11_701_aouf.dvi EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS Vol. 3, No. 6, 2010, 1070-1085 ISSN 1307-5543 – www.ejpam.com SPECIAL ISSUE ON COMPLEX ANALYSIS: THEORY AND APPLICATIONS DEDICATED TO PROFESSOR HARI M. SRIVASTAVA, ON THE OCCASION OF HIS 70TH BIRTHDAY Differential Subordination and Superordination on p-Valent Meromorphic Functions Defined by Extended Multiplier Transformations R. M. El-Ashwah 1,∗, M. K. Aouf 2 1 Department of Mathematics, Faculty of Science (Damietta Branch), Mansoura University, New Damietta 34517, Egypt 2 Department of Mathematics, Faculty of Science, Mansoura University, Mansoura 35516, Egypt Abstract. In this paper we derive some differential subordination and superordination results for p- valent meromorphic functions in the punctured unit disc, which are acted upon by a class of extended multiplier transformations. These results are obtained by investigating appropriate classes of admissi- ble functions. Sandwich-type results are also obtained. 2000 Mathematics Subject Classifications: 30C45 Key Words and Phrases: Meromorphic functions, extended multiplier transformations, sandwich the- orems 1. Introduction Let H(U) be the class of analytic functions in the open unit disc 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 H = H[1,1]. If f (z) and g(z) are members of H(U), we say that f (z) is subordinate to g(z) written symbolically as follows: f ≺ g or f (z)≺ g(z) (z ∈ U), ∗Corresponding author. Email addresses: r_elashwah�yahoo. om (R. El-Ashwah), mkaouf127�yahoo. om (M. Aouf) http://www.ejpam.com 1070 c© 2010 EJPAM All rights reserved. R. El-Ashwah, M. Aouf / Eur. J. Pure Appl. Math, 3 (2010), 1070-1085 1071 if there exists a Schwarz function w(z), which (by definition) is analytic in U with w(0) = 0 and |w(z)| < 1 (z ∈ U) such that f (z) = g(w(z)) (z ∈ U). Indeed it is known that f (z) ≺ g(z) (z ∈ U) ⇒ f (0) = g(0) and f (U) ⊂ g(U). Further, if the function g(z) is univalent in U , then we have the following equivalent (cf., e.g., [13]; see also [14, p.4]) f (z) ≺ g(z) ⇔ f (0) = g(0) and f (U)⊂ g(U). Denote by D the set of all functions q(z) that are analytic and injective on U\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 of D for which q(0) = a be denoted by D(a), and D(1) = D1. The following classes of admissible functions will be required. Definition 1 (14, Definition 2.3a, p. 27). Let Ω be a set in C,q ∈ Dand n be a positive integer. The class of admissible functions Ψn[Ω,q] consists of these functions ψ : C3×U → C that satisfy the admissibility condition ψ(r, s, t; z) /∈ Ω whenever r = q(ζ), s = kζq ′ (ζ) and Re § t s + 1 ª ≥ kRe ( 1+ ζq ′′ (ζ) q ′ (ζ) ) , where z ∈ U , ζ ∈ ∂ U\E(q) and k ≥ n. We write Ψ1[Ω,q] as Ψ[Ω,q]. In particular when q(z) = M Mz + a M + az , with M > 0 and |a| < M , then q(U) = UM = {w : |w| < M} , q(0) = a, E(q) = φ and q ∈ D(a). 