4_486_aouf.dvi EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS Vol. 3, No. 1, 2010, 26-44 ISSN 1307-5543 – www.ejpam.com Differential Subordination and Superordination of Analytic Func- tions Defined by an Integral Operator M. K. Aouf1∗ and T. M. Seoudy2 1 Department of Mathematics, Faculty of Science, Mansoura 35516, Egypt 2 Department of Mathematics, Faculty of Science, Fayoum 63514, Egypt Abstract. Differential subordination and superordination results are obtained for analytic functions in the open unit disk which are associated with the integral operator. These results are obtained by investigating appropriate classes of admissible functions. Sandwich-type results are also obtained. Some of the results established in this paper would provide extensions of those given in earlier works. 2000 Mathematics Subject Classifications: 30C45 Key Words and Phrases: Analytic function, integral operator, Hadamard product, differential subor- dination, superordination. 1. Introduction Let H(U) be the class of functions analytic in U = {z : z ∈ C and |z| < 1} and H[a, n] be the subclass of H(U) consisting of functions of the form f (z) = a+ anzn+ an+1zn+1+ ..., with H0 = H[0,1] and H = H[1,1]. Let A � p � denote the class of all analytic functions of the form f (z) = zp + ∞ ∑ n=1 ap+nzp+n � p ∈ N = {1,2,3, ...} ; z ∈ U � (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 F(z), or F(z) is said to be superordinate to f (z), if there exists a function ω(z) analytic in U with ω(0) = 0 and |ω(z)| < 1(z ∈ U), such that f (z) = F(ω(z)). In such a case we write f (z) ≺ F(z). If F is univalent, then f (z) ≺ F(z) if and only if f (0) = F(0) and f (U) ⊂ F(U) (see [8] and [9]. For two functions f (z) given by (1) and g(z) = zp + ∞ ∑ n=1 bp+nzp+n, ∗Corresponding author. Email addresses: mkaouf127�yahoo. om (M. Aouf), tmseoudy�gmail. om (T. Seoudy) http://www.ejpam.com 26 c© 2009 EJPAM All rights reserved. M. Aouf, T. Seoudy / Eur. J. Pure Appl. Math, 3 (2010), 26-44 27 the Hadamard product (or convolution) of f and g is defined by � f ∗ g � (z) = zp + ∞ ∑ n=1 ap+n bp+n zp+n = � g ∗ f � (z) . Motivated essentially by Jung et al. Liu [5] and Owa [7] introduced the integral operator Qα β ,p : A � p � → A � p � as follows: Qαβ ,p f (z) = � p+α+ β − 1 p+ β − 1 � α zβ ∫ z 0 � 1− t z �α−1 tβ−1 f (t) d t, � α > 0;β > −1; p ∈ N � , (2) and Q0 β ,p f (z) = f (z), � α= 0;β > −1 � . For f ∈ A � p � given by (1), then from (2), we deduce that Qαβ ,p f (z) = zp + Γ � α+ β + p � Γ � β + p � ∞ ∑ n=1 Γ � β + p+ n � Γ � α+ β + p+ n �ap+nzp+n � α ≥ 0;β > −1; p ∈ N � . (3) It is easily verified from the definition (3) that (see [7]) z � Qαβ ,p f (z) �′ = � α+ β + p− 1 � Qα−1 β ,p f (z)− � α+ β − 1 � Qαβ ,p f (z). (4) We note that the one-parameter family of integral operator Qα β ,1 f (z) = Qα β was defined by Jung et al. [5]. To prove our results, we need the following definitions and Lemmas. Denote by F the set of all functions q(z) that are analytic and injective on Ū\E(q) where E(q) = � ζ ∈ ∂ U : lim z→ζ q(z) =∞ � , and are such that q′(ζ) 6= 0 for ζ ∈ ∂ U\E(q). Further let the subclass ofF for which q(0) = a be denoted by F (a), F (0)≡F0 and F (1)≡F1. Definition 1 ([8], Definition 2.3a, p. 27). Let Ω be a set in C, q ∈ F and n be a positive integer. The class of admissible functions Ψn[Ω,q], consists of those functions ψ : C3 × U → C that satisfy the admissibility condition: ψ(r, s, t; z) /∈ Ω whenever r = q(ζ), s = kζq′(ζ), ℜ § t s + 1 ª ≥ kℜ ( 1+ ζq ′′ (ζ) q′ (ζ) ) , where z ∈ U , ζ ∈ ∂ U\E(q) and k ≥ n. We write Ψ1[Ω,q] as