h 2, q 2 ] Φ I [h 1, q 1 ].Iffz) A p,i p n, λ )f z) H [0, p] Q 0 and I p n, λ )f z), I p n + 1, λ )f z), I p n + 2, λ )f z); z) is univalent in U, h 1 z) I p n, λ )f z), I p n + 1, λ )f z), I p n + 2, λ )f z); z) h 2 z), q 1 z) I p n, λ )f z) q 2 z).

136 ROSIHAN M. ALI, V.RAVICHANDRAN AND N. SEENIVASAGAN DEFINITION 3.2. Let Ω be a set in C and qz) H 0 with zq z) 0. Theclass of admissible functions Φ I,1 [Ω, q] consists of those functions : C3 U C that satisfy the admissibility condition u, v, w; ζ) Ω whenever u = qz), v = zq z)/m)+ 1)qz), { ) 2 w 1) 2 } u R 2 1) 1 { zq } )v 1)u m R z) q z) + 1, z U, ζ U and m 1. Now we will give the dual result of Theorem 2.5 for differential superordination. THEOREM 3.4. Let Φ I,1 [Ω, q].iffz) A p, Ip n, λ )f z), I pn + 1, λ )f z) is univalent in U, { Ip n, λ )f z) Ω, I pn + 1, λ )f z) qz) I pn, λ )f z). Proof. From 2.19) and 3.4),wehave I pn,λ)f z), I pn + 2, λ )f z) ; z Q 0 and ), I ) } pn + 2, λ )f z) ; z : z U Ω { ψ pz), zp z), z 2 p z); z ) : z U }. 3.4) From 2.17), we see that the admissibility condition for Φ I,1 [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) pz) or qz) I pn, λ )f z). If Ω C is a simply connected domain, and Ω = hu) for some conformal mapping hz) of U onto Ω and the class Φ I,1 [hu), q] is written as Φ I,1 [h, q]. Proceeding similarly as in the previous section, the following result is an immediate consequence of Theorem 3.4. THEOREM 3.5. Let qz) H 0, hz) is analytic on U and Φ I,1 [h, q]. ) Ipn,λ)f z) Ipn+1,λ)f z) Ipn+2,λ)f z) If f z) A p, I p n, λ )f z) Q 0 and,, ; z is univalent in U, Ip n, λ )f z) hz), I pn + 1, λ )f z), I ) pn + 2, λ )f z) ; z 3.5)

SUBORDINATION AND SUPERORDINATION FOR ANALYTIC FUNCTIONS 137 qz) I pn, λ )f z). Combining Theorems 2.6 and 3.5, we obtain the following sandwich theorem. COROLLARY 3.2. Let h 1 z) and q 1 z) be analytic functions in U, h 2 z) be univalent function in U, q 2 z) Q 0 with q 1 0) =q 2 0) =0 and Φ I,1 [h 2, q 2 ] Φ I,1 [h 1, q 1 ].Iffz) A p, H 0 Q 0 and Ip n, λ )f z) is univalent in U, Ip n, λ )f z) h 1 z) I pn,λ)f z), I pn + 1, λ )f z), I pn + 1, λ )f z) q 1 z) I pn, λ )f z) q 2 z)., I ) pn + 2, λ )f z) ; z, I ) pn + 2, λ )f z) ; z h 2 z), Now we will give the dual result of Theorem 2.7 for the differential superordination. DEFINITION 3.3. Let Ω be a set in C, qz) 0, zq z) 0andqz) H.The class of admissible functions Φ I,2 [Ω, q] consists of those functions : C3 U C that satisfy the admissibility condition whenever u, v, w; ζ) Ω u = qz), v = 1 { )vw v) R )2u v) v u z U, ζ U and m 1. )qz)+ zq z) mqz) } ), 1 m R { zq z) q z) + 1 }, THEOREM 3.6. Let Φ I,2 [Ω, q].iffz) A p, I pn+1,λ)f z) I pn,λ)f z) Ip n + 1, λ )f z), I pn + 2, λ )f z) I p n, λ )f z) I p n + 1, λ )f z), I pn + 3, λ )f z) I p n + 2, λ )f z) ; z is univalent in U, { Ip n + 1, λ )f z) Ω I p n, λ )f z) Q 1 and, I pn + 2, λ )f z) I p n + 1, λ )f z), I ) } pn + 3, λ )f z) I p n + 2, λ )f z) ; z : z U qz) I pn + 1, λ )f z). I p n, λ )f z) ) 3.6)

138 ROSIHAN M. ALI, V.RAVICHANDRAN AND N. SEENIVASAGAN Proof. From 2.29) and 3.6 ), wehave Ω { ψ pz), zp z), z 2 p z); z ) : z U }. From 2.28), we see that the admissibility condition for Φ I,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) pz) or qz) I pn + 1, λ )f z). I p n, λ )f z) If Ω C is a simply connected domain, Ω = hu) for some conformal mapping hz) of U onto Ω. In this case the class Φ I,2 [hu), q] is written as Φ I,2 [h, q]. The following result is an immediate consequence of Theorem 3.6. THEOREM 3.7. Let hz) be analytic in U and Φ I,2 [h, q]. If f z) A p, ) I pn+1,λ)f z) Ipn+1,λ)f z) Ipn+2,λ)f Ipn+3,λ)f z) I pn,λ)f z) Q 1, and I pn,λ)f z), I pn+1,λ)f z), I pn+2,λ)f z) ; z is univalent in U, Ip n + 1, λ )f z) hz), I pn + 2, λ )f z) I p n, λ )f z) I p n + 1, λ )f z), I ) pn + 3, λ )f z) I p n + 2, λ )f z) ; z, 3.7) qz) I pn + 1, λ )f z). I p n, λ )f z) Combining Theorems 2.8 and 3.7, we obtain the following sandwich theorem. COROLLARY 3.3. Let h 1 z) and q 1 z) be analytic functions in U, h 2 z) be univalent function in U, q 2 z) Q 1 with q 1 0) =q 2 0) =1 and Φ I,2 [h 2, q 2 ] Φ I,2 [h 1, q 1 ].Iffz) A p, H Q 1,I p n, λ )f z) 0 and I pn+1,λ)f z) I pn,λ)f z) Ip n + 1, λ )f z), I pn + 2, λ )f z) I p n, λ )f z) I p n + 1, λ )f z), I ) pn + 3, λ )f z) I p n + 2, λ )f z) ; z is univalent in U, Ip n + 1, λ )f z) h 1 z), I pn + 2, λ )f z) I p n, λ )f z) I p n + 1, λ )f z), I ) pn + 3, λ )f z) I p n + 2, λ )f z) ; z h 2 z), q 1 z) I pn + 1, λ )f z) I p n, λ )f z) q 2 z).

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