Sandwich-Type Theorems for a Class of Multiplier Transformations Associated with the Noor Integral Operators

and Applied Analysis 3 and let f c be defined such that f c z ∗ f c z z 1 − z μ ( μ > 0; z ∈ U), 1.9 where the symbol ∗ stands for the Hadamard product or convolution . Then, motivated essentially by the Noor integral operator 7 see also 8–11 , we now introduce the operator I λ,μ : A → A, which are defined here by I λ,μ c f z ( f λ,μ c ∗ f ) z ( f ∈ A; Re{c} > 0; λ ≥ 0; μ > 0). 1.10 In view of 1.9 and 1.10 , we obtain the following relations: z ( I λ,μ c f z )′ cI 1,μ c f z − c − 1 I c f z , 1.11 z ( I λ,μ c f z )′ μI 1 c f z − ( μ − 1)Iλ,μ c f z . 1.12 Making use of the principle of subordination between analytic functions, Miller et al. 12 investigated some subordination theorems involving certain integral operators for analytic functions in U see, also 13 . Moreover, Miller and Mocanu 2 considered differential superordinations, as the dual concept of differential subordinations see also 14 . In the present paper, we obtain the subordinationand superordination-preserving properties of the multiplier transformations I c defined by 1.10 with the sandwich-type theorems. The following lemmas will be required in our present investigation. Lemma 1.4 see 15 . Suppose that the functionH : C2 → C satisfies the condition: Re{H is, t } ≤ 0, 1.13 for all real s and t ≤ −n 1 s2 /2, where n is a positive integer. If the function p z 1 pnz · · · is analytic in U and Re { H ( p z , zp′ z )} > 0 z ∈ U , 1.14 then Re{p z } > 0 in U. Lemma 1.5 see 16 . Let β, γ ∈ C with β / 0 and let h ∈ H U with h 0 c. If Re{βh z γ} > 0 z ∈ U , then the solution of the differential equation: q z zq′ z βq z γ h z ( z ∈ U; q 0 c) 1.15 is analytic in U and satisfies Re{βq z γ} > 0 z ∈ U . 4 Abstract and Applied Analysis Lemma 1.6 see 1 . Let p ∈ Q with p 0 a and let q z a anz · · · be analytic in U with q z /≡a and n ≥ 1. If q is not subordinate to p, then there exist points z0 r0e ∈ U and ζ0 ∈ ∂U \ E f , for which q Ur0 ⊂ p U , p z0 q ζ0 , z0p′ z0 mζ0q′ ζ0 m ≥ n . 1.16 Lemma 1.7 see 2 . Let q ∈ H a, 1 , let φ : C2 → C, and set φ q z , zq′ z ≡ h z . If L z, t φ q z , tzq′ z is a subordination chain and p ∈ H a, 1 ∩ Q, then h z ≺ φ(p z , zp′ z ) z ∈ U 1.17 implies that q z ≺ p z z ∈ U . 1.18 Furthermore, if φ q z , zp′ z h z has a univalent solution q ∈ Q, then q is the best subordinant. A function L z, t defined on U × 0,∞ is the subordination chain or Löwner chain if L ·, t is analytic and univalent in U for all t ∈ 0,∞ ; L z, · is continuously differentiable on 0,∞ for all z ∈ U and L z, s ≺ L z, t for z ∈ U and 0 ≤ s < t. Lemma 1.8 see 17 . The function L z, t a1 t z · · · with a1 t / 0 and limt→∞|a1 t | ∞. Suppose that L ·; t ia analytic in U for all t ≥ 0, L z; · is continuously differentiable on 0,∞ for all z ∈ U. If L z; t satisfies |L z; t | ≤ K0|a1 t | |z| < r0 < 1; 0 ≤ t <∞ 1.19 for some positive constants K0 and r0 and R { z∂L z, t /∂z ∂L z, t /∂t } > 0 z ∈ U; 0 ≤ t <∞ , 1.20 then L z; t is a subordination chain. 2. Main Results Firstly, we begin by proving the following subordination theorem involving the multiplier transformation I c defined by 1.10 . Theorem 2.1. Let f, g ∈ A. Suppose that Re { 1 zφ′′ z φ′ z } > −δ ( φ z : 1 − α I 1,μ c g z αI c g z ; 0 ≤ α < 1; z ∈ U ) , 2.1 Abstract and Applied Analysis 5 whereand Applied Analysis 5 where δ 1 − α 2 |c − 1 α| − ∣∣ 1 − α 2 − c − 1 α 2 ∣∣ 4 1 − α Re{c − 1 α} Re{c − 1 α} > 0 . 2.2 Then the subordination: 1 − α I 1,μ c f z αI c f z ≺ φ z z ∈ U 2.3