Ψ[Ω,q]. M. Aouf, T. Seoudy / Eur. J. Pure Appl. Math, 3 (2010), 26-44 28 In particular when q(z) = M Mz+a M+āz , with M > 0 and |a|< M , then q(U) = UM = {w : |w| < M}, q(0) = a, E(q) = ∅ and q ∈ F . In this case, we set Ψn[Ω, M , a] = Ψn[Ω,q], and in the special case when the set Ω = UM , the class is simply denoted by Ψn[M , a]. Definition 2 ([9], Definition 3, p. 817). Let Ω be a set in C, q(z) ∈ H[a, n] with q′(z) 6= 0. The class of admissible functions Ψ′n[Ω,q] consists of those functions ψ : C3× Ū → C that satisfy the admissibility condition ψ(r, s, t;ζ) ∈ Ω whenever r = q(z), s = zq′(z) m , ℜ § t s + 1 ª ≥ 1 m ℜ ( 1+ zq ′′ (z) q′ (z) ) , where z ∈ U ,ζ ∈ ∂ U and m ≥ n≥ 1. In particular, we write Ψ′1[Ω,q] as Ψ′[Ω,q]. Lemma 1 ([8],Theorem 2.3b, p. 28). Letψ ∈Ψn � Ω,q � with q(0) = a. If the analytic function g(z) = a+ anzn + an+1zn+1 + ... satisfies ψ(g(z), zg′(z), z2 g ′′ (z); z) ∈ Ω, then g(z) ≺ q(z). Lemma 2 ([9],Theorem 1, p. 818). . Let ψ ∈Ψ′n[Ω,q] with q(0) = a. If g(z) ∈ F (a) and ψ(g(z), zg′(z), z2 g ′′ (z); z) is univalent in U then Ω⊂ {ψ(g(z), zg′(z), z2 g ′′ (z); z) : z ∈ U}, implies q(z)≺ g(z). In the present investigation, the differential subordination result of Miller and Mocanu [8,Theorem 2.3b, p.28] is extended for functions associated with the integral operator Qα β ,p , and we obtain certain other related results. A similar problem for analytic functions was studied by Aghalary et al. [1], Ali et al. [2], Aouf [3], Aouf et al. [4], and Kim and Srivastava [6]. Additionally, the corresponding differential superordination problem is investigated, and several sandwich-type results are obtained. 2. Subordination Results Involving the Integral Operator Definition 3. Let Ω be a set in C and q(z) ∈ F0 ∩ H[0, p]. The class of admissible functions ΦQ � Ω,q � consists of those functions φ : C3 × U → C that satisfy the admissibility condition φ (u, v, w; z) /∈ Ω M. Aouf, T. Seoudy / Eur. J. Pure Appl. Math, 3 (2010), 26-44 29 whenever u = q (ζ) , v = kζq′ (ζ) + � α+ β − 1 � q (ζ) α+ β + p− 1 , ℜ ¨� α+ β + p− 1 �� α+ β + p− 2 � w − � α+ β − 1 �� α+ β − 2 � u � α+ β + p− 1 � v − � α+ β − 1 � u − 2 � α+ β � + 3 « ≥ kℜ ( 1+ ζq ′′ (ζ) q′ (ζ) ) , where z ∈ U , ζ ∈ ∂ U\E � q � , and k ≥ p. Theorem 1. Let φ ∈ ΦQ � Ω,q � . If f (z) ∈ A � p � satisfies n φ � Qαβ ,p f (z),Qα−1 β ,p f (z),Qα−2 β ,p f (z); z � : z ∈ U o ⊂ Ω � α > 2; β > −1; p ∈ N � , (5) then Qαβ ,p f (z) ≺ q (z) (z ∈ U) . Proof. Define the analytic function g(z) in U by g(z) = Qαβ ,p f (z) � α > 2; β > −1; p ∈ N; z ∈ U � . (6) In view of the relation (4) from (6), we get Qα−1 β ,p f (z) = zg′ (z) + � α+ β − 1 � g (z) α+β + p− 1 . (7) Further computations show that Qα−2 β ,p f (z) = z2 g ′′ (z) + 2 � α+ β − 1 � zg′ (z) + � α+ β − 1 �� α+ β − 2 � g (z) � α+ β + p− 1 �� α+ β + p− 2 � . (8) Define the transformations from C3 to C by u = r, v = s+ � α+ β − 1 � r α+ β + p− 1 , w = t + 2 � α+ β − 1 � s+ � α+ β − 1 �� α+ β − 2 � r � α+ β + p− 1 �� α+ β + p− 2 � . (9) Let ψ(r, s, t; z) = φ (u, v, w; z) = φ � r, s+ � α+ β − 1 � r α+ β + p− 1 , t + 2 � α+ β − 1 � s+ � α+ β − 1 �� α+ β − 2 � r � α+ β + p− 1 �� α+ β + p− 2 � ; z � . (10) M. Aouf, T. Seoudy / Eur. J. Pure Appl. Math, 3 (2010), 26-44 30 The proof shall make use of Lemma 1. Using equations (6), (7) and (8), from (10), we obtain ψ(p(z), zp′(z), z2p ′′ (z); z) = φ � Qαβ ,p f (z),Qα−1 β ,p f (z),Qα−2 β ,p f (z); z � (11) � α > 2; β > −1; p ∈ N; z ∈ U � . Hence (5) becomes ψ(p(z), zp′(z), z2p ′′ (z); z) ∈ Ω. The proof is completed if it can be shown that the admissibility condition for φ ∈ ΦQ � Ω,q � is