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 (15, Definition 3, p. 817). Let Ω be a set in C,q ∈ H[a, n] with q ′ (z) 6= 0. The class of admissible functions Ψ ′ n[Ω,q] consists of these functions ψ : C3× U → C that satisfy the admissibility condition ψ(r, s, t;ζ) ∈ Ω whenever r = q(z), s = zq ′ (z) m , and Re § t s + 1 ª ≤ 1 m Re ( 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]. In our investigations we shall need the following lemmas. Lemma 1 (14, Theorem 2.3b, p. 28). Let ψ ∈ Ψn[Ω,q] with q(0) = a. If the analytic function p(z) = a+ anzn + an+1zn+1 + . . . satisfies ψ(p(z), zp ′ (z), z2p ′′ (z); z) ∈ Ω, then p(z) ≺ q(z). R. El-Ashwah, M. Aouf / Eur. J. Pure Appl. Math, 3 (2010), 1070-1085 1072 Lemma 2 (15, Theorem 1, p. 818). Let ψ ∈ Ψ ′ n[Ω,q] with q(0) = a. If p(z) ∈ D(a) and ψ(p(z), zp ′ (z), z2p ′′ (z); z) is univalent in U then Ω⊂ ¦ ψ(p(z), zp ′ (z), z2p ′′ (z); z) : z ∈ U © implies q(z)≺ p(z). Let ∑ (p) denote the class of functions of the form: f (z) = z−p + ∞ ∑ k=1−p akzk (p ∈ N = {1,2, . . . .}; z ∈ U∗ = U\{0}), (1) which are analytic and p-valent in U∗. For functions f j(z) ∈ ∑ (p), given by f j(z) = z−p + ∞ ∑ k=1−p ak, jz k ( j = 1,2), (2) we define the Hadamard product (or convolution) of f1(z) and f2(z) by ( f1 ∗ f2)(z) = z−p + ∞ ∑ k=1−p ak,1ak,2zk = ( f2 ∗ f1)(z). (3) Now, using the linear operator Im p (λ,ℓ) (λ ≥ 0,ℓ > 0, m ∈ N0 = N ⋃ {0}) introduced by El-Ashwah [9] for a function f (z) ∈ ∑ (p) given by (1) as follows: Im p (λ,ℓ) f (z) = z−p + ∞ ∑ k=1−p � ℓ+λ(k+ p) ℓ �m akzk, (4) we can write (4) in the form: Im p (λ,ℓ) f (z) = (Φ p,m λ,ℓ ∗ f )(z), where Φ p,m λ,ℓ (z) = z−p + ∞ ∑ k=1−p � ℓ+λ(k+ p) ℓ �m zk. (5) It is easily verified from (4) that λz(Im p (λ,ℓ) f (z)) ′ = ℓIm+1 p (λ,ℓ) f (z)− (λp+ ℓ)Im p (λ,ℓ) f (z) (λ > 0). (6) We note that: I0 p(λ,ℓ) f (z) = f (z) and I1 p(1,1) f (z) = (zp+1 f (z)) ′ zp = (p+ 1) f (z) + z f ′ (z). Also by specializing the parameters λ,ℓ and p, we obtain the following operators studied by various authors: R. El-Ashwah, M. Aouf / Eur. J. Pure Appl. Math, 3 (2010), 1070-1085 1073 (i) Im 1 (1,ℓ) f (z) = I(m,ℓ) f (z) (see Cho et al. [7,8] ); (ii) Im p (1,1) f (z) = Dm p f (z) (see Aouf and Hossen [6], Liu and Owa [11], Liu and Srivastava [12] and Srivastava and Patel [16]); (iii) Im 1 (1,1) f (z) = Im f (z) (see Uralegaddi and Somanatha [17]). Also we note that: (i) Im p (1,ℓ) f (z) = Ip(m,ℓ) f (z), where Ip(m,ℓ) f (z) is defined by Ip(m,ℓ) f (z) = z−p + ∞ ∑ k=1−p � ℓ+ k+ p ℓ �m akzk (ℓ > 0; m ∈ N0); (7) (ii) Im p (λ, 1) f (z) = Dm λ,p f (z), where Dm λ,p f (z) is defined by Dm λ,p f (z) = z−p + ∞ ∑ k=1−p � 1+λ(k+ p) �m akzk (λ≥ 0; m ∈ N0). (8) Aghalary et al. [1,2], Ali et al. [3,4,5], Aouf and Hossen [6] and Kim and Srivestava [10] obtained sufficient conditions for certain differential subordination implications to hold. In the present paper, the differential subordination result of Miller and Mocanu [14, The- orem 2.3b, p. 28] is extended for functions associated with the operator Im p (λ,ℓ), and we obtain certain other related results. Additionally, the corresponding differential superordina- tion problem is investigated, and several sandwich-type results are obtained. 