A Class of Nonlinear Integral Operators Preserving Double Subordinations

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 functionw analytic in U, withw 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 : C2 → C and let h be univalent in U. If p is analytic in U and satisfies the differential subordination


Introduction
Let H H U denote the class of analytic functions in the open unit disk U {z ∈ C : |z| < 1}.For a ∈ C, let 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 φ p z , zp z ≺ h z , 1.2 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 .One denotes by Q the class of functions f that are analytic and injectiveon U \ E f , where and are such that Let Σ denote the class of functions of the form which are analytic in the punctured open unit disk D U \ {0}.Let Σ * and Σ k be the subclasses of Σ consisting of all functions which are, respectively, meromorphic starlike and meromorphic convex in D see, for details, 3-5 .
For a function f ∈ Σ, we introduce the following integral operators I β,γ defined by 1.6 The integral operator I β,γ f defined by 1.6 has been extensively studied by many authors 6-10 with suitable restrictions on the parameters β and γ, and for f belonging to some favored classes of meromorphic functions.In particular, Bajpai 6 showed that the integral operator I 1,1 f belongs to the classes Σ * and Σ k , whenever f belongs to the classes Σ * and Σ k , respectively.Moreover, the operator I β,γ for the case β 1 is related to the generalized Libera transform introduced by Stević see, e.g., 11-13 .Making use of the principle of subordination between analytic functions, Miller et al. 14 obtained some subordination-preserving properties for certain integral operators see also 15 .Moreover, Miller and Mocanu 2 considered differential superordinations as the dual concept of differential subordinations see also 16 .In the present paper, we obtain the subordination-and superordination-preserving properties of the integral operator I β,γ defined by 1.6 with the sandwich-type theorem.
Throughout this paper, we denote the class Σ β,γ by where I β,γ is the integral operator defined by 1.6 .For various interesting developments involving functions in the class Σ β,γ , the reader may be referred, for example, to the work of Dwivedi et al. 9 .

A set of lemmas
The following lemmas will be required in our present investigation.
Lemma 2.1 see 17 .Suppose that the function H : C 2 → C satisfies the following condition: 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 A function L z, t defined on U × 0, ∞ is the subordination chain or L öwner 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 0 ≤ s < t.
implies that q z ≺ p z .

2.6
Furthermore, if ϕ q z , zp z h z has a univalent solution q ∈ Q, then q is the best subordinant.

Main results
Subordination theorem involving the integral operator I β,γ defined by 1.6 is contained in Theorem 3.1 below. where Then, the subordination where I β,γ is the integral operator defined by 1.6 .Moreover, the function z zI β,γ g z β is the best dominant.
Proof.Let us define the functions F and G by respectively.Without loss of generality, we can assume that G is analytic and univalent on U and that We first show that if the function q is defined by In terms of the function φ involved in 3.1 , the definition 1.6 readily yields We also have By a simple calculation with 3.9 and 3.10 , we obtain the relationship and by using Lemma 2.2, we conclude that the differential equation 3.11 has a solution q ∈ H U with q 0 h 0 1.

3.13
Let us put where δ is given by 3.2 .From 3.1 , 3.11 , and 3.14 , we obtain Re H q z , zq z > 0 z ∈ U .

3.15
Now, we proceed to show that Re{H is, t } ≤ 0 for all real s and t ≤ − 1 s 2 /2.From 3.14 , we have where

3.17
For δ given by 3.2 , we note that the coefficient of s 2 in the quadratic expression E δ s given by 3.17 is positive or equal to zero.Moreover, for the assumed value of δ given by 3.2 , the quadratic expression E δ s by s in 3.17 is a perfect square.Hence, from 3.16 , we see that Re{H is, t } ≤ 0 for all real s and t ≤ − 1 s 2 /2.Thus, by using Lemma 2.1, we conclude that Next, we prove that the subordination condition 3.3 implies that for the functions F and G defined by 3.5 .For this purpose, we consider the function L z, t given by L z, t : Since G is convex in U and Re{γ − β} > 0, we obtain that ∂L z, t ∂z

3.21
Therefore, by virtue of Lemma 2.5, L z, t is a subordination chain.We observe from the definition of a subordination chain that

3.22
This implies that

3.23
Now suppose that F is not subordinate to G, then by Lemma 2.3, there exist points z 0 ∈ U and ζ 0 ∈ ∂U such that

3.24
Hence, we have by virtue of the subordination condition 3.3 .This contradicts the above observation that L ζ 0 , t / ∈ φ U .Therefore, the subordination condition 3.3 must imply the subordination given by 3.19 .Considering F z G z , we see that the function G z is the best dominant.This evidently completes the proof of Theorem 3.1.Remark 3.2.We note that δ given by 3.2 in Theorem 3.1 satisfies the inequality 0 < δ ≤ 1/2.
We next prove a dual problem of Theorem 3.1 in the sense that the subordinations are replaced by superordinations.
where δ is given by 3.2 , z zf z β is univalent in U, and z zI β,γ f z β ∈ Q, where I β,γ is the integral operator defined by 1.6 .Then, the superordination

3.28
Moreover, the function z zI β,γ g z β is the best subordinant.
Proof.The first part of the proof is similar to that of Theorem 3.1 and so we will use the same notation as in the proof of Theorem 3.1.Now let us define the functions F and G, respectively, by 3.5 .We first note that by using 3.9 and 3.10 , we obtain

3.29
After a simple calculation, 3.29 yields the relationship where the function q is defined by 3.7 .Then, by using the same method as in the proof of Theorem 3.1, we can prove that Re q z > 0 z ∈ U , 3.31 that is, G defined by 3.5 is convex univalent in U.
Next, we prove that the superordination condition 3.27 implies that Now, consider the function L z, t defined by Since G is convex and Re{γ − β} > 0, we can prove easily that L z, t is a subordination chain as in the proof of Theorem 3.1.Therefore, according to Lemma 2.4, we conclude that the superordination condition 3.27 must imply the superordination given by 3.32 .Furthermore, since the differential equation 3.29 has the univalent solution G, it is the best subordinant of the given differential superordination.Therefore, we complete the proof of Theorem 3.3.
If we combine Theorems 3.1 and 3.3, then we can obtain the following sandwich-type theorem.
where δ is given by 3.2 , z zf z β is univalent in U, and z zI β,γ f z β ∈ Q, where I β,γ is the integral operator defined by 1.6 .Then, the subordination

3.36
Moreover, the functions z zI β,γ g 1 z β and z zI β,γ g 2 z β are the best subordinant and the best dominant, respectively.
The assumption of Theorem 3.4, that the functions z zf z β and z zI β,γ f z β need to be univalent in U, may be replaced by another condition in the following result.Corollary 3.5.Let f, g k ∈ Σ β,γ k 1, 2 .Suppose that the condition 3.34 is satisfied and where δ is given by 3.2 .Then, the subordination where I β,γ is the integral operator defined by 1.6 .Moreover, the functions z zI β,γ g 1 z β and z zI β,γ g 2 z β are the best subordinant and the best dominant, respectively.
Proof.In order to prove Corollary 3.5, we have to show that the condition 3.37 implies the univalence of ψ z and F z : z zI β,γ f z β .

3.40
Since 0 < δ ≤ 1/2 from Remark 3.2, the condition 3.37 means that ψ is a close-to-convex function in U see 19 and hence ψ is univalent in U. Furthermore, by using the same techniques as in the proof of Theorem 3.4, we can prove the convexity univalence of F and so the details may be omitted.Therefore, by applying Theorem 3.
4, we obtain Corollary 3.5.By setting γ −β 1 in Theorem 3.4, so that δ 1/2, we deduce the following consequence of Theorem 3.4.Let f, g k ∈ Σ β,β 1 k 1, 2 .Suppose that is univalent in U, and z zI β,β 1 f z β ∈ Q, where I β,β 1 is the integral operator defined by 1.6 with γ β 1.Then, the subordination Moreover, the functions z zI β,β 1 g 1 z β and z zI β,β 1 g 2 z β are the best subordinant and the best dominant, respectively.If we take γ − β 1 i in Theorem 3.4, then we are easily led to the following result.Moreover, the functions z zI β,β 1 i g 1 z β and z zI β,β 1 i g 2 z β are the best subordinant and the best dominant, respectively.