Introduction
Let H H U denote the class of analytic functions in the open unit disk U {z ∈ C : |z| < 1}.For a ∈ C and nonnegative integer n, let We also denote A by the subclass of H a, 1 with the usual normalization f 0 f 0 − 1 0. Let f and F be members of H.The function f is said to be subordinate to F, or F is said to be superordinate to f, if there exists a function w analytic in U, with w 0 0 and |w z | < 1, and such that f z F w z .In such a case, we write f ≺ F or f z ≺ F z .If the function F is univalent in U, then we have f ≺ F if and only if f 0 F 0 and f U ⊂ F U cf. 1 .Definition 1.1 see 1 .Let φ : C 2 → C and let h be univalent in U.If p is analytic in U and satisfies the differential subordination: Abstract and Applied Analysis then p is called a solution of the differential subordination.The univalent function q is called a dominant of the solutions of the differential subordination, or more simply a dominant if p ≺ q for all p satisfying 1.2 .A dominant q that satisfies q ≺ q for all dominants q of 1.2 is said to be the best dominant.
Definition 1.2 see 2 .Let ϕ : C 2 → C and let h be analytic in U.If p and ϕ p z , zp z are univalent in U and satisfy the differential superordination: then p is called a solution of the differential superordination.An analytic function q is called a subordinant of the solutions of the differential superordination, or more simply a subordinant if q ≺ p for all p satisfying 1.3 .A univalent subordinant q that satisfies q ≺ q for all subordinants q of 1.3 is said to be the best subordinant.
Definition 1.3 see 2 .We denote by Q the class of functions f that are analytic and injective on U \ E f , where and are such that f ζ / 0 for ζ ∈ ∂U \ E f .Following Komatu 3 , we introduce the integral operator L λ c : A → A defined by where the symbol Γ stands the Gamma function.We also note that the operator L λ c f z defined by 1.5 can be expressed by the series expansion as follows: Obviously, we have, for λ, ν ≥ 0, In particular, the operator L λ 2 is closely related to the multiplier transformation studied earlier by Flett 4 .Various interesting properties of the operator L λ 2 have been studied by Jung et al. 5 and Liu 6 .We also note from 1.6 that we can define the operator L λ c for any real number λ.Let be defined such that where the symbol * stands for the Hadamard product or convolution .Then, motivated essentially by the Noor integral operator 7 see also 8-11 , we now introduce the operator I κ λ,μ : A → A, which are defined here by In view of 1.9 and 1.10 , we obtain the following relations: Making use of the principle of subordination between analytic functions, Miller et al. 12 investigated some subordination theorems involving certain integral operators for analytic functions in U see, also 13 .Moreover, Miller and Mocanu 2 considered differential superordinations, as the dual concept of differential subordinations see also 14 .In the present paper, we obtain the subordination-and superordination-preserving properties of the multiplier transformations I λ,μ c defined by 1.10 with the sandwich-type theorems.
The following lemmas will be required in our present investigation.
is analytic in U and satisfies Re{βq z γ} > 0 z ∈ U .
Lemma 1.6 see 1 .Let p ∈ Q with p 0 a and let q z a a n z n • • • be analytic in U with q z / ≡ a and n ≥ 1.If q is not subordinate to p, then there exist points z 0 r 0 e iθ ∈ U and Furthermore, if ϕ q z , zp z h z has a univalent solution q ∈ Q, then q is the best subordinant.
for some positive constants K 0 and r 0 and

Main Results
Firstly, we begin by proving the following subordination theorem involving the multiplier transformation I λ,μ c defined by 1.10 . where Then the subordination: Moreover, the function I λ,μ c g z is the best dominant.
Proof.Let us define the functions F and G, respectively, by We first show that if the function q is defined by Taking the logarithmic differentiation on both sides of the second equation in 2.5 and using 1.11 for g ∈ A, we obtain which, in conjunction with 2.8 , yields the relationship: 2.9 From 2.1 , we have and by using Lemma 1.5, we conclude that the differential equation 2.9 has a solution q ∈ H U with q 0 h 0 1.

Let us put
where δ is given by 2.2 .From 2.1 , 2.9 , and 2.11 , we obtain Re H q z , zq z > 0 z ∈ U .

2.12
Now we proceed to show that Re{H is, t } ≤ 0 for all real s and t ≤ − 1 s 2 /2.From 2.11 , we have

2.14
For δ given by 2.2 , we can prove easily that the expression E δ s given by 2.14 is positive or equal to zero.Moreover, the quadratic expression by s in 2.14 is a perfect square for the assumed value of δ.Hence from 2.13 , we see that Re{H is, t } ≤ 0 for all real s and t ≤ − 1 s 2 /2.Thus, by using Lemma 1.4, we conclude that Re{q z } > 0 for all z ∈ U.That is, q is convex in U.
Next, we prove that the subordination condition 2.3 implies that for the functions F and G defined by 2.5 .Without loss of generality, we can assume that G is analytic and univalent on U and that G ζ / 0 |ζ| 1 .Now we consider the function L z, t given by We note that Abstract and Applied Analysis 7 2.17 This shows that the function satisfies the condition a 1 t / 0 for all t ∈ 0, ∞ .By using the well-known growth and distortion theorems for convex functions, it is easy to check that the first part of Lemma 1.8 is satisfied.Furthermore, we have since G is convex and Re{c − 1 α} > 0. Therefore, by virtue of Lemma 1.8, L z, t is a subordination chain.We observe from the definition of a subordination chain that