equivalent to the admissibility condition for ψ as given in Definition 1. Note that t s + 1= � α+ β + p− 1 �� α+ β + p− 2 � w − � α+ β − 1 �� α+ β − 2 � u � α+ β + p− 1 � v − � α+ β − 1 � u − 2 � α+ β � + 3, and hence ψ ∈Ψp � Ω,q � . By Lemma 1, g(z)≺ q(z) or Qαβ ,p f (z)≺ q (z) (z ∈ U) . If Ω 6= C is a simply connected domain, then Ω = h(U) for some conformal mapping h(z) of U onto Ω. In this case the class ΦQ[h(U),q] is written as ΦQ[h,q]. The following result is an immediate consequence of Theorem 1. Theorem 2. Let φ ∈ ΦQ[h,q]. If f (z) ∈ A � p � satisfies φ � Qαβ ,p f (z),Qα−1 β ,p f (z),Qα−2 β ,p f (z); z � ≺ h(z) � α > 2;β > −1; p ∈ N; z ∈ U � , (12) then Qαβ ,p f (z) ≺ q (z) (z ∈ U) . Our next result is an extension of Theorem 1 to the case where the behavior of q(z) on ∂ U is not known. Corollary 1. Let Ω ⊂ C and let q(z) be univalent in U, q(0) = 0. Let φ ∈ ΦQ[Ω,qρ] for some ρ ∈ (0,1) where qρ(z) = q(ρz). If f (z) ∈ A � p � and φ � Qαβ ,p f (z),Qα−1 β ,p f (z),Qα−2 β ,p f (z); z � ∈ Ω � α > 2; β > −1; p ∈ N; z ∈ U � , then Qαβ ,p f (z) ≺ q (z) (z ∈ U) . Proof. Theorem 1 yields Qα β ,p f (z) ≺ qρ (z). The result is now deduced from qρ(z) ≺ q(z). Theorem 3. Let h(z) and q(z) be univalent in U with q(0) = 0 and set qρ(z) = q(ρz) and hρ(z) = h(ρz). Letφ : C3× U → C satisfy one of the following conditions: (1) φ ∈ ΦQ[h,qρ], for some ρ ∈ (0,1), or M. Aouf, T. Seoudy / Eur. J. Pure Appl. Math, 3 (2010), 26-44 31 (2) there exists ρ0 ∈ (0,1) such that φ ∈ ΦQ[hρ,qρ], for all ρ ∈ (ρ0, 1). If f (z) ∈ A � p � satisfies (12), then Qαβ ,p f (z) ≺ q (z) (z ∈ U) . Proof. The proof is similar to the proof of [8, Theorem 2.3d, p.30] and is therefore omitted. The next theorem yields the best dominant of the differential subordination (12). Theorem 4. Let h(z) be univalent in U. Let φ : C3 × U → C. Suppose that the differential equation φ(q(z), zq′(z), z2q ′′ (z); z) = h(z) (13) has a solution q(z) with q(0) = 0 and satisfy one of the following conditions: (1) q(z) ∈ F0 and φ ∈ ΦQ[h,q], (2) q(z) is univalent in U and φ ∈ ΦQ[h,qρ], for some ρ ∈ (0,1), or (3) q(z) is univalent in U and there exists ρ0 ∈ (0,1) such that φ ∈ ΦQ[hρ,qρ], for all ρ ∈ (ρ0, 1). If f (z) ∈ A � p � satisfies (12), then Qαβ ,p f (z) ≺ q (z) (z ∈ U) , and q(z) is the best dominant. Proof. Following the same arguments in [8, Theorem 2.3e, p. 31], we deduce that q(z) is a dominant from Theorems 2 and 3. Since q(z) satisfies (13) it is also a solution of (12) and therefore q(z) will be dominated by all dominants. Hence q(z) is the best dominant. In the particular case q(z) = Mz, M > 0, and in view of the Definition 1, the class of admissible functions ΦQ[Ω,q], denoted by ΦQ[Ω, M], is described below. Definition 4. Let Ω be a set in C and M > 0. The class of admissible functions ΦQ[Ω, M] consists of those functions φ : C3 × U → C such that φ � Meiθ , k+α+ β − 1 α+ β + p− 1 Meiθ , L + � 2 � α+ β − 1 � k+ � α+ β − 1 �� α+ β − 2 �� Meiθ � α+ β + p− 1 �� α+ β + p− 2 � ; z � /∈ Ω (14) whenever z ∈ U, θ ∈ R, ℜ � Le−iθ � ≥ (k− 1)kM for all real θ , α > 2, β > −1, p ∈ N and k ≥ p. Corollary 2. Let φ ∈ ΦQ[Ω, M]. If f (z) ∈ A � p � satisfies φ � Qαβ ,p f (z),Qα−1 β ,p f (z),Qα−2 β ,p f (z); z � ∈ Ω � α > 2; β > −1; p ∈ N; z ∈ U � , then � � �Qαβ ,p f (z) � � � < M (z ∈ U) . M. Aouf, T. Seoudy / Eur. J. Pure Appl. Math, 3 (2010), 26-44 32 In the special case Ω = q(U) = {ω : |ω| < M}, the class ΦQ[Ω, M] is simply denoted by ΦQ[M]. Corollary 3. Let φ ∈ ΦQ[M]. If f (z) ∈ A � p � satisfies � � �φ � Qαβ ,p f (z),Qα−1 β ,p f (z),Qα−2 β ,p f (z); z � � � � < M � α > 2;β > −1; p ∈ N; z ∈ U � , then � � �Qαβ ,p f (z) � � � < M (z ∈ U) . Remark 1. Putting M = 1 in the Corollary 3 we obtain the result obtained by Aouf [3, Theorem 2]. Corollary 4. If k ≥ p and f (z) ∈ A � p � satisfies � � �Qα−1 β ,p f (z) � � � < M � α > 1;β > −1; p ∈ N; z ∈ U � . then � � �Qαβ ,p f (z) � � � < M (z ∈ U) . Proof. This follows from Corollary 3 by taking φ (u, v, w; z) = v = k+α+β−1 α+β+p−1 Meiθ . Remark 2. For M = 1, Corollary 4 yields the result obtained by Aouf [3, Corollary 2]. Definition 5. Let Ω be a set in C and q(z) ∈ F0 ∩ H0. The class of admissible functions ΦQ,1 � Ω,q � consists of those functions φ : C3 × U → C that satisfy the admissibility condition: φ (u, v, w; z) /∈ Ω whenever u= q (ζ) , v = kζq′ (ζ) + � α+ β + p− 2 � q (ζ) α+ β + p− 1 , ℜ ¨� α+ β + p− 2 ��� α+ β + p− 1 � w − � α+ β + p− 3 � u � � α+ β + p− 1 � v − � α+ β + p− 2 � u − 2 � α+ β � + 5 « ≥ kℜ ( 1+ ζq ′′ (ζ) q′ (ζ) ) , where z ∈ U ,ζ ∈ ∂ U\E � q � ,α > 2,β > −1; p ∈ N and k ≥ 1. Theorem 5. Let φ ∈ ΦQ,1 � Ω,q � . If f (z) ∈ A � p � satisfies ( φ Qα β ,p f (z) zp−1 , Qα−1 β ,p f (z) zp−1 , Qα−2 β ,p f (z) zp−1 ; z ! : z ∈ U ) ⊂ Ω � α > 2;β > −1 � , (15) then Qα β ,p f (z) zp−1 ≺ q (z) (z ∈ U) . M. Aouf, T. Seoudy / Eur. J. Pure Appl. Math, 3 (2010), 26-44 33 Proof. Define an analytic function g(z) in U by g (z) = Qα β ,p f (z) zp−1 � α > 2; β > −1; p ∈ N; z ∈ U � . (16) By making use of (4) and (16), we get Qα−1 β ,p f (z) zp−1 = zg′ (z) + � α+ β + p− 2 � g (z) α+ β + p− 1 . (17) Further computations show that Qα−2 β ,p f (z) zp−1 = z2 g ′′ (z) + 2 � α+ β + p− 2 � zg′ (z) + � α+ β + p− 2 �� α+ β + p− 3 � g (z) � α+ β + p− 1 �� α+ β + p− 2 � . (18) Define the transformations from C3 to C by u = r, v = s+ � α+ β + p− 2 � r α+ β + p− 1 , w = t + 2 � α+ β + p− 2 � s+ � α+ β + p− 2 �� α+ β + p− 3 � r � α+ β + p− 1 �� α+β + p− 2 � . (19) Let ψ (r, s, t; z) = φ (u, v, w; z) = φ � r, s+ � α+ β + p− 2 � r α+β + p− 1 , t + 2 � α+ β + p− 2 � s+ � α+ β + p− 2 �� α+β + p− 3 � r � α+ β + p− 1 �� α+ β + p− 2 � ; z � . (20) The proof shall make use of Lemma 1. Using equations (16)-(18), and from (20), we obtain ψ � g (z) , zg′ (z) , z2 g ′′ (z) ; z � = φ � Qα β ,p f (z) zp−1 , Qα β ,p f (z) zp−1 , Qα β ,p f (z) zp−1 ; z � . (21) Hence (15) becomes ψ � g (z) , zg′ (z) , z2 g ′′ (z) ; z � ∈ Ω. The proof is completed if it can be shown that the admissibility condition for φ ∈ ΦQ,1 � Ω,q � is equivalent to the admissibility condition for ψ as given in Definition 1. Note that t s + 1= � α+ β + p− 2 ��� α+ β + p− 1 � w − � α+ β + p− 3 � u � � α+ β + p− 1 � v − � α+ β + p− 2 � u − 2 � α+ β � + 5, and hence ψ ∈Ψ � Ω,q � . By Lemma 1, g (z) ≺ q (z) or Qα β ,p f (z) zp−1 ≺ q (z) (z ∈ U) . M. Aouf, T. Seoudy / Eur. J. Pure Appl. Math, 3 (2010), 26-44 34 If Ω 6= C is a simply connected domain, then Ω = h(U), for some conformal mapping h(z) of U onto Ω. In this case the class ΦQ,1 � h(U) ,q � is written as ΦQ,1 � h,q � . In the particular case q(z) = Mz, M > 0, the class of admissible functions ΦQ,1 � Ω,q � , denoted by ΦQ,1 [Ω, M]. Proceeding similarly as in the previous section, the following result is an immediate con- sequence of Theorem 5. Theorem 6. Let φ ∈ ΦQ,1 � h,q � . If f (z) ∈ A � p � satisfies φ Qα β ,p f (z) zp−1 , Qα−1 β ,p f (z) zp−1 , Qα−2 β ,p f (z) zp−1 ; z ! ≺ h(z) � α > 2;β > −1; p ∈ N; z ∈ U � , (22) then Qα β ,p f (z) zp−1 ≺ q (z) (z ∈ U) . Definition 6. Let Ω be a set in C and M > 0. The class of admissible functions ΦQ,1 [Ω, M] consists of those functions φ : C3 × U → C such that φ � Meiθ , k+α+ β + p− 2 α+ β + p− 1 Meiθ , L + � α+ β + p− 2 �� 2k+α+ β + p− 3 � Meiθ � α+ β + p− 1 �� α+ β + p− 2 � ; z � /∈ Ω (23) whenever z ∈ U, θ ∈ R, ℜ � Le−iθ � ≥ (k− 1)kM for all real θ , p ∈ N and k ≥ 1. Corollary 5. Let φ ∈ ΦQ,1 [Ω, M]. If f (z) ∈ A � p � satisfies φ Qα β ,p f (z) zp−1 , Qα−1 β ,p f (z) zp−1 , Qα−2 β ,p f (z) zp−1 ; z ! ∈ Ω � α > 2;β > −1; p ∈ N; z ∈ U � , then � � � � � Qα β ,p f (z) zp−1 � � � � � < M (z ∈ U) . In the special case Ω = {ω : |ω| < M}, the class ΦQ,1 [Ω, M] is simply denoted by ΦQ,1 [M]. Corollary 6. Let φ ∈ ΦQ,1 [M]. If f (z) ∈ A � p � satisfies � � � � � φ Qα β ,p f (z) zp−1 , Qα−1 β ,p f (z) zp−1 , Qα−2 β ,p f (z) zp−1 ; z ! � � � � � < M � α > 2; β > −1; p ∈ N; z ∈ U � , then � � � � � Qα β ,p f (z) zp−1 � � � � � < M (z ∈ U) . M. Aouf, T. Seoudy / Eur. J. Pure Appl. Math, 3 (2010), 26-44 35 Corollary 7. If k ≥ 1 and f (z) ∈ A � p � satisfies � � � � � Qα−1 β ,p f (z) zp−1 � � � � � < M � α > 1; β > −1; p ∈ N; z ∈ U � . then � � � � � Qα β ,p f (z) zp−1 � � � � � < M (z ∈ U) . Proof. This follows from Corollary 6 by taking φ(u, v, w; z) = v = k+α+β+p−2 α+β+p−1 Meiθ . Definition 7. Let Ω be a set in C and q(z) ∈ F1∩H. The class of admissible functions ΦQ,2 � Ω,q � consists of those functions φ : C3 × U → C that satisfy the admissibility condition φ (u, v, w; z) /∈ Ω whenever u = q (ζ) , v = 1 α+ β + p− 2 ¨ −1+ � α+β + p− 1 � q (ζ) + kζq′ (ζ) q (ζ) « , ℜ � �� α+ β + p− 3 � w − � α+ β + p− 2 � v+ 1 � v � α+ β + p− 2 � v − � α+ β + p− 1 � u+ 1 + � α+ β + p− 2 � v− 2 � α+ β + p− 1 � u+ 1 � ≥ kℜ ( 1+ ζq ′′ (ζ) q′ (ζ) ) , where z ∈ U ,ζ ∈ ∂ U\E � q � , p ∈ N and k ≥ 1. Theorem 7. Let φ ∈ ΦQ,2 � Ω,q � and Qα β ,p f (z) 6= 0. If f (z) ∈ A � p � satisfies ( φ Qα−1 β ,p (z) Qα β ,p f (z) , Qα−2 β ,p f (z) Qα−1 β ,p f (z) , Qα−3 β ,p f (z) Qα−2 β ,p f (z) ; z ! : z ∈ U ) ⊂ Ω � α > 3;β > −1; p ∈ N � , (24) then Qα−1 β ,p f (z) Qα β ,p f (z) ≺ q (z) (z ∈ U) . Proof. Define an analytic function g(z) in U by g (z) = Qα−1 β ,p f (z) Qα β ,p f (z) � α > 3;β > −1; p ∈ N; z ∈ U � . (25) M. Aouf, T. Seoudy / Eur. J. Pure Appl. Math, 3 (2010), 26-44 36 Using (25), we get zg′ (z) g (z) = z � Qα−1 β ,p f (z) �′ Qα−1 β ,p f (z) − z � Qα β ,p f (z) �′ Qα β ,p f (z) . (26) By making use of (4) in (26), we get Qα−2 β ,p f (z) Qα−1 β ,p f (z) = 1 α+ β + p− 2 ¨ −1+ � α+ β + p− 1 � g (z) + zg′ (z) g (z) « . (27) Further computations show that Qα−3 β ,p f (z) Qα−2 β ,p f (z) = 1 α+ β + p− 2 ¨ −2+ � α+ β + p− 1 � g (z) + zg′ (z) g (z) + � α+ β + p− 1 � zg′ (z) + zg ′(z) g(z) + z2 g ′′ (z) g(z) − � zg ′(z) g(z) �2 −1+ � α+ β + p− 1 � g (z) + zg ′(z) g(z)    . (28) Define the transformations from C3 to C by u = r, v = 1 α+ β + p− 2 § −1+ � α+ β + p− 1 � r + s r ª , (29) w = 1 α+ β + p− 2    −2+ � α+ β + p− 1 � r + s r + � α+ β + p− 1 � s+ s r + l r − � s r �2 −1+ � α+ β + p− 1 � r + s r    . Let ψ (r, s, t; z) = φ (u, v, w; z) = φ � r, 1 α+ β + p− 2 § −1+ � α+β + p− 1 � r + s r ª , 1 α+ β + p− 2    −2+ � α+ β + p− 1 � r + s r + � α+ β + p− 1 � s+ s r + l r − � s r �2 −1+ � α+ β + p− 1 � r + s r    ; z � . (30) The proof shall make use of Lemma 1. Using equations (25), (27) and (28), from (30), we obtain ψ(p(z), zp′(z), z2p ′′ (z); z) = φ Qα−1 β ,p (z) Qα β ,p f (z) , Qα−2 β ,p f (z) Qα−1 β ,p f (z) , Qα−3 β ,p f (z) Qα−2 β ,p f (z) ; z ! . (31) M. Aouf, T. Seoudy / Eur. J. Pure Appl. Math, 3 (2010), 26-44 37 Hence (24) becomes ψ(p(z), zp′(z), z2p ′′ (z); z) ∈ Ω. The proof is completed if it can be shown that the admissibility condition for φ ∈ ΦI ,2 � Ω,q � is equivalent to the admissibility condition for ψ as given in Definition 1. Note that t s + 1 = �� α+ β + p− 3 � w − � α+ β + p− 2 � v + 1 � v � α+ β + p− 2 � v− � α+ β + p− 1 � u+ 1 + � α+ β + p− 2 � v− 2 � α+ β + p− 1 � u+ 1, and hence ψ ∈Ψ � Ω,q � . By Lemma 1, g(z)≺ q(z) or Qα−1 β ,p (z) Qα β ,p f (z) ≺ q(z) (z ∈ U) . If Ω 6= C is a simply connected domain, then Ω = h(U), for some conformal mapping h(z) of U onto Ω. In this case the class ΦQ,2 � h(U) ,q � is written as ΦQ,2 � h,q � . In the particular case q(z) = Mz, M > 0, the class of admissible functions ΦQ,2 � Ω,q � becomes the class ΦQ,2 [Ω, M]. Proceeding similarly as in the previous section, the following result is an immediate con- sequence of Theorem 7. Theorem 8. Let φ ∈ ΦQ,2 � h,q � . If f (z) ∈ A � p � satisfies φ Qα−1 β ,p (z) Qα β ,p f (z) , Qα−2 β ,p f (z) Qα−1 β ,p f (z) , Qα−3 β ,p f (z) Qα−2 β ,p f (z) ; z ! ≺ h(z) � α > 3, β > −1; p ∈ N; z ∈ U � , (32) then Qα−1 β ,p (z) Qα β ,p f (z) ≺ q (z) (z ∈ U) . Definition 8. Let Ω be a set in C and M > 0. The class of admissible functions ΦQ,2 [Ω, M] consists of those functions φ : C3 × U → C such that φ � Meiθ , k− 1+ � α+ β + p− 1 � Meiθ α+ β + p− 2 , 1 α+ β + p− 2 ¦ k− 2+ � α+ β + p− 1 � Meiθ + � α+ β + p− 1 � kM2eiθ + kM + Le−iθ − k2M (k− 1)M + � α+ β + p− 1 � M2eiθ « ; z � /∈ Ω, (33) whenever z ∈ U , θ ∈ R, ℜ � Le−iθ � ≥ (k− 1)kM for all real θ , α > 3,β > −1, p ∈ N and k ≥ 1. M. Aouf, T. Seoudy / Eur. J. Pure Appl. Math, 3 (2010), 26-44 38 Corollary 8. Let φ ∈ ΦQ,2 [Ω, M]. If f (z) ∈ A � p � satisfies φ Qα−1 β ,p (z) Qα β ,p f (z) , Qα−2 β ,p f (z) Qα−1 β ,p f (z) , Qα−3 β ,p f (z) Qα−2 β ,p f (z) ; z ! ∈ Ω � α > 3;β > −1; p ∈ N; z ∈ U � , then � � � � � Qα−1 β ,p (z) Qα β ,p f (z) � � � � � < M (z ∈ U) . In the special caseΩ = q (U) = {ω : |ω| < M}, the classΦQ,2 [Ω, M] is denoted by ΦQ,2 [M]. Corollary 9. Let φ ∈ ΦQ,2 [M]. If f (z) ∈ A � p � satisfies � � � � � φ Qα−1 β ,p (z) Qα β ,p f (z) , Qα−2 β ,p f (z) Qα−1 β ,p f (z) , Qα−3 β ,p f (z) Qα−2 β ,p f (z) ; z ! � � � � � < M � α > 3; β > −1; p ∈ N; z ∈ U � , then � � � � � Qα−1 β ,p (z) Qα β ,p f (z) � � � � � < M (z ∈ U) . Remark 3. The result in the Corollary 9 is extension of the result obtained by Aouf [3, Theorem 4]. 3. Superordination of the Integral Operator The dual problem of differential subordination, that is, differential superordination of the integral operator Qα β ,p is investigated in this section. For this purpose the class ofvadmissible functions is given in the following definition. Definition 9. Let Ω be a set in C and q(z) ∈ H[0, p] with zq′(z) 6= 0. The class of admissible functions Φ′Q � Ω,q � consists of those functions φ : C3 × Ū → C that satisfy the admissibility condition: φ (u, v, w;ζ) ∈ Ω whenever u= q (z) , v = ζq′ (z) +m � α+ β − 1 � q (z) m � α+ β + p− 1 � , ℜ ¨� α+ β + p− 1 �� α+ β + p− 2 � w − � α+ β − 1 �� α+ β − 2 � u � α+ β + p− 1 � v − � α+ β − 1 � u − 2 � α+ β � + 3 « ≤ 1 m ℜ ( 1+ zq ′′ (z) q′ (z) ) , where z ∈ U , ζ ∈ ∂ U , α > 2,β > −1, p ∈ N and m ≥ p. M. Aouf, T. Seoudy / Eur. J. Pure Appl. Math, 3 (2010), 26-44 39 Theorem 9. Let φ ∈ Φ′Q � Ω,q � . If f (z) ∈ A � p � , Qα β ,p f (z) ∈ F0 and φ � Qαβ ,p f (z),Qα−1 β ,p f (z),Qα−2 β ,p f (z); z � is univalent in U, then Ω⊂ n φ � Qαβ ,p f (z),Qα−1 β ,p f (z),Qα−2 β ,p f (z); z � : z ∈ U o � α > 2;β > −1; p ∈ N � , (34) implies q (z) ≺ Qαβ ,p f (z) (z ∈ U) . Proof. From (11) and (34), we have Ω⊂ ¦ ψ(g(z), zg′(z), z2 g ′′ (z); z) : z ∈ U © . From (9), we see that the admissibility condition for φ ∈ Φ′Q � Ω,q � is equivalent to the admissibility condition for ψ as given in Definition 2. Hence ψ ∈Ψ′p � Ω,q � , and by Lemma 2, q(z)≺ g(z) or q (z) ≺ Qαβ ,p f (z) (z ∈ U) . If Ω 6= C is a simply connected domain, then Ω = h(U) for some conformal mapping h(z) of U onto Ω. In this case the class Φ′Q � h(U) ,q � is written as Φ′Q � h,q � . Proceeding similarly as in the previous section, the following result is an immediate con- sequence of Theorem 9. Theorem 10. Let h(z) is analytic on U and φ ∈ Φ′Q � h,q � . If f (z) ∈ A � p � , Qα β ,p f (z) ∈ F0 and φ � Qαβ ,p f (z),Qα−1 β ,p f (z),Qα−2 β ,p f (z); z � is univalent in U, then h(z) ≺ φ � Qαβ ,p f (z),Qα−1 β ,p f (z),Qα−2 β ,p f (z); z � � α > 2; β > −1; p ∈ N; z ∈ U � , (35) implies q (z) ≺ Qαβ ,p f (z) (z ∈ U) . Theorems 9 and 10 can only be used to obtain subordinants of differential superordination of the form (34) or (35). The following theorem proves the existence of the best subordinant of (35) for certain φ. Theorem 11. Let h(z) be analytic in U and φ : C3 × Ū → C. Suppose that the differential equation φ � q(z), zq′(z), z2q ′′ (z); z � = h(z) has a solution q(z) ∈ F0. If φ ∈ Φ′Q � h,q � , f (z) ∈ A � p � , Qα β ,p f (z) ∈ F0 and φ � Qαβ ,p f (z),Qα−1 β ,p f (z),Qα−2 β ,p f (z); z � M. Aouf, T. Seoudy / Eur. J. Pure Appl. Math, 3 (2010), 26-44 40 is univalent in U, then h(z) ≺ φ � Qαβ ,p f (z),Qα−1 β ,p f (z),Qα−2 β ,p f (z); z � � α > 2; β > −1; p ∈ N; z ∈ U � implies q (z) ≺ Qαβ ,p f (z) (z ∈ U) . and q(z) is the best subordinant. Proof. The proof is similar to the proof of Theorem 4 and is therefore omitted. Combining Theorems 2 and 10, we obtain the following sandwich-type theorem. Corollary 10. Let h1(z) and q1(z) be analytic functions in U, h2(z) be univalent function in U, q2(z) ∈ F0 with q1(0) = q2(0) = 0 and φ ∈ ΦQ � h2,q2 � ∩ Φ′Q � h1,q1 � . If f (z) ∈ A � p � , Qα β ,p f (z) ∈ H[0, p] ∩F0 and φ � Qαβ ,p f (z),Qα−1 β ,p f (z),Qα−2 β ,p f (z); z � � α > 2; β > −1; p ∈ N; z ∈ U � is univalent in U, then h1(z) ≺ φ � Qαβ ,p f (z),Qα−1 β ,p f (z),Qα−2 β ,p f (z); z � ≺ h2(z) � α > 2; p ∈ N; z ∈ U � , implies q1(z) ≺ Qαβ ,p f (z) ≺ q2(z) (z ∈ U) . Definition 10. Let Ω be a set in C and q(z) ∈ H0 with zq′(z) 6= 0. The class of admissible functions Φ′Q,1 � Ω,q � consists of those functions φC3 × Ū → C that satisfy the admissibility condition: φ (u, v, w;ζ) ∈ Ω (36) whenever u = q (z) , v = zq′ (z) +m � α+β + p− 2 � q (z) m � α+ β + p− 1 � , ℜ ¨� α+ β + p− 2 ��� α+ β + p− 1 � w − � α+ β + p− 3 � u � � α+ β + p− 1 � v − � α+ β + p− 2 � u − 2 � α+ β � + 5 « ≤ 1 m ℜ ( 1+ zq ′′ (z) q′ (z) ) , where z ∈ U , ζ ∈ ∂ U and m ≥ 1 Now we will give the dual result of Theorem 5 for differential superordination. M. Aouf, T. Seoudy / Eur. J. Pure Appl. Math, 3 (2010), 26-44 41 Theorem 12. Let φ ∈ Φ′Q,1 � Ω,q � . If f (z) ∈ A � p � , Qα β ,p f (z) zp−1 ∈ F0 and φ Qα β ,p f (z) zp−1 , Qα−1 β ,p f (z) zp−1 , Qα−2 β ,p f (z) zp−1 ; z ! is univalent in U, then Ω⊂ ( φ Qα β ,p f (z) zp−1 , Qα−1 β ,p f (z) zp−1 , Qα−2 β ,p f (z) zp−1 ; z ! : z ∈ U ) � α > 2;β > −1; p ∈ N � (37) implies q(z) ≺ Qα β ,p f (z) zp−1 (z ∈ U) . Proof. From (21) and (37), we have Ω⊂ ¦ ψ � g (z) , zg′ (z) , z2 g ′′ (z) ; z � : z ∈ U © � α > 2;β > −1; p ∈ N � . From (19), we see that the admissibility condition for φ ∈ Φ′Q,1 � Ω,q � is equivalent to the admissibility condition for ψ as given in Definition 2. Hence ψ ∈Ψ′ � Ω,q � , and by Lemma 2 q(z) ≺ p(z) or q(z) ≺ Qα β ,p f (z) zp−1 � α > 2;β > −1; p ∈ N; z ∈ U � . If Ω 6= C is a simply connected domain, and Ω = h(U) for some conformal mapping h(z) of U onto Ω and the class Φ′Q,1 � h(U) ,q � is written as Φ′Q,1 � h,q � . Proceeding similarly as in the previous section, the following result is