Korean J. Math. 23 2015, No. 4, pp. 503 519 http://dx.doi.org/10.11568/jm.2015.23.4.503 STRONG DIFFERENTIAL SUBORDINATION AND SUPERORDINATION OF NEW GENERALIZED DERIVATIVE OPERATOR Anessa Oshah and Maslina Darus Abstract. In this wor, certain classes of admissible functions are considered. Some strong differential subordination and superordination properties of analytic functions associated with new generalized derivative operator B µ,q,s λ 1,λ 2,l,d are investigated. New strong differential sandwich-type results associated with the generalized derivative operator are also given. 1. Introduction Let H = HU denote the class of analytic functions in the open unit dis U = {z C : z < 1}. For n N and a C, let H[a, n] be the subclass of H consisting of functions of the form 1 fz = a + a n z n + a n+1 z n+1 +, a C with H 0 H[0, 1] and H 1 H[1, 1], and let A denote the class of all normalized analytic functions of the form Received February 27, 2015. Revised September 28, 2015. Accepted October 6, 2015. 2010 Mathematics Subject Classification: 30C45. Key words and phrases: Analytic function, Derivative operator, Strong differential subordination, Strong differential superodination. * Corresponding author. The wor here is supported by FRGSTOPDOWN/2013/ST06/UKM/01/1. c The Kangwon-Kyungi Mathematical Society, 2015. This is an Open Access article distributed under the terms of the Creative commons Attribution Non-Commercial License http://creativecommons.org/licenses/by -nc/3.0/ which permits unrestricted non-commercial use, distribution and reproduction in any medium, provided the original wor is properly cited.
504 A. Oshah and M. Darus 2 fz = z + a z, =2 z U. If f and F are members of H and there exists the Schwarz function wz, analytic in U with w0 = 0 and wz < 1 such that fz = F wzz U, then we say that f is subordinate to F or F superordinate to f, and we write fz F zz U. In particular, if F is univalent in U, then fz F z is equivalent to f0 = F 0 and fu F U cf. [9]. If fz of the form 2 and gz = z + =2 b z are two functions in A, then the Hadamard product or convolution of fz and gz is denoted by fz gz and defined as 3 fz gz = z + a b z, z U. =2 For parameters α i Ci = 1,..., q, and β j C\{0, 1, 2,...}j = 1,..., s, the generalized hypergeometric function q F s α 1,..., α q ; β 1,..., β s ; z is defined as: qf s α 1,..., α q ; β 1,..., β s ; z = =0 α 1,..., α q z β 1,..., β s!, q s + 1, q, s N 0 = N {0}, z U, where a is the Pochhammer symbol or the shifted factorial defined in terms of the Gamma function by a = Γa + Γa = { 1, = 0, a C\{0}; aa + 1a + 2...a +, N = {1, 2, 3,...}. Dzio and Srivastava [6] defined the linear operator 4 Hα 1,..., α q ; β 1,..., β s ; zfz = z + Υ q sa z, where 5 Υ q s = =2 α 1 1,..., α q 1 β 1 1,..., β s 1!.