2. Subordination Results Involving the Operator I m p (λ,ℓ) Unless otherwise mentioned, we assume throughout this paper that ℓ > 0, λ > 0, p ∈ N and m ∈ N0. Definition 3. Let Ω be a set in C and q(z) ∈ D1 ∩H. The class of admissible functions ΦH[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) + � ℓ λ � q(ζ) � ℓ λ � , Re (� ℓ λ � (w − u) v− u − 2 � ℓ λ � ) ≥ kRe ( 1+ ζq ′′ (ζ) q ′ (ζ) ) , where z ∈ U , ζ ∈ ∂ U\E(q) and k ≥ 1. R. El-Ashwah, M. Aouf / Eur. J. Pure Appl. Math, 3 (2010), 1070-1085 1074 Theorem 1. Let ϕ ∈ ΦH[Ω,q]. If f (z) ∈ ∑ (p) satisfies n ϕ(zp Im p (λ,ℓ) f (z), zp Im+1 p (λ,ℓ) f (z), zp Im+2 p (λ,ℓ) f (z) ; z) : z ∈ U o ∈ Ω, (9) then zp Im p (λ,ℓ) f (z) ≺ q(z). Proof. Define the analytic function p(z) in U by p(z) = zp Im p (λ,ℓ) f (z). (10) From (6) and (10), we have zp Im+1 p (λ,ℓ) f (z) = � zp ′ (z) + � ℓ λ � p(z) � � ℓ λ � . (11) Further computations show that zp Im+2 p (λ,ℓ) f (z) = z2p ′′ (z) + � 1+ 2 � ℓ λ �� zp ′ (z) + � ℓ λ �2 p(z) � ℓ λ �2 . (12) Define the transformations from C3 to C by u(r, s, t) = r, v(r, s, t) = s+ � ℓ λ � r � ℓ λ � , w(r, s, t) = t + � 1+ 2 � ℓ λ �� s+ � ℓ λ �2 r � ℓ λ �2 . (13) Let ψ(r, s, t; z) = ϕ(u, v, w; z) = ϕ   r, s+ � ℓ λ � r � ℓ λ � , t + � 1+ 2 � ℓ λ �� s+ � ℓ λ �2 r � ℓ λ �2 ; z    . (14) The proof will make use of Lemma 1. Using (10), (11) and (12), from (14), we obtain ψ(p(z), zp ′ (z), z2p ′′ (z); z) = ϕ � zp Im p (λ,ℓ) f (z), zp Im+1 p (λ,ℓ) f (z), zp Im+2 p (λ,ℓ) f (z); z � . (15) Hence (9) 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 ∈ ΦH[Ω,q] is equivalent to the admissibility condition for ψ as given in Definition 1. Note that t s + 1= � ℓ λ � (w − u) v − u − 2 � ℓ λ � , R. El-Ashwah, M. Aouf / Eur. J. Pure Appl. Math, 3 (2010), 1070-1085 1075 and hence ψ ∈Ψ[Ω,q]. By Lemma 1, p(z) ≺ q(z) or zp Im p (λ,ℓ) f (z)≺ q(z). 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 ΦH[h(U),q] is written as ΦH[h,q]. The following result is an immediate consequence of Theorem 1. Theorem 2. Let ϕ ∈ ΦH[h,q] with q(0) = 1. If f (z) ∈ ∑ (p) satisfies ϕ(zp Im p (λ,ℓ) f (z) , zp Im+1 p (λ,ℓ) f (z), zp Im+2 p (λ,ℓ) f (z); z)≺ h(z), (16) then zp Im p (λ,ℓ) f (z) ≺ q(z). 