an immediate con- sequence of Theorem 12. Theorem 13. Let q(z) ∈ H0, h(z) is analytic on U and φ ∈ Φ′Q,1 � h,q � . If f (z) ∈ A � p � , Qα β ,p f (z) zp−1 ∈ F0 and φ Qα β ,p f (z) zp−1 , Qα−1 β ,p f (z) zp−1 , Qα−2 β ,p f (z) zp−1 ; z ! is univalent in U, then h(z) ≺ φ Qα β ,p f (z) zp−1 , Qα−1 β ,p f (z) zp−1 , Qα−2 β ,p f (z) zp−1 ; z ! � α > 2; β > −1; p ∈ N; z ∈ U � (38) implies q(z)≺ Qα β ,p f (z) zp−1 (z ∈ U) . Combining Theorems 6 and 13, we obtain the following sandwich-type theorem. M. Aouf, T. Seoudy / Eur. J. Pure Appl. Math, 3 (2010), 26-44 42 Corollary 11. Let h1(z) and q1(z) be analytic functions in U, h2(z) be univalent function in U, q2(z) ∈ F0 with q1(0) = q2(0) = 0 and φ ∈ ΦQ,1 � h2,q2 � ∩ Φ′Q,1 � h1,q1 � . If f (z) ∈ A � p � , Qα β ,p f (z) zp−1 ∈ H0 ∩F0 and φ Qα β ,p f (z) zp−1 , Qα−1 β ,p f (z) zp−1 , Qα−2 β ,p f (z) zp−1 ; z ! is univalent in U, then h1 (z) ≺ φ Qα β ,p f (z) zp−1 , Qα−1 β ,p f (z) zp−1 , Qα−2 β ,p f (z) zp−1 ; z ! ≺ h2 (z) � α > 2;β > −1; p ∈ N; z ∈ U � , implies q1 (z) ≺ Qα β ,p f (z) zp−1 ≺ q2 (z) (z ∈ U) . Definition 11. Let Ω be a set in C, q(z) 6= 0, zq′(z) 6= 0 and q(z) ∈ H. The class of admissible functions φ ∈ Φ′Q,2 � Ω,q � consists of those functions φ : C3×Ū → C that satisfy the admissibility condition: φ (u, v, w;ζ) ∈ Ω whenever u = q (z) , v = 1 α+ β + p+ 2 ¨ −1+ � α+ β + p− 1 � g (z) + zg′ (z) mg (z) « , ℜ � �� α+ β + p− 3 � w − � α+ β + p− 2 � v + 1 � v � α+ β + p− 2 � v− � α+ β + p− 1 � u+ 1 + � α+ β + p− 2 � v − 2 � α+β + p− 1 � u+ 1 � ≤ 1 m ℜ ( 1+ zq ′′ (z) q′ (z) ) , where z ∈ U , ζ ∈ ∂ U , p ∈ N and m≥ 1. Now we will give the dual result of Theorem 7 for the differential superordination. Theorem 14. Let φ ∈ Φ′I ,2 � Ω,q � . If f (z) ∈ A � p � , Qα−1 β ,p f (z) Qα β ,p f (z) ∈ F1 and φ Qα−1 β ,p f (z) Qα β ,p f (z) , Qα−2 β ,p f (z) Qα−1 β ,p f (z) , Qα−3 β ,p f (z) Qα−2 β ,p f (z) ; z ! M. Aouf, T. Seoudy / Eur. J. Pure Appl. Math, 3 (2010), 26-44 43 is univalent in U, then Ω⊂ ( φ Qα−1 β ,p f (z) Qα β ,p f (z) , Qα−2 β ,p f (z) Qα−1 β ,p f (z) , Qα−3 β ,p f (z) Qα−2 β ,p f (z) ; z ! : z ∈ U ) � α > 3; β > −1; p ∈ N � (39) implies q (z) ≺ Qα−1 β ,p f (z) Qα β ,p f (z) (z ∈ U) . Proof. From (31) and (39), we have Ω⊂ ¦ ψ � g (z) , zg′ (z) , z2 g ′′ (z) ; z � : z ∈ U © . In view of (29), the admissibility condition for φ ∈ Φ′Q,2 � Ω,q � is equivalent to the admissibil- ity condition for ψ as given in Definition 2. Hence ψ ∈Ψ′ � Ω,q � , and by Lemma 2 q(z)≺ g(z) or q(z) ≺ Qα−1 β ,p f (z) Qα β ,p f (z) (z ∈ U) . If Ω 6= C is a simply connected domain, then Ω = h(U) for some conformal mapping h(z) of U onto Ω. In this case the class Φ′Q,2 � h(U) ,q � is written as Φ′Q,2 � h,q � . Proceeding similarly as in the previous section, The following result is an immediate con- sequence of Theorem 14. Theorem 15. Let q (z) ∈ H, h(z) be analytic in U and φ ∈ Φ′Q,2 � h,q � . If f (z) ∈ A � p � , Qα−1 β ,p f (z) Qα β ,p f (z) ∈ F1 and φ Qα−1 β ,p f (z) Qα β ,p f (z) , Qα−2 β ,p f (z) Qα−1 β ,p f (z) , Qα−3 β ,p f (z) Qα−2 β ,p f (z) ; z ! is univalent in U, then h(z) ≺ φ Qα−1 β ,p f (z) Qα β ,p f (z) , Qα−2 β ,p f (z) Qα−1 β ,p f (z) , Qα−3 β ,p f (z) Qα−2 β ,p f (z) ; z ! � α > 3;β >−1; p ∈ N; z ∈ U � , (40) implies q (z) ≺ Qα−1 β ,p f (z) Qα β ,p f (z) (z ∈ U) . Combining Theorems 8 and 15, we obtain the following sandwich-type theorem. REFERENCES 44 Corollary 12. Let h1(z) and q1(z) be analytic functions in U, h2(z) be univalent function in U, q2(z) ∈ F1 with q1(0) = q20) = 1 and φ ∈ ΦQ,2 � h2,q2 � ∩ Φ′Q,2 � h1,q1 � . 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