Strong Differential Subordination and Superordination 505 Oshah and Darus [15] introduced a function M µ λ 1,λ 2,l,d as follows [ 6 M µ λ 1,λ 2,l,d z = z + + lλ ] µ 1 z, where =2 7 = l1 + λ 2 + d, µ, d N 0 = {0, 1, 2,...}, λ 2 λ 1 0, l 0, and l + d > 0. By maing use of Hadamard product, we define linear operator B µ,q,s λ 1,λ 2,l,d : A A as follows B µ,q,s λ 1,λ 2,l,d fz = Mµ λ 1,λ 2,l,d z Hα 1,..., α q ; β 1,..., β s ; zfz, then, from 6 and 4, we have [ 8 B µ,q,s λ 1,λ 2,l,d fz = z + + lλ ] µ 1 Υ q sa z, =2 where Υ q s and are defined in 5,7, respectively. One can easily verify from 8 that [ 9 ] B µ+1,q,s λ 1,λ 2,l,d fz = z B µ,q,s λ 1,λ 2,l,d fz + [ ] B µ,q,s λ 1,λ 2,l,d fz. Note that, for µ = 0 or d = 1, l = 0, we obtain B 0,q,s λ 1,λ 2,l,dfz = Bµ,q,s λ 1,λ 2,0,1 fz = Hα 1,..., α q ; β 1,..., β s ; zfz which was introduced and studied by Dzio and Srivastava see [6], which includes various other linear operators introduced and studied earlier in the literature. For example, when q = 2, s = 1, Hohlov in [7] studied this operator for α 1, α 2, and β 1, also for α 2 = 1, this operator becomes the Carlson-Shaffer operator [4], and Ruscheweyh [16] studied this operator for α 1 = n + 1, α 2 = 1, and β 1 = 1. Further, if q = 2, s = 1, α 1 = n+1, α 2 = 1, β 1 = 1, we get B µ,2,1 λ 1,λ 2,l,d fz = D n,µ λ 1,λ 2,l,dfz which was introduced and studied by Oshah and Darus see [15]. Antonino and Romaguera in [1] have introduced the concept of strongly differential subordination which refered to the generalization of the notion of differential subordination developed by Oros and Oros [13], and
506 A. Oshah and M. Darus Oros [14] of strong differential subordination and superordination. In the present investigation, by maing use of that notion of strong differential subordination, which is indeed an extension version of the theory of differential subordination introduced and developed by Miller and Mocanu [9, 10], we consider certain suitable classes of admissible functions. Here we investigate some strong differential subordination and strong differential superordination properties of analytic functions associated with the new generalized derivative operator, defined above in 8. New strong differential sandwich-type results associated with the generalized derivative operator are also obtained. Using various linear operators, strong differential subordinations were investigated by Jeyaraman et al. [8], Cho [5], and AL-Shaqsi [3] and of course many others. To prove our results, we need the following definition and theorems considered by Antonino and Romaguera [1, 2], and Oros and Oros [13, 14]. Definition 1.1. [1, 2, 13] Let Hz, ζ be analytic in U U and let fz be analytic and univalent in U. Then, the function Hz, ζ is said to be strongly subordinate to fz, or fz is said to be strongly superordinate to Hz, ζ, written as Hz, ζ fz, if, for ζ U, Hz, ζ as the function of z is subordinate to fz. We note that Hz, ζ fz if and only if H0, ζ = f0 and HU U fu. Definition 1.2. [9, 13] Let φ : C 3 U U C and let hz be univalent in U. If pz is analytic in U and satisfies the second-order differential subordination 10 φpz, zp z, zp z; z; ζ hz, then pz is called a solution of the strong differential subordination. The univalent function qz is called a dominant of the solution of the strong differential subordination, or more simply a dominant, if pz qz for all pz satisfying 10. A dominant qz that satisfies qz qz for all dominants qz of 10 is said to be best dominant. Recently, Oros [14] introduced the following strong differential superordinations as dual concept of strong differential subordination. Definition 1.3. [12,14] Let ϕ : C 3 U U C and let hz be analytic in U. If pz and φpz, zp z, z 2 p z; z; ζ are univalent in U for ζ U and satisfy the second-order strong differential superordination 11 hz φpz, zp z, z 2 p z; z; ζ,
Strong Differential Subordination and Superordination 507 then pz is called a solution of the strong differential superordination. An analytic function qz is called a subordinant of the solution of the strong differential superordination, or more simply a subordinant, if qz pz for all pz satisfying 11. A univalent subordinant qz that satisfies qz qz for all subordinantes qz of 11 is said to be best subordinant. We denote by Q the class of functions q that are analytic and injective on U \ Eq, where Eq = { η U : lim qz = } z η and are such that q η 0, η U \ Eq. Further, let the subclass of Q for which q0 = a be denoted by Qa, Q0 Q 0 and Q1 Q 1. Definition 1.4. [13] Let Ω be a set in C, qz Q and n be a positive integer. The class of admissible functions Ψ n [Ω, q] consists of those functions ψ : C 3 U U C that satisfy the admissibility condition ψr, s, t; z, ζ / Ω, whenever r = qη, s = ηq η and { } { } t ηq Re s + 1 η Re q η + 1, z U, η U \ Eq, ζ U, n. We write Ψ 1 [Ω, q] as Ψ[Ω, q]. Definition 1.5. [14] Let Ω be a set in C, and q H[a, n] with q z 0. The class of admissible functions Ψ n[ω, q] consists of those functions ψ : C 3 U U C that satisfy the admissibility condition ψr, s, t; η, ζ Ω, whenever r = qz, s = zq z and m { } t Re s + 1 1 { } zq m Re z q z + 1, z U, η U, ζ U, m n 1. We write Ψ 1[Ω, q] as Ψ [Ω, q].
508 A. Oshah and M. Darus In order to prove the main results, we need the following theorem which was proved by Oros and Oros [13]. Theorem 1.1. [13] Let ψ Ψ n [Ω, q] with q0 = a. If p H[a, n] satisfies then pz qz. ψpz, z pz, z 2 p z; z, ζ Ω, Furthermore, Oros [14] proved the following theorem. Theorem 1.2. [14] Let ψ Ψ n[ω, q] with q0 = a. If pz Qa and is univalent in U for ζ U, then ψpz, z pz, z 2 p z; z, ζ, Ω {ψpz, z pz, z 2 p z; z, ζ : z U, ζ U} implies that qz pz. 2. The Main Subordination Result First, we prove the subordination theorem by using the derivative operator B µ,q,s λ 1,λ 2,l,dfz. For this purpose, we need the following class of admissible functions. Definition 2.1. Let Ω be a set in C, λ 2 λ 1 > 0, l > 0, µ 1, d N 0, and qz Q 0 H 0. The class of admissible functions Φ B [Ω, q] consists of those functions φ : C 3 U U C that satisfy the admissibility condition φu, v, w; z, ζ / Ω, whenever u = qη, v = ηq η + λ 2,l,d qη,
and Re Strong Differential Subordination and Superordination 509 { 2 w lλ } 1 2 u [ v λ 2,l,d u ] 2 z U, η U, ζ U, d N 0 ; 1. Re { } ηq η q η + 1, Theorem 2.1. Let φ Φ B [Ω, q]. If f A satisfies 12 {φ B µ 2,q,s λ 1,λ 2,l,dfz, Bµ 1,q,s λ 1,λ 2,l,dfz, Bµ,q,s λ 1,λ 2,l,d fz; z, ζ : z U, ζ U} Ω, then B µ 2,q,s λ 1,λ 2,l,dfz qz, z U. Proof. From 9, we can see z B µ 1,q,s 13 B µ,q,s λ λ 1,λ 2,l,d fz = 1,λ 2,l,d fz + λ 2,l,d and hence z B µ 2,q,s 14 B µ 1,q,s λ λ 1,λ 2,l,d fz = 1,λ 2,l,d fz + λ 2,l,d Define the function p in U by B µ 1,q,s λ 1,λ 2,l,d fz, B µ 2,q,s λ 1,λ 2,l,d fz. 15 pz = B µ 2,q,s λ 1,λ 2,l,d fz. Maing use of 14 and 15, we get 16 B µ 1,q,s λ 1,λ 2,l,d fz = zp z + pz.
510 A. Oshah and M. Darus Also, maing use of 13 and 15, and simple calculation we get 17 z 2 p 2 z + λ 2,l,d B µ,q,s λ 1,λ 2,l,d fz = zp z + Define the transformation from C 3 to C by 18 s + λ 2,l,d r t + u = r, v = λ 2,l,d, w = 2 pz λ 2,l,d 2. 2 s + 2 r. 2 Let ψr, s, t; z, ζ = φu.v, w; z, ζ = 19 φ s + r, r λ 2,l,d, t + s + 2 2 r ; z, ζ 2 Using 15, 16 and 17, from 19 we obtain 20 ψpz, zp z, z 2 p z; z, ζ = φ B µ 2,q,s λ 1,λ 2,l,dfz, Bµ 1,q,s λ 1,λ 2,l,dfz, Bµ,q,s λ 1,λ 2,l,d fz; z, ζ. Hence, 12 becomes 21 ψpz, zp z, z 2 p z; z, ζ Ω. We note that 22 t s + 1 = λ 2,l,d 2 w 2 u [ v λ 2,l,d u ], and so the admissibility condition for φ Φ B [Ω, q] in Definition 2.1 is equivalent to the admissibility condition for ψ Ψ[Ω, q]. Therefore, by Theorem 1.1, we have pz qz or equivalently 23 B µ 2,q,s λ 1,λ 2,l,dfz qz, which evidently completes the proof of Theorem 2.1..
Strong Differential Subordination and Superordination 511 Next, if we consider the special situation when Ω C is a simply connected domain, then Ω = hu for some conformal mapping h of U onto Ω. In this case, the class Φ B [hu, q] is written as Φ B [h, q]. The following result is an immediate consequence of Theorem 2.1. Theorem 2.2. Let φ Φ B [h, q]. If f A satisfies 24 φ B µ 2,q,s λ 1,λ 2,l,dfz, Bµ 1,q,s λ 1,λ 2,l,dfz, Bµ,q,s λ 1,λ 2,l,d fz; z, ζ hz, then 25 B µ 2,q,s λ 1,λ 2,l,dfz qz. Our next result is an extension of Theorem 2.1 to the case where the behavior of q on U is not nown. Corollary 2.1. Let Ω C and q be univalent in U with q0 = 1. Let φ Φ B [Ω, q ρ ] for some ρ 0, 1 where q ρ z = qρz. If f A satisfies then φb µ 2,q,s λ 1,λ 2,l,dfz, Bµ 1,q,s λ 1,λ 2,l,dfz, Bµ,q,s λ 1,λ 2,l,dfz; z, ζ Ω, 26 B µ 2,q,s λ 1,λ 2,l,dfz qz. Proof. From Theorem 2.1, we see B µ 2,q,s λ 1,λ 2,l,d fz q ρz, and the result is deduced from q ρ z qz. Theorem 2.3. Let h and q be univalent in U with q0 = 0 and set q ρ z = qρz and h ρ z = hρz. Let φ : C 3 U U C satisfy one of the following conditions: i: φ Φ B [h, q ρ ] for some ρ 0, 1, or ii: there exists ρ 0 0, 1 such that φ Φ B [h ρ, q ρ ] for all ρ ρ 0, 1. If f A satisfies 24, then 27 B µ 2,q,s λ 1,λ 2,l,dfz qz. Proof. The proof is similar to the one in [11, Theorem 2.3d, page 30] and therefore is omitted. Now, our next results give the best dominant of the strong differential subordination 24.
512 A. Oshah and M. Darus Theorem 2.4. Let h be univalent in U and φ : C 3 U U. Suppose that the differential equation zq z + λ 2,l,d qz 28 φ qz, z 2 p z + 2, zp z + 2 pz ; z, ζ 2 = hz has a solution q with q0 = 0 and satisfies one of the following conditions: i: q Q 0 and φ Φ B [h, q], or ii: q is univalent in U and φ Φ B [h, q ρ ], for some ρ 0, 1, or iii: q is univalent in U and there exists ρ 0 0, 1 such that φ Φ B [h ρ, q ρ ] for all ρ ρ 0, 1. If f A satisfies 24 and is analytic in U, then φb µ 2,q,s λ 1,λ 2,l,dfz, Bµ 1,q,s λ 1,λ 2,l,dfz, Bµ,q,s λ 1,λ 2,l,dfz; z, ζ and q is the best dominant. B µ 2,q,s λ 1,λ 2,l,dfz qz Proof. Using the same method given by [11, Theorem 2.3e, p.31], we deduce that from Theorems 2.2 and 2.3, q is a dominant of 24. Since q satisfies 28, q is also a solution of 24 and therefore q will be dominated by all dominants. Hence q should be the best dominant of 24. In the particular case, qz = Mz, M > 0, and in view of Definition 2.1, the class of admissible function Φ B [hu, q], denoted by Φ B [hu, M], is described below.
Strong Differential Subordination and Superordination 513 Definition 2.2. Let Ω be a set in C, λ 2 λ 1 > 0, l > 0, µ 1, d N 0 and M > 0. The class of admissible functions Φ B [Ω, M] consists of those functions such that + 29 φ Me iθ, φ : C 3 U U C, [ Me iθ, 2 L + λ 2,l,d + 2 ] Me iθ ; z, ζ 2 / Ω, whenever z U, η U, Re{Le iθ } M, θ is real number, and 1. then Corollary 2.2. Let φ Φ B [Ω, M]. If f A satisfies φb µ 2,q,s λ 1,λ 2,l,dfz, Bµ 1,q,s λ 1,λ 2,l,dfz, Bµ,q,s λ 1,λ 2,l,dfz; z, ζ Ω, 30 B µ 2,q,s λ 1,λ 2,l,dfz < M. For the special case Ω = qu = {w : w < M}, the class Φ B [Ω, M] is simply denoted by Φ B [M]. then Corollary 2.3. Let φ Φ B [M]. If f A satisfies φb µ 2,q,s λ 1,λ 2,l,dfz, Bµ 1,q,s λ 1,λ 2,l,dfz, Bµ,q,s λ 1,λ 2,l,d fz; z, ζ < M, 31 B µ 2,q,s λ 1,λ 2,l,dfz < M. Corollary 2.4. Let λ 2 λ 1 > 0, l > 0, M > 0, and let Cη be an analytic function in U with Re{ζCη} 0 for ζ U. If f A
514 A. Oshah and M. Darus satisfies 2 B µ,q,s then λ 1,λ 2,l,d fz λ 2,l,d 2 B µ 2,q,s B µ 1,q,s λ 1,λ 2,l,d fz 32 B µ 2,q,s λ 1,λ 2,l,dfz < M. Proof. Let 2 φu, v, w; z, η = w λ 2,l,d v λ 1,λ 2,l,d fz + Cη < M, 2 u + Cη, and Ω = hu, where hz = λ 2,l,d Mz. Before using Corollary 2.2, we need to show that φ Φ B [Ω, M], that means the admissible condition 29 is satisfied. This follows since + Me φ iθ, Me iθ, [ 2 L + λ 2,l,d + 2 ] Me iθ ; z, ζ 2 [ 2 2 λ = L + 2,l,d ] + Me iθ 2 + λ 2,l,d Me iθ Me iθ + Cη = L + 2 Me iθ + Cη
Strong Differential Subordination and Superordination 515 M + 2 M + Re{Le iθ } + Re{Cηe iθ } 2 M + Re{Cηe iθ } whenever z U, η U, Re{Le iθ } M, θ is real number, and 1. Hence, the required result now follows from Corollary 2.2. M, 3. Superordination and Sandwich Results In this section, the dual problem of strong differential subordination that is, strong differential superordination of the differential operator B µ,q,s λ 1,λ 2,l,dfz is investigated. We will also give sandwich-type results, but first we will define the class of admissible functions as follows: Definition 3.1. Let Ω be a set in C, q H 0 with zq z 0, λ 2 λ 1 > 0, l > 0, µ 1, d N 0. The class of admissible functions Φ B [Ω, q] consists of those functions φ : C 3 U U C that satisfy the admissibility condition φu, v, w; η, ζ Ω whenever and Re u = qz, v = zq z/m + λ 2,l,d qz, { 2 w lλ } 1 2 u [ v λ 2,l,d u ] 1 { } zq m Re z q z + 1, where z U, η U, ζ U, and m 1. Theorem 3.1. Let φ Φ B [Ω, q]. If f A, Bµ 2,q,s λ 1,λ 2,l,d fz Q 0, and φb µ 2,q,s λ 1,λ 2,l,dfz, Bµ 1,q,s λ 1,λ 2,l,dfz, Bµ,q,s λ 1,λ 2,l,dfz; z, ζ
516 A. Oshah and M. Darus is univalent in U, then 33 Ω {φb µ 2,q,s λ 1,λ 2,l,dfz, Bµ 1,q,s λ 1,λ 2,l,dfz, Bµ,q,s λ 1,λ 2,l,dfz; z, ζ : z U, η U}, which implies that 34 qz B µ 2,q,s λ 1,λ 2,l,d fz. Proof. For p defined by 15 and φ by 19, the equations 20 and 33 yield Ω {ψpz, zp z, z 2 p z; z, ζ : z U, ζ U}. From 18, the admissibility condition for φ Φ B [Ω, q] is equivalent to the admissibility condition for ψ as given in Definition 1.5. Hence ψ Ψ B [Ω, q] and by Theorem 1.2, qz pz or equivalently qz B µ 2,q,s λ 1,λ 2,l,d fz. Similar to the previous section, if we consider the special situation when Ω C is a simply connected domain, then Ω = hu for some conformal mapping h of U onto Ω. In this case, the class Φ B [hu, q] is written as Φ B [h, q]. The following result is an immediate consequence of Theorem 3.1. Theorem 3.2. Let h be analytic in the open unit dis U,q H 0, and we let φ Φ B [h, q]. If f A, Bµ 2,q,s λ 1,λ 2,l,d fz Q 0, also 35 φ B µ 2,q,s λ 1,λ 2,l,dfz, Bµ 1,q,s λ 1,λ 2,l,dfz, Bµ,q,s λ 1,λ 2,l,dfz; z, ζ is univalent in U, then 36 hz φ B µ 2,q,s λ 1,λ 2,l,dfz, Bµ 1,q,s λ 1,λ 2,l,dfz, Bµ,q,s λ 1,λ 2,l,d fz; z, ζ, which implies that 37 qz B µ 2,q,s λ 1,λ 2,l,d fz. Theorems 3.1 and 3.2 can only be used to obtain subordinants of differential superordination of the form 33 or 36. The following theorem proves the existence of the best subordinant of 36 for an appropriate φ.
Strong Differential Subordination and Superordination 517 Theorem 3.3. Let h be analytic in the open unit dis U, and ψ : C 3 U U C. Suppose that the differential equation zq z + λ 2,l,d qz φ qz, z 2 p z +, 2 zp z + 2 pz ; z, ζ 2 has a solution q Q 0. If φ Φ B [h, q], and φ B µ 2,q,s λ 1,λ 2,l,dfz, Bµ 1,q,s λ 1,λ 2,l,dfz, Bµ,q,s λ 1,λ 2,l,dfz; z, ζ is univalent in U, then hz φ B µ 2,q,s λ 1,λ 2,l,dfz, Bµ 1,q,s λ 1,λ 2,l,dfz, Bµ,q,s λ 1,λ 2,l,d fz; z, ζ, which implies that qz B µ 2,q,s λ 1,λ 2,l,d fz and q is the best subordinant. = hz Proof. The proof is similar to that of Theorem 2.4, and so it is being omitted here. Combining Theorems 2.2 and 3.2, we obtain the following sandwich-type result. Theorem 3.4. Let h 1 and q 1 be analytic functions in U, and let h 2 be analytic function in U, q 2 Q 0 with q 1 0 = q 2 0 = 0 and φ Φ B [h 1, q 1 ] Φ B [h 2, q 2 ]. If f A, B µ 2,q,s λ 1,λ 2,l,d fz H 0 Q 0 and φ B µ 2,q,s λ 1,λ 2,l,dfz, Bµ 1,q,s λ 1,λ 2,l,dfz, Bµ,q,s λ 1,λ 2,l,dfz; z, ζ is univalent in U, then h 1 z φ B µ 2,q,s λ 1,λ 2,l,dfz, Bµ 1,q,s λ 1,λ 2,l,dfz, Bµ,q,s λ 1,λ 2,l,d fz; z, ζ h 2 z, which implies that q 1 z B µ 2,q,s λ 1,λ 2,l,d fz q 2z.
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Strong Differential Subordination and Superordination 519 Anessa Oshah School of Mathematical Sciences Faculty of Science and Technology Universiti Kebangsaan Malaysia Bangi, Selangor, Malaysia E-mail: anessa.oshah@yahoo.com Maslina Darus School of Mathematical Sciences Faculty of Science and Technology Universiti Kebangsaan Malaysia Bangi, Selangor, Malaysia E-mail: maslina@um.edu.my