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) = 1. Let ϕ ∈ ΦH[Ω,qρ] for some ρ ∈ (0,1), where, qρ(z) = q(ρz). If f ∈ ∑ (p) and ϕ(zp Im p (λ,ℓ) f (z) , zp Im+1 p (λ,ℓ) f (z) , zp Im+2 p (λ,ℓ) f (z); z) ∈ Ω, then zp Im p (λ,ℓ) f (z) ≺ q(z). Proof. Theorem 1 yields zp Im p (λ,ℓ) 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) = 1 and set qρ(z) = q(ρz) and hρ(z) = h(ρz). Let ϕ : C3× U → C satisfy one of the following conditions: (1) ϕ ∈ ΦH[h,qρ], for some ρ ∈ (0,1), or (2) there exists ρ0 ∈ (0,1) such that ϕ ∈ ΦH[hρ,qρ], for all ρ ∈ (ρ0, 1). If f (z) ∈ ∑ (p) satisfies (16), then zp Im p (λ,ℓ) f (z) ≺ q(z). Proof. The proof is similar to [14, Theorem 2.3d, p. 30] and is therefore omitted. The next theorem yields the best dominant of the differential subordination (16). Theorem 4. Let h(z) be univalent in U, andϕ : C3 × U → C. Suppose that the differential equation ϕ � p(z), zp ′ (z)+ � ℓ λ � p(z) � ℓ λ � , z2p ′′ (z)+ � 1+2 � ℓ λ �� zp ′ (z)+ � ℓ λ �2 p(z) � ℓ λ �2 ; z � = h(z) (17) has a solution q(z) with q(0) = 1 and satisfy one of the following conditions: R. El-Ashwah, M. Aouf / Eur. J. Pure Appl. Math, 3 (2010), 1070-1085 1076 (1) q(z) ∈ D1 and ϕ ∈ ΦH[h,q], (2) q(z) is univalent in U and ϕ ∈ ΦH[h,qρ], for some ρ ∈ (0,1), or (3) q(z) is univalent in U and there exists ρ0 ∈ (0,1) such that ϕ ∈ ΦH[hρ,qρ], for all ρ ∈ (ρ0, 1). If f (z) ∈ ∑ (p) satisfies (16), then zp Im p (λ,ℓ) f (z) ≺ q(z), and q(z) is the best dominant. Proof. Following the same arguments in [14, Theorem 2.3e, p. 31], we deduce that q(z) is a dominant from Theorems 2 and 3. Since q(z) satisfies (17) it is also a solution of (16) and therefore q(z) will be dominated by all dominants. Hence q(z) is the best dominant. In the particular case q(z) = 1 + Mz, M > 0, and in view of Definition 3, the class of admissible functions ΦH[Ω,q], denoted by ΦH[Ω, M], is described below. Definition 4. Let Ω be a set in C and M > 0. The class of admissible functions ΦH[Ω, M] consists of those functions ϕ : C3× U → C such that ϕ   1+Meiθ , 1+ k+ � ℓ λ � � ℓ λ � Meiθ , 1+ L + h � 1+ 2 � ℓ λ �� k+ � ℓ λ �2 i Meiθ � ℓ λ �2 ; z    /∈ Ω (18) whenever z ∈ U, θ ∈ R, Re � Le−iθ � ≥ (k− 1)kM for all real θ and k ≥ 1. Corollary 2. Let ϕ ∈ ΦH[Ω, M]. If f (z) ∈ ∑ (p) satisfies ϕ(zp Im p (λ,ℓ) f (z), zp Im+1 p (λ,ℓ) f (z), zp Im+2 p (λ,ℓ) f (z) ; z) ∈ Ω, then � � �zp Im p (λ,ℓ) f (z)− 1 � � �< M . In the special case Ω = q(U) = {w : |w − 1|< M}, the class ΦH[Ω, M] is simply denoted by ΦH[M]. Corollary 2 can be written as: Corollary 3. Let ϕ ∈ ΦH[M]. If f (z) ∈ ∑ (p) satisfies � � �ϕ(zp Im p (λ,ℓ) f (z), zp Im+1 p (λ,ℓ) f (z), zp Im+2 p (λ,ℓ) f (z); z)− 1 � � �< M , then � � �zp Im p (λ,ℓ) f (z)− 1 � � �< M . R. El-Ashwah, M. Aouf / Eur. J. Pure Appl. Math, 3 (2010), 1070-1085 1077 Corollary 4. If M > 0 and f (z) ∈ ∑ (p) satisfies � � �zp Im+1 p (λ,ℓ) f (z)− zp Im p (λ,ℓ) f (z) � � � < M � ℓ λ � , then � � �zp Im p (λ,ℓ) f (z)− 1 � � �< M . (19) Proof. The proof follows from Corollary 2 by taking ϕ(u, v, w; z) = v − u and Ω = h(U), where h(z) = Mz � ℓ λ � , M > 0. To use Corollary 2, we need to show that ϕ ∈ ΦH[Ω, M], that is, the admissible condition (18) is satisfied. This follows since � � � � � ϕ � 1+Meiθ , 1+ k+ � ℓ λ � � ℓ λ � Meiθ , 1+ L+ n � 2 � ℓ λ � +1 � k+ � ℓ λ �2 o Meiθ � ℓ λ �2 ; z � � � � � � = kM � ℓ λ � ≥ M � ℓ λ � , where z ∈ U , θ ∈ R, and k ≥ 1. Hence by Corollary 2, we deduce the required result. Theorem 4 shows that the result is sharp. The differential equation zq ′ (z) � ℓ λ � = M � ℓ λ �z (ℓ < λM) has a univalent solution q(z) = 1+Mz. It follows from Theorem 4 that q(z) = 1+Mz is the best dominant. Definition 5. Let Ω be a set in C and q(z) ∈ D1∩H. The class of admissible functions ΦH,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 = 1 � ℓ λ � � ℓ λ � q(ζ) + kζq ′ (ζ) q(ζ) ! (q(ζ) 6= 0), Re (� ℓ λ � v(w− v) v − u − � ℓ λ � (2u− v) ) ≥ kRe ( 1+ ζq ′′ (ζ) q ′ (ζ) ) , where z ∈ U, ζ ∈ ∂ U\E(q) and k ≥ 1. Theorem 5. Let ϕ ∈ ΦH,1[Ω,q]. If f (z) ∈ ∑ (p) satisfies ( ϕ Im+1 p (λ,ℓ) f (z) Im p (λ,ℓ) f (z) , Im+2 p (λ,ℓ) f (z) Im+1 p (λ,ℓ) f (z) , Im+3 p (λ,ℓ) f (z) Im+2 p (λ,ℓ) f (z) ; z ! : z ∈ U ) ⊂ Ω, (20) then Im+1 p (λ,ℓ) f (z) Im p (λ,ℓ) f (z) ≺ q(z). R. El-Ashwah, M. Aouf / Eur. J. Pure Appl. Math, 3 (2010), 1070-1085 1078 Proof. Define an analytic function p(z) in U by p(z) = Im+1 p (λ,ℓ) f (z) Im p (λ,ℓ) f (z) . (21) By making use of (6) and (21), we obtain Im+2 p (λ,ℓ) f (z) Im+1 p (λ,ℓ) f (z) = p(z) + 1 � ℓ λ �   zp ′ (z) p(z)   . (22) Further computations show that Im+3 p (λ,ℓ) f (z) Im+2 p (λ,ℓ) f (z) = p(z) + 1 � ℓ λ �   zp ′ (z) p(z) + � ℓ λ � zp ′ (z) + zp ′ (z) p(z) − � zp ′ (z) p(z) �2 + z2p ′′ (z) p(z) � ℓ λ � p(z) + zp ′ (z) p(z)      . (23) Define the transformations from C3 to C by u = r, v = r + 1 � ℓ λ � � s r � , w = r + 1 � ℓ λ �    s r + � ℓ λ � s+ s r − � s r �2 + t r � ℓ λ � r + s r    . (24) Let ψ(r, s, t; z) = ϕ(u, v, w; z) = ϕ   r, 1 � ℓ λ � � � ℓ λ � r + s r � , 1 � ℓ λ �    � ℓ λ � r + s r + � ℓ λ � s+ s r − � s r �2 + t r � ℓ λ � r + s r    ; z    . (25) Using equations (21), (22) and (23), from (25), we obtain ψ(p(z), zp ′ (z), z2p ′′ (z); z) = ϕ Im+1 p (λ,ℓ) f (z) Im p (λ,ℓ) f (z) , Im+2 p (λ,ℓ) f (z) Im+1 p (λ,ℓ) f (z) , Im+3 p (λ,ℓ) f (z) Im+2 p (λ,ℓ) f (z) ; z ! . (26) Hence (20) implies ψ(p(z), zp ′ (z), z2p ′′ (z); z) ∈ Ω. The proof is completed if it can be shown that the admissibility condition for ϕ ∈ ΦH,1[Ω,q] is equivalent to the admissibility condition for ψ as given in Definition 1. Note that t s + 1= � ℓ λ � v(w− v) v − u − � ℓ λ � (2u− v), and hence ψ ∈Ψ[Ω,q]. By Lemma 1, p(z) ≺ q(z) or Im+1 p (λ,ℓ) f (z) Im p (λ,ℓ) f (z) ≺ q(z). R. El-Ashwah, M. Aouf / Eur. J. Pure Appl. Math, 3 (2010), 1070-1085 1079 If Ω 6= C is a simply connected domain, with Ω = h(U), for some conformal mapping h(z) of U onto Ω. In this case ΦH,1[h(U),q] is written as ΦH,1[h,q]. The following theorem is an immediate consequence of Theorem 5. Theorem 6. Let ϕ ∈ ΦH,1[h,q] with q(0) = 1. If f (z) ∈ ∑ (p) satisfies ϕ Im+1 p (λ,ℓ) f (z) Im p (λ,ℓ) f (z) , Im+2 p (λ,ℓ) f (z) Im+1 p (λ,ℓ) f (z) , Im+3 p (λ,ℓ) f (z) Im+2 p (λ,ℓ) f (z) ; z ! ≺ h(z), (27) then Im+1 p (λ,ℓ) f (z) Im p (λ,ℓ) f (z) ≺ q(z). In the particular case q(z) = 1+Mz, M > 0, the class of admissible functions ΦH,1[Ω,q] becomes the class ΦH,1[Ω, M]. Definition 6. Let Ω be a set in C and M > 0. The class of admissible functions ΦH,1[Ω, M] consists of those functions ϕ : C3× U → C such that ϕ 1+Meiθ , 1+ k+ � ℓ λ � (1+Meiθ ) � ℓ λ � (1+Meiθ ) Meiθ , 1+ k+ � ℓ λ � (1+Meiθ ) � ℓ λ � (1+Meiθ ) Meiθ+ (M + e−iθ ) ¦ Le−iθ + �� ℓ λ � + 1 � kM + � ℓ λ � kM2eiθ © − k2M2 � ℓ λ � (M + e−iθ ) ¦� ℓ λ � e−iθ + � 2 � ℓ λ � + k � M + � ℓ λ � M2eiθ © ; z ! /∈ Ω, (28) where z ∈ U, θ ∈ R, Re � Le−iθ � ≥ (k− 1)kM for all real θ and k ≥ 1. Corollary 5. Let ϕ ∈ ΦH,1[Ω, M]. If f (z) ∈ ∑ (p) satisfies ϕ Im+1 p (λ,ℓ) f (z) Im p (λ,ℓ) f (z) , Im+2 p (λ,ℓ) f (z) Im+1 p (λ,ℓ) f (z) , Im+3 p (λ,ℓ) f (z) Im+2 p (λ,ℓ) f (z) ; z ! ∈ Ω, then � � � � � Im+1 p (λ,ℓ) f (z) Im p (λ,ℓ) f (z) − 1 � � � � � < M . In the special case Ω = q(U) = {w : |w − 1| < M}, the class ΦH,1[Ω, M] is simply denoted by ΦH,1[M], and Corollary 5 takes the following form: Corollary 6. Let ϕ ∈ ΦH,1[M]. If f (z) ∈ ∑ (p) satisfies � � � � � ϕ Im+1 p (λ,ℓ) f (z) Im p (λ,ℓ) f (z) , Im+2 p (λ,ℓ) f (z) Im+1 p (λ,ℓ) f (z) , Im+3 p (λ,ℓ) f (z) Im+2 p (λ,ℓ) f (z) ; z ! − 1 � � � � � < M , then � � � � � Im+1 p (λ,ℓ) f (z) Im p (λ,ℓ) f (z) − 1 � � � � � < M . R. El-Ashwah, M. Aouf / Eur. J. Pure Appl. Math, 3 (2010), 1070-1085 1080 Corollary 7. If M > 0 and f (z) ∈ ∑ (p) satisfies � � � � � Im+2 p (λ,ℓ) f (z) Im+1 p (λ,ℓ) f (z) − Im+1 p (λ,ℓ) f (z) Im p (λ,ℓ) f (z) � � � � � < M � ℓ λ � (1+M) , then � � � � � Im+1 p (λ,ℓ) f (z) Im p (λ,ℓ) f (z) − 1 � � � � � < M . Proof. This follows from Corollary 6 by taking ϕ(u, v, w; z) = v − u and Ω = h(U), where h(z) = M � ℓ λ � (1+M) z, M > 0. To use Corollary 6, we need to show that ϕ ∈ ΦH,1[M], that is, the admissible condition (28) is satisfied. This follows since � �ϕ(u, v, w; z) � � = � � � � � −1−Meiθ + 1+ k+ � ℓ λ � (1+Meiθ ) � ℓ λ � (1+Meiθ ) Meiθ � � � � � = � � � � � kMeiθ � ℓ λ � (1+Meiθ ) � � � � � ≥ M � ℓ λ � (1+M) , for z ∈ U , θ ∈ R, λ > 0, ℓ > 0 and k ≥ 1. Hence by Corollary 6, we deduce the required result. 3. Superordination Results Involving the Operator I m p (λ,ℓ) In this section we obtain differential superordination for the operator Im p (λ,ℓ). For this purpose the class of admissible functions is given in the following definition. Definition 7. Let Ω be a set in C and q(z) ∈ H with zq ′ (z) 6= 0. The class of admissible functions Φ ′ H[Ω,q] consists of those functions ϕ : C3× U → C that satisfy the admissibility condition ϕ(u, v, w;ζ) ∈ Ω whenever u= q(z), v = zq ′ (z) +m � ℓ λ � q(z) m � ℓ λ � , Re (� ℓ λ � (w − u) v − u − 2 � ℓ λ � ) ≤ 1 m Re ( 1+ zq ′′ (z) q ′ (z) ) , where z ∈ U, ζ ∈ ∂ U and m ≥ 1. R. El-Ashwah, M. Aouf / Eur. J. Pure Appl. Math, 3 (2010), 1070-1085 1081 Theorem 7. Let ϕ ∈ Φ ′ H[Ω,q]. If f (z) ∈ ∑ (p), zp Im p (λ,ℓ) f (z) ∈ D1 and ϕ � zp Im p (λ,ℓ) f (z) , zp Im+1 p (λ,ℓ) f (z), zp Im+2 p (λ,ℓ) f (z) ; z � is univalent in U, then Ω⊂ n ϕ � zp Im p (λ,ℓ) f (z), zp Im+1 p (λ,ℓ) f (z), zp Im+2 p (λ,ℓ) f (z); z � : z ∈ U o (29) implies q(z) ≺ zp Im p (λ,ℓ) f (z). Proof. Let p(z) defined by (10) and ψ(z) defined by (15). Since ϕ ∈ Φ ′ H[Ω,q], from (15) and (29), we have Ω⊂ ¦ ψ(p(z), zp ′ (z), z2p ′′ (z); z) : z ∈ U © . From (14), we see that the admissibility condition for ϕ ∈ Φ ′ H[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) ≺ zp Im p (λ,ℓ) f (z). If Ω 6= C is a simply connected domain, then Ω = h(U) for some conformal mapping h(z) for U onto Ω. In this case the class Φ ′ H[h(U),q] is written as Φ ′ H[h,q]. Proceeding similarly as in Section 2, the following result is an immediate consequence of Theorem 7. Theorem 8. Let q(z) ∈ H, h(z) is analytic on U and ϕ ∈ Φ ′ H[h,q]. If f (z) ∈ ∑ (p), zp Im p (λ,ℓ) f (z) ∈ D1 and ϕ(zp Im p (λ,ℓ) f (z), zp Im+1 p (λ,ℓ) f (z), zp Im+2 p (λ,ℓ) f (z); z)is univalent in U, then h(z)≺ ϕ(zp Im p (λ,ℓ) f (z), zp Im+1 p (λ,ℓ) f (z), zp Im+2 p (λ,ℓ) f (z) ; z) (30) implies q(z) ≺ zp Im p (λ,ℓ) f (z). Theorem 7 and Theorem 8 can only be used to obtain subordinants of differential super- ordination of the form (29) or (30). The following theorem proves the existence of the best subordinant of (30) for certain ϕ. Theorem 9. Let h(z) be analytic in U and ϕ : C3 × U → C. Suppose that the differential equation ϕ   p(z), zp ′ (z) + � ℓ λ � p(z) � ℓ λ � , z2p ′′ (z) + � 2 � ℓ λ � + 1 � zp ′ (z) + � ℓ λ �2 p(z) � ℓ λ �2 ; z    = h(z) (31) R. El-Ashwah, M. Aouf / Eur. J. Pure Appl. Math, 3 (2010), 1070-1085 1082 has a solution q(z) ∈ D1. If ϕ ∈ Φ ′ H[h,q], f (z) ∈ ∑ (p), zp Im p (λ,ℓ) f (z) ∈ D1 and ϕ � zp Im p (λ,ℓ) f (z), zp Im+1 p (λ,ℓ) f (z), zp Im+2 p (λ,ℓ) f (z); z � is univalent in U, then h(z) ≺ ϕ � zp Im p (λ,ℓ) f (z), zp Im+1 p (λ,ℓ) f (z), zp Im+2 p (λ,ℓ) f (z); z � implies q(z) ≺ zp Im p (λ,ℓ) f (z) 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 8, we obtain the following sandwich theorem. Corollary 8. Let h1(z) and q1(z) be analytic functions in U, h2(z) be univalent function in U, q2(z) ∈ D1 with q1(0) = q2(0) = 1 and ϕ ∈ ΦH[h2,q2] ∩ Φ ′ H[h1,q1]. If f (z) ∈ ∑ (p), zp Im p (λ,ℓ) f (z) ∈ H ∩ D1 and ϕ � zp Im p (λ,ℓ) f (z) , zp Im+1 p (λ,ℓ) f (z), zp Im+2 p (λ,ℓ) f (z); z � is univalent in U, then h1(z) ≺ ϕ � zp Im p (λ,ℓ) f (z) , zp Im+1 p (λ,ℓ) f (z), zp Im+2 p (λ,ℓ) f (z); z � ≺ h2(z), implies q1(z)≺ zp Im p (λ,ℓ) f (z) ≺ q2(z). Definition 8. Let Ω be a set in C with q(z) ∈ H and zq ′ (z) 6= 0. The class of admissible functions Φ ′ H,1[Ω,q] consists of those functions ϕ : C3 × U → C that satisfy the admissibility condition ϕ(u, v, w;ζ) ∈ Ω whenever u= q(z), v = q(z) + 1 � ℓ λ � zq ′ (z) mq(z) ! (q(z) 6= 0) Re (� ℓ λ � v(w− v) v− u − � ℓ λ � (2u− v) ) ≤ 1 m Re ( 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. R. El-Ashwah, M. Aouf / Eur. J. Pure Appl. Math, 3 (2010), 1070-1085 1083 Theorem 10. Let ϕ ∈ Φ ′ H,1[Ω,q]. If f (z) ∈ ∑ (p), Im+1 p (λ,ℓ) f (z) Im p (λ,ℓ) f (z) ∈ D1 and ϕ Im+1 p (λ,ℓ) f (z) Im p (λ,ℓ) f (z) , Im+2 p (λ,ℓ) f (z) Im+1 p (λ,ℓ) f (z) , Im+3 p (λ,ℓ) f (z) Im+2 p (λ,ℓ) f (z) ; z ! is univalent in U, then Ω⊂ ( ϕ Im+1 p (λ,ℓ) f (z) Im p (λ,ℓ) f (z) , Im+2 p (λ,ℓ) f (z) Im+1 p (λ,ℓ) f (z) , Im+3 p (λ,ℓ) f (z) Im+2 p (λ,ℓ) f (z) ; z ! : z ∈ U ) . (32) implies q(z)≺ Im+1 p (λ,ℓ) f (z) Im p (λ,ℓ) f (z) . Proof. Let p(z) defined by (21) and ψ defined by (25). Since ϕ ∈ Φ ′ H,1[Ω,q], from (26) and (32), we have Ω ⊂ ¦ ψ(p(z), zp ′ (z), z2p ′′ (z); z) : z ∈ U © . From (25), we see that the admissibility condition for ϕ ∈ Φ ′ H,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)≺ Im+1 p (λ,ℓ) f (z) Im p (λ,ℓ) f (z) . 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 ′ H,1[h(U),q] is written as Φ ′ H,1[h,q]. The following result is an immediate consequence of Theorem 10. Theorem 11. Let q(z) ∈ H, h(z) be analytic in U and ϕ ∈ Φ ′ H,1[h,q]. If f (z) ∈ ∑ (p), Im+1 p (λ,ℓ) f (z) Im p (λ,ℓ) f (z) ∈ D1 and ϕ Im+1 p (λ,ℓ) f (z) Im p (λ,ℓ) f (z) , Im+2 p (λ,ℓ) f (z) Im+1 p (λ,ℓ) f (z) , Im+3 p (λ,ℓ) f (z) Im+2 p (λ,ℓ) f (z) ; z ! is univalent in U, then h(z) ≺ ϕ Im+1 p (λ,ℓ) f (z) Im p (λ,ℓ) f (z) , Im+2 p (λ,ℓ) f (z) Im+1 p (λ,ℓ) f (z) , Im+3 p (λ,ℓ) f (z) Im+2 p (λ,ℓ) f (z) ; z ! , (33) implies q(z)≺ Im+1 p (λ,ℓ) f (z) Im p (λ,ℓ) f (z) . REFERENCES 1084 Combining Theorems 6 and 11, we obtain the following sandwich-type theorem. Corollary 9. Let h1(z) and q1(z) be analytic functions in U , h2(z) be univalent function in U , q2(z) ∈ D1 with q1(0) = q2(0) = 1 and ϕ ∈ ΦH,1[h2,q2] ∩ Φ ′ H,1[h1,q1]. If f (z) ∈ ∑ (p), Im+1 p (λ,ℓ) f (z) Im p (λ,ℓ) f (z) ∈ H ∩ D1 and ϕ Im+1 p (λ,ℓ) f (z) Im p (λ,ℓ) f (z) , Im+2 p (λ,ℓ) f (z) Im+1 p (λ,ℓ) f (z) , Im+3 p (λ,ℓ) f (z) Im+2 p (λ,ℓ) f (z) ; z ! is univalent in U, then h1(z)≺ ϕ Im+1 p (λ,ℓ) f (z) Im p (λ,ℓ) f (z) , Im+2 p (λ,ℓ) f (z) Im+1 p (λ,ℓ) f (z) , Im+3 p (λ,ℓ) f (z) Im+2 p (λ,ℓ) f (z) ; z ! ≺ h2(z), implies q1(z) ≺ Im+1 p (λ,ℓ) f (z) Im p (λ,ℓ) f (z) ≺ q2(z). Remark 1. (i) Putting λ = 1 in the above results we obtain results associated with the operator Ip(m,ℓ) which defined by (7); (ii) Putting ℓ = 1 in the above results we obtain results associated with the operator Dm λ,p which defined by (8). References [1] R. Aghalary, R. M. Ali, S. B. Joshi and V. Ravichandran, Inequalities for analytic functions defined by certain linear operator, Internat. J. Math. Sci., 4, no. 2, 267-274. 2005. [2] R. Aghalary, S. B. Joshi, R.N. Mohapatra and V. 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