2.20
This implies that

2.21
Now suppose that F is not subordinate to G, then by Lemma 1.6, there exists points z 0 ∈ U and ζ 0 ∈ ∂U such that

2.22
Hence we have by virtue of the subordination condition 2.3 .This contracts the above observation that L ζ 0 , t / ∈ φ U .Therefore, the subordination condition 2.3 must imply the subordination given by 2.15 .Considering F z G z , we see that the function G z is the best dominant.This evidently completes the proof of Theorem 2.1.
We next prove a dual problem of Theorem 2.1, in the sense that the subordinations are replaced by superordinations.
where δ is given by 2.2 , and the function Then the superordination:

2.26
Moreover, the function I λ,μ c g z is the best subordinant.
Proof.Let us define the functions F and G, respectively, by 2.5 .We first note that, if the function q is defined by 2.6 , by using 2.8 , then we obtain : ϕ G z , zG z .

2.27
After a simple calculation, 2.27 yields the relationship: Then by using the same method as in the proof of Theorem 2.1, we can prove that Re{q z } > 0 for all z ∈ U.That is, G defined by 2.6 is convex univalent in U.
Next, we prove that the subordination condition 2.27 implies that for the functions F and G defined by 2.5 .Now consider the function L z, t defined by Since G is convex and Re{c−1 α} > 0, we can prove easily that L z, t is a subordination chain as in the proof of Theorem 2.1.Therefore according to Lemma 1.7, we conclude that the superordination condition 2.27 must imply the superordination given by 2.29 .Furthermore, since the differential equation 2.27 has the univalent solution G, it is the best subordinant of the given differential superordination.Therefore we complete the proof of Theorem 2.3.
If we combine this Theorems 2.1 and 2.3, then we obtain the following sandwich-type theorem.
where δ is given by 2.2 , and the function Then the subordination relation:

2.33
Moreover, the functions I λ,μ c g 1 z and I λ,μ c g 2 z are the best subordinant and the best dominant, respectively.
The assumption of Theorem 2.4, that the functions I λ 1,μ c f z and I λ,μ c f z need to be univalent in U, may be replaced by another conditions in the following result.
Corollary 2.5.Let f, g k ∈ A k 1, 2 .Suppose that the condition 2.31 is satisfied and where δ is given by 2.2 .Then the subordination relation:

2.36
Moreover, the functions I λ,μ c g 1 z and I λ,μ c g 2 z are the best subordinant and the best dominant, respectively.
Proof.In order to prove Corollary 2.5, we have to show that the condition 2.34 implies the univalence of ψ z and F z : I λ,μ c f z .Since 0 < δ ≤ 1/2 from Remark 2.2, the condition 2.34 means that ψ is a close-to-convex function in U see 18 and hence ψ is univalent in U. Furthermore, by using the same techniques as in the proof of Theorem 2.4, we can prove the convexity univalence of F and so the details may be omitted.Therefore, by applying Theorem 2.4, we obtain Corollary 2.5.
By setting c 2 and α 0 in Theorem 2.4, so that δ 1/2, we deduce the following consequence of Theorem 2.4.
and the function Then the subordination relation:

2.39
Moreover, the functions I λ,μ 2 g 1 z and I λ,μ 2 g 2 z are the best subordinant and the best dominant, respectively.
If we take c 2 i and α 0 in Theorem 2.4, then we easily to lead to the following result.

Corollary 2.7. Let f, g
Then the subordination relation:

2.42
Moreover, the functions I λ,μ 2 i g 1 z and I λ,μ 2 i g 2 z are the best subordinant and the best dominant, respectively.
The proof of Theorem 2.8 below is similar to that of Theorem 2.4 by using 1.12 and so the details may be omitted.
where δ is given by 2.2 with c μ, and the function Then the subordination relation: Moreover, the functions I λ,μ c g 1 z and I λ,μ c g 2 z are the best subordinant and the best dominant, respectively.
By using a similar method given in the proof of Theorems 2.4 and 2.8, we have the corresponding two theorems below.

Theorem 2 . 9 .where δ is given by δ 1 − α 2 |c| 2 − 1 − α 2 − c 2 4Theorem 2 . 10 .
Let f, g k ∈ A k 1, 2 with the additional condition g k z / 0. Suppose that z /z are the best subordinant and the best dominant, respectively.Let f, g k ∈ A k 1, 2 with the additional condition g k z / 0. Suppose that z z ; k 1, 2; 0 ≤ α < 1; z ∈ U ,2.50where δ is given by 2.47 with c μ, and the functionI λ,μ 1 c f z /z is univalent in U and I λ,μ c f z /z ∈ Q. Thenthe subordination relation: z /z and I λ,μ c g 2 z /z are the best subordinant and the best dominant, respectively.

Lemma 1.4 see
15. Suppose that the function H : C 2 → C satisfies the condition: