Research Article Subordination and Superordination on Schwarzian Derivatives

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1 Hindawi Publishing Corporation Journal of Inequalities and Applications Volume 008, Article ID 7138, 18 pages doi: /008/7138 Research Article Subordination and Superordination on Schwarzian Derivatives Rosihan M. Ali, 1 V. Ravichandran, and N. Seenivasagan 3 1 School of Mathematical Sciences, Universiti Sains Malaysia USM, Penang, Malaysia Department of Mathematics, University of Delhi, Delhi , India 3 Department of Mathematics, Rajah Serfoji Government College, Thanjavur , India Correspondence should be addressed to Rosihan M. Ali, rosihan@cs.usm.my Received 4 September 008; Accepted 30 October 008 Recommended by Paolo Ricci Let the functions q 1 be analytic and let q be analytic univalent in the unit disk. Using the methods of differential subordination and superordination, sufficient conditions involving the Schwarzian derivative of a normalized analytic function f are obtained so that either q 1 zf /f q or q 1 1 zf /f q. As applications, sufficient conditions are determined relating the Schwarzian derivative to the starlikeness or convexity of f. Copyright q 008 Rosihan M. Ali et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. 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 a a n z n a n 1 z n 1. We will write H H 1, 1. Denote by A the subclass of H 0, 1 consisting of normalized functions f of the form f z a k z k z U. 1.1 k Let S and K, respectively, be the familiar subclasses of A consisting of starlike and convex functions in U. The Schwarzian derivative {f, z} of an analytic, locally univalent function f is defined by {f, z} : f f 1 f f. 1.

2 Journal of Inequalities and Applications Owa and Obradović 1 proved that if f Asatisfies [ 1 R 1 zf f z {f, z}] > 0, 1.3 then f K. Miller and Mocanu proved that if f A satisfies one of the following conditions: [ R 1 zf f ] αz {f, z} > 0 Rα 0, [ R 1 zf f z {f, z}] > 0, 1.4 or [ R 1 zf f ] e z {f,z} > 0, 1.5 then f K. In fact, Miller and Mocanu found conditions on φ : C U C such that { R φ 1 zf } f,z {f, z}; z > implies f K. Each of the conditions mentioned above readily followed by choosing an appropriate φ. Miller and Mocanu also found conditions on φ : C 3 U C such that { zf R φ f, 1 zf } f,z {f, z}; z > implies f S. As applications, if f Asatisfies either [ R α zf f β 1 zf f zf ] z {f, z} > 0 α, β R, 1.8 f or [ zf R 1 zf ] f f z {f, z} > 1, 1.9 then f S. Let f and F be members of H U. The function f is said to be subordinate to F, orf is said to be superordinate to f, written f F, if there exists a function w analytic in U with w 0 0and w < 1 z U, such that f F w. IfF is univalent, then f F if and only if f 0 F 0 and f U F U.

3 Rosihan M. Ali et al. 3 In this paper, sufficient conditions involving the Schwarzian derivatives are obtained for functions f Ato satisfy either q 1 zf f q or q 1 1 zf f q, 1.10 where the functions q 1 are analytic and q is analytic univalent in U. InSection, a class of admissible functions is introduced. Sufficient conditions on functions f Aare obtained so that zf /f is subordinated to a given analytic univalent function q in U. Asa consequence, we obtained the result 1.7 of Miller and Mocanu relating the Schwarzian derivatives to the starlikeness of functions f A. Recently, Miller and Mocanu 3 investigated certain first- and second-order differential superordinations, which is the dual problem to subordination. Several authors have continued the investigation on superordination to obtain sandwich-type results 4 0. In Section 3, superordination is investigated on a class of admissible functions. Sufficient conditions involving the Schwarzian derivatives of functions f A are obtained so that zf /f is superordinated to a given analytic subordinant q in U. For q 1 analytic and q analytic univalent in U, sandwich-type results of the form q 1 zf f q 1.11 are obtained. This result extends earlier works by several authors. Section 4 is devoted to finding sufficient conditions for functions f Ato satisfy q 1 1 zf f q. 1.1 As a consequence, we obtained the result 1.6 of Miller and Mocanu. To state our results, we need the following preliminaries. Denote by Q the set of all functions q that are analytic and injective on U \ E q, where E q { } ζ U : limq z ζ, 1.13 and are such that q ζ / 0forζ U \ E q. Further, let the subclass of Q for which q 0 a be denoted by Q a and Q 1 Q 1. Definition 1.1 see, Definition.3a, page 7. LetΩ be a set in C,q Qand let 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 ψ r, s, t; z / Ω 1.14

4 4 Journal of Inequalities and Applications whenever r q ζ, s kζq ζ,and { } { t ζq } R s 1 ζ kr q ζ 1, 1.15 z U, ζ U \ E q,andk n. We write Ψ 1 Ω,q as Ψ Ω,q. If ψ : C U C, then the admissibility condition 1.14 reduces to ψ q ζ,kζq ζ ; z / Ω, 1.16 z U, ζ U \ E q,andk n. Definition 1. see 3, Definition 3, page 817. LetΩ be a set in C,q H a, n with q / 0. The class of admissible functions Ψ n Ω,q consists of those functions ψ : C 3 U C that satisfy the admissibility condition ψ r, s, t; ζ Ω 1.17 whenever r q, s zq /m,and { } t R s 1 1 { zq } m R q 1, 1.18 z U, ζ U, andm n 1. In particular, we write Ψ 1 Ω,q as Ψ Ω,q. If ψ : C U C, then the admissibility condition 1.17 reduces to ψ q, zq m ; ζ Ω, 1.19 z U, ζ U and m n. Lemma 1.3 see, Theorem.3b, page 8. Let ψ Ψ n Ω,q with q 0 a. If the analytic function p a a n z n a n 1 z n 1 satisfies ψ p,zp,z p ; z Ω, 1.0 then p q. Lemma 1.4 see 3, Theorem 1, page 818. Let ψ Ψ n Ω,q with q 0 a. Ifp Q a and ψ p,zp,z p ; z is univalent in U, then Ω { ψ p,zp,z p ; z : z U } 1.1 implies q p.

5 Rosihan M. Ali et al. 5. Subordination and starlikeness We first define the following class of admissible functions that are required in our first result. Definition.1. Let Ω be a set in C and q Q 1. The class of admissible functions Φ S Ω,q consists of those functions φ : C 3 U C that satisfy the admissibility condition φ u, v, w; z / Ω.1 whenever u q ζ, v q ζ kζq ζ q ζ / 0, q ζ { w u 1 3 v u } { ζq } ζ R kr v u q ζ 1,. z U, ζ U \ E q,andk 1. Theorem.. Let f Awith ff /z / 0. Ifφ Φ S Ω,q and { zf φ f, 1 zf } f,z {f, z}; z : z U Ω,.3 then zf f q..4 Proof. Define the function p by p : zf f..5 A simple calculation yields 1 zf f p zp p..6 Further computations show that z {f, z} zp z p p 3 [ zp p ] 1 p..7

6 6 Journal of Inequalities and Applications Define the transformation from C 3 to C 3 by u r, v r s s t, w 3 [ ] s 1 r..8 r r r Let ψ r, s, t; z φ u, v, w; z φ r, r s r, s t 3 [ s r r ] 1 r ; z..9 The proof will make use of Lemma 1.3.Using.5,.6,and.7, from.9 we obtain ψ p,zp,z p ; z zf φ f, 1 zf f,z {f, z}; z..10 Hence.3 becomes ψ p,zp,z p ; z Ω..11 A computation using.8 yields t s 1 w u 1 3 v u..1 v u Thus the admissibility condition for φ Φ S Ω,q in Definition.1 is equivalent to the admissibility condition for ψ as given in Definition 1.1. Hence ψ Ψ Ω,q and by Lemma 1.3, p q or zf f q..13 If Ω / C is a simply connected domain, then Ω h U for some conformal mapping h of U onto Ω. In this case, the class Φ S h U,q is written as Φ S h, q. The following result is an immediate consequence of Theorem.. Theorem.3. Let φ Φ S h, q. Iff Awith ff /z / 0 satisfies zf φ f, 1 zf f,z {f, z}; z h,.14 then zf f q..15

7 Rosihan M. Ali et al. 7 Following similar arguments as in, Theorem.3d, page 30, Theorem.3 can be extended to the following theorem where the behavior of q on U is not known. Theorem.4. Let h and q be univalent in U with q 0 1, and set q ρ q ρz and h ρ h ρz.letφ : C 3 U C satisfy one of the following conditions: i φ Φ S h, q ρ for some ρ 0, 1, or ii there exists ρ 0 0, 1 such that φ Φ S h ρ,q ρ for all ρ ρ 0, 1. If f Awith ff /z / 0 satisfies.14, then zf f q..16 The next theorem yields the best dominant of the differential subordination.14. Theorem.5. Let h be univalent in U, and φ : C 3 U C. Suppose that the differential equation φ q,q zq q, zq z q 3 zq q q 1 q ; z h.17 has a solution q with q 0 1 and one of the following conditions is satisfied: 1 q Q 1 and φ Φ S h, q, q is univalent in U and φ Φ S 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 S h ρ,q ρ for all ρ ρ 0, 1. If f Awith ff /z / 0 satisfies.14, then zf f q,.18 and q is the best dominant. Proof. Applying the same arguments as in, Theorem.3e, page 31, wefirstnotethatq is a dominant from Theorems.3 and.4. Since q satisfies.17, it is also a solution of.14, and therefore q will be dominated by all dominants. Hence q is the best dominant. We will apply Theorem. to two specific cases. First, let q 1 Mz, M > 0. Theorem.6. Let Ω be a set in C, and φ : C 3 U C satisfy the admissibility condition φ 1 Me iθ, 1 Me iθ kmeiθ,l; z / Ω.19 1 Meiθ

8 8 Journal of Inequalities and Applications whenever z U, θ R, with { L R 1 Me iθ 1 e iθ M 3k M } k M e iθ M.0 for all real θ and k 1. If f Awith ff /z / 0 satisfies zf φ f, 1 zf f,z {f, z}; z Ω,.1 then zf f 1 <M.. Proof. Let q 1 Mz, M > 0. A computation shows that the conditions on φ implies that it belongs to the class of admissible functions Φ S Ω, 1 Mz. The result follows immediately from Theorem.. In the special case Ω q U {ω : ω 1 <M}, the conclusion of Theorem.6 can be written as zf φ f, 1 zf f,z {f, z}; z 1 <M zf f 1 <M..3 Example.7. The functions φ 1 u, v, w; z : 1 α u αv, α M 1 0 and φ u, v, w; z : v/u, 0 < M satisfy the admissibility condition.19 and hence Theorem.6 yields 1 α zf f α 1 zf f 1 zf /f zf /f 1 <M zf f 1 <M α M 1 0, 1 <M zf f 1 <M 0 <M..4 By considering the function φ u, v, w; z : u v 1 λ u 1 with 0 <M 1, λ M 0, it follows again from Theorem.6 that z f f zf λ f 1 M λ M zf f 1 <M..5 This above implication was obtained in 1, Corollary, page 583. A second application of Theorem. is to the case q U being the half-plane q U {w : Rw >0} : Δ.

9 Rosihan M. Ali et al. 9 Theorem.8. Let Ω be a set in C and let the function φ : C 3 U C satisfy the admissibility condition φ iρ, iτ, ξ iη; z / Ω.6 for all z U and for all real ρ, τ, ξ and η with ρτ 1 1 3ρ, ρη 0..7 Let f Awith f f/z / 0. If zf φ f, 1 zf f,z {f, z}; z Ω,.8 then f S. Proof. Let q : 1 z / 1 z ; then q 0 1, E q {1} and q Q 1. For ζ : e iθ U \{1}, we obtain q ζ iρ, ζq ζ 1 ρ, ζ q ζ 1 ρ 1 iρ,.9 where ρ : cot θ/. Notethat ζq ζ R q ζ 1 0 ζ / We next describe the class of admissible functions Φ S Ω, 1 z / 1 z in Definition.1. For ζ / 1, u q ζ : iρ, v q ζ kζq ζ q ζ [ i ρ k 1 ρ ρ ] : iτ, w ξ iη.31 with { w u 1 3 v u } ρη R v u k 1 ρ..3 Thus the admissibility condition for functions in Φ S Ω, 1 z / 1 z is equivalent to.6, whence φ Φ S Ω, 1 z / 1 z. From Theorem., we deduce that f S. When h 1 z / 1 z, then h U Δ q U. Writing the class of admissible functions Φ S h U, Δ as Φ S Δ, the following result is a restatement of 1.7, which is an immediate consequence of Theorem.8.

10 10 Journal of Inequalities and Applications Corollary.9 see, Theorem 4.6a, page 44. Let φ Φ S Δ.Iff Awith ff /z / 0 satisfies { zf R φ f, 1 zf } f,z {f, z}; z > 0,.33 then f S. 3. Superordination and starlikeness Now we will give the dual result of Theorem. for differential superordination. Definition 3.1. Let Ω be a set in C, q Hwith zq / 0. The class of admissible functions Φ S Ω,q consists of those functions φ : C3 U C that satisfy the admissibility condition φ u, v, w; ζ Ω 3.1 whenever u q, v q zq q / 0, zq / 0, mq { w u 1 3 v u } R 1 { zq } v u m R q 1, 3. z U, ζ U and m 1. Theorem 3.. Let φ Φ S Ω,q, and f Awith f f/z / 0. Ifzf /f Q 1 and φ zf /f, 1 zf /f,z {f, z}; z is univalent in U, then Ω { zf φ f, 1 zf } f,z {f, z}; z : z U 3.3 implies q zf f. 3.4 Proof. With p zf /f,and ψ r, s, t; z φ r, r s r, s t r 3 s r 1 r ; z φ u, v, w; z, 3.5 equations.10 and 3.3 yield Ω { ψ p,zp,z p ; z : z U }. 3.6

11 Rosihan M. Ali et al. 11 Since t s 1 w u 1 3 v u, 3.7 v u the admissibility condition for φ Φ S Ω,q is equivalent to the admissibility condition for ψ as given in Definition 1.. Hence ψ Ψ Ω,q, and by Lemma 1.4, q p or q zf f. 3.8 If Ω / C is a simply connected domain, then Ω h U for some conformal mapping h of U onto Ω.WithΦ S h U,q as Φ S h, q, Theorem 3. can be written in the following form. Theorem 3.3. Let q H, h be analytic in U and φ Φ S h, q. Iff A, f f/z / 0, zf /f Q 1 and φ zf /f, 1 zf /f,z {f, z}; z is univalent in U, then zf h φ f, 1 zf f,z {f, z}; z 3.9 implies q zf f Theorems 3. and 3.3 can only be used to obtain subordinants of differential superordinations of the form 3.3 or 3.9. The following theorem proves the existence of the best subordinant of 3.9 for an appropriate φ. Theorem 3.4. Let h be analytic in U and φ : C 3 U C. Suppose that the differential equation φ q,q zq q, zq z q 3 zq q q 1 q ; z h 3.11 has a solution q Q 1.Letφ Φ S h, q, and f Awith f f/z / 0.Ifzf /f Q 1 and zf φ f, 1 zf f,z {f, z}; z 3.1 is univalent in U, then zf h φ f, 1 zf f,z {f, z}; z 3.13

12 1 Journal of Inequalities and Applications implies q zf f, 3.14 and q is the best subordinant. Proof. The proof is similar to the proof of Theorem.5, and is therefore omitted. Combining Theorems.3 and 3.3, we obtain the following sandwich-type theorem. Corollary 3.5. Let h 1 and q 1 be analytic functions in U, let h 1 be an analytic univalent function in U, q Q 1 with q 1 0 q 0 1 and φ Φ S h,q Φ S h 1,q 1.Letf Awith f f/z / 0. If zf /f H Q 1 and φ zf /f, 1 zf /f, z {f, z}; z is univalent in U, then zf h 1 φ f, 1 zf f,z {f, z}; z h 3.15 implies q 1 zf f q Schwarzian derivatives and convexity We introduce the following class of admissible functions. Definition 4.1. Let Ω be a set in C and q Q 1 H. The class of admissible functions Φ Sc Ω,q consists of those functions φ : C U C that satisfy the admissibility condition φ q ζ,kζq ζ 1 q ζ ; z / Ω, 4.1 z U, ζ U \ E q,andk 1. Theorem 4.. Let φ Φ Sc Ω,q, and f Awith f / 0. If { φ 1 zf } f,z {f, z}; z : z U Ω, 4. then 1 zf f q. 4.3

13 Rosihan M. Ali et al. 13 Proof. Define the function p by p : 1 zf f. 4.4 Clearly p A, and a simple calculation yields z {f, z} zp 1 p. 4.5 Define the transformation from C to C by u r, v s 1 r. 4.6 Let ψ r, s; z φ u, v; z φ r, s 1 r ; z. 4.7 The proof will make use of Lemma 1.3.Using 4.4 and 4.5,from 4.7, weobtain ψ p,zp ; z φ 1 zf f,z {f, z}; z. 4.8 Hence 4. becomes ψ p,zp ; z Ω. 4.9 From 4.7, we see that the admissibility condition for φ Φ Sc Ω,q is equivalent to the admissibility condition for ψ as given in Definition 1.1. Hence ψ Ψ Ω,q and by Lemma 1.3, p q or 1 zf f q We will denote by Φ Sc h, q the class Φ Sc h U,q, where h is the conformal mapping of U onto Ω / C. Proceeding similarly as in the previous section, the following results can be established, which we state without proof. Theorem 4.3. Let φ Φ Sc h, q.iff Awith f / 0 satisfies φ 1 zf f,z {f, z}; z h, 4.11

14 14 Journal of Inequalities and Applications then 1 zf f q. 4.1 We extend Theorem 4.3 to the case where the behavior of q on U is not known. Theorem 4.4. Let Ω C and let q be univalent in U with q 0 1. Letφ Φ Sc h, q ρ for some ρ 0, 1 where q ρ q ρz.iff Awith f / 0 satisfies 4., then 4.1 holds. Theorem 4.5. Let Ω be a set in C, q 1 Mz, M > 0, and φ : C U C satisfy φ 1 Me iθ, whenever z U, θ R and k 1. Let f Awith f / 0. If k 1 Meiθ Me iθ ; z / Ω 4.13 φ 1 zf f,z {f, z}; z Ω, 4.14 then zf f <M In the special case Ω q U {ω : ω 1 <M}, Theorem 4.5 gives the following: let φ : C U C satisfy φ 1 Me iθ, k 1 Meiθ Me iθ ; z 1 M whenever z U, θ R, andk 1; if f Awith f / 0satisfies φ 1 zf f,z {f, z}; z 1 <M, then zf f <M With φ u, v; z u v, we get the following: Example 4.6. If 0 <M<, and f Awith f / 0satisfies zf f z {f, z} <M, 4.19

15 Rosihan M. Ali et al. 15 then zf f <M. 4.0 We next apply Theorem 4. to the particular case corresponding to q U being a halfplane q U Δ. Theorem 4.7. Let Ω be a set in C. Letφ : C U C satisfy the admissibility condition φ iρ, η; z / Ω 4.1 for all z U, and for all real ρ and η with η 0. Letf Awith f / 0. If φ 1 zf f,z {f, z}; z Ω, 4. then f K. Let h 1 z / 1 z. Clearly, h U Δ. Writing the class of admissible functions Φ Sc h U, Δ as Φ Sc Δ, the following result is a restatement of 1.6, which is an immediate consequence of Theorem 4.7. Corollary 4.8 see, Theorem 4.6b, page 46. Let φ Φ Sc Δ. Iff A with f / 0 satisfies { R φ 1 zf } f,z {f, z}; z > 0, 4.3 then f K. Definition 4.9. Let Ω be a set in C and q H. The class of admissible functions Φ Sc Ω,q consists of those functions φ : C U C that satisfy the admissibility condition φ q, zq m 1 q ; ζ Ω, 4.4 z U, ζ U, andm 1. Now we will give the dual result of Theorem 4. for differential superordination. Theorem Let φ Φ Sc Ω,q, and f A with f / 0. If1 zf /f Q 1 and φ 1 zf /f,z {f, z}; z is univalent in U, then Ω { φ 1 zf } f,z {f, z}; z : z U 4.5

16 16 Journal of Inequalities and Applications implies q 1 zf f. 4.6 Proof. With p 1 zf /f and ψ r, s; z φ r, s 1 r ; z φ u, v; z, 4.7 from 4.8 and 4.5, we have Ω { ψ p,zp ; z : z U }. 4.8 From 4.6, we see that the admissibility condition for φ Φ Sc Ω,q is equivalent to the admissibility condition for ψ as given in Definition 1.. Hence ψ Ψ Ω,q, and by Lemma 1.4, q p or q 1 zf f. 4.9 Proceeding similarly as in the previous section, the following result is an immediate consequence of Theorem Theorem Let q H,leth be analytic in U and φ Φ Sc h, q. Letf Awith f / 0. If 1 zf /f Q 1 and φ 1 zf /f,z {f, z}; z is univalent in U, then h φ 1 zf f,z {f, z}; z 4.30 implies q 1 zf f Combining Theorems 4.3 and 4.11, we obtain the following sandwich-type theorem. Corollary 4.1. Let h 1 and q 1 be analytic functions in U, let h 1 be analytic univalent in U, q Q 1 with q 1 0 q 0 1 and φ Φ Sc h,q Φ Sc h 1,q 1.Letf A with f / 0. If 1 zf /f H Q 1 and φ 1 zf /f,z {f, z}; z is univalent in U, then h 1 φ 1 zf f,z {f, z}; z h 4.3

17 Rosihan M. Ali et al. 17 implies q 1 1 zf f q Acknowledgments The work presented here is supported by the FRGS and Science Fund research grants, and it was completed during V. Ravichandran visit to USM. The University s support is gratefully acknowledged. References 1 S. Owa and M. Obradović, An application of differential subordinations and some criteria for univalency, Bulletin of the Australian Mathematical Society, vol. 41, no. 3, pp , S. S. Miller and P. T. Mocanu, Differential Subordinations: Theory and Application, vol. 5 of Monographs and Textbooks in Pure and Applied Mathematics, Marcel Dekker, New York, NY, USA, S. S. Miller and P. T. Mocanu, Subordinants of differential superordinations, Complex Variables. Theory and Application, vol. 48, no. 10, pp , R. M. Ali and V. Ravichandran, Classes of meromorphic α-convex functions, to appear in Taiwanese Journal of Mathematics. 5 R. M. Ali, V. Ravichandran, M. Hussain Khan, and K. G. Subramanian, Differential sandwich theorems for certain analytic functions, Far East Journal of Mathematical Sciences, vol.15,no.1,pp , R. M. Ali, V. Ravichandran, M. Hussain Khan, and K. G. Subramanian, Applications of first order differential superordinations to certain linear operators, Southeast Asian Bulletin of Mathematics, vol. 30, no. 5, pp , R. M. Ali, V. Ravichandran, and S. K. Lee, Subclasses of multivalent starlike and convex functions, to appear in Bulletin of the Belgian Mathematical Society. Simon Stevin. 8 R. M. Ali, V. Ravichandran, and N. Seenivasagan, Differential subordination and superordination of analytic functions defined by the Dziok-Srivastava linear operator, preprint. 9 R. M. Ali, V. Ravichandran, and N. Seenivasagan, Differential subordination and superordination of analytic functions defined by the multiplier transformation, to appear in Mathematical Inequalities & Applications. 10 R. M. Ali, V. Ravichandran, and N. Seenivasagan, Subordination and superordination of the Liu- Srivastava linear operator on meromorphic functions, Bulletin of the Malaysian Mathematical Sciences Society, vol. 31, no., pp , T. Bulboacă, Sandwich-type theorems for a class of integral operators, Bulletin of the Belgian Mathematical Society. Simon Stevin, vol. 13, no. 3, pp , T. Bulboacă, A class of double subordination-preserving integral operators, Pure Mathematics and Applications, vol. 15, no. -3, pp , T. Bulboacă, A class of superordination-preserving integral operators, Indagationes Mathematicae, vol. 13, no. 3, pp , T. Bulboacă, Generalized Briot-Bouquet differential subordinations and superordinations, Revue Roumaine de Mathématiques Pures et Appliquées, vol. 47, no. 5-6, pp , T. Bulboacă, Classes of first-order differential superordinations, Demonstratio Mathematica, vol. 35, no., pp. 87 9, N. E. Cho and S. Owa, Double subordination-preserving properties for certain integral operators, Journal of Inequalities and Applications, vol. 007, Article ID 83073, 10 pages, N. E. Cho and H. M. Srivastava, A class of nonlinear integral operators preserving subordination and superordination, Integral Transforms and Special Functions, vol. 18, no. 1-, pp , S. S. Miller and P. T. Mocanu, Briot-Bouquet differential superordinations and sandwich theorems, Journal of Mathematical Analysis and Applications, vol. 39, no. 1, pp , 007.

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