Differential Subordination and Superordination Results for Certain Subclasses of Analytic Functions by the Technique of Admissible Functions

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1 Filomat 28:10 (2014), DOI /FIL J Published by Faculty of Sciences and Mathematics, University of Niš, Serbia Available at: htt:// Differential Subordination and Suerordination Results for Certain Subclasses of Analytic Functions by the Technique of Admissible Functions R.Jayasankar a, Maslina Darus b, S. Sivasubramanian c a Deartment of Mathematics, GRT Institute of Engineering and Technology, Tiruttani , India. b School of Mathematical Sciences, Faculty of Science and Technology Universiti Kebangsaan Malaysia, Bangi c Deartment of Mathematics, University College of Engineering Tindivanam, Anna University Chennai, Melakkam , India. Abstract. By investigating aroriate classes of admissible functions, various Differential subordination and suerordination results for analytic functions in the oen unit disk are obtained using Cho-Kwon- Srivastava oerator. As a consequence of these results, Sandwich-tye results are also obtained. 1. Introduction and Motivations 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 (z) = a + a n z n + a n+1 z n , with a C, H 0 H [0, 1] and H H [1, 1]. Let A denote the class of all analytic functions of the form f (z) = z + a k z k (z U) (1) and let A 1 := A. k= Mathematics Subject Classification. Primary 30C80; Secondary 30C45 Keywords. Analytic, Univalent, Subordination, Suerordination. Received: 06 March 2013; Acceted: 08 May 2013 Communicated by Hari M. Srivastava addresses: jaymaths@gmail.com (R.Jayasankar), maslina@krisc.cc.ukm.my (Maslina Darus), sivasaisastha@rediffmail.com (S. Sivasubramanian)

2 R.Jayasankar et al. / Filomat 28:10 (2014), 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 suerordinate to f (z), if there exists a function w(z) analytic in U with w(0) = 0 and w(z) < 1 (z U), such that f (z) = F(w(z)). In such a case we write f (z) F(z). If F is univalent, and f (U) F(U). For two functions f (z) given by (1) and (z) = z + and is defined by ( f )(z) := z + f (z) F(z) if and only if f (0) = F(0) b k z k, the Hadamard roduct(or convolution) of f k=+1 a k b k z k := ( f )(z). (2) k=+1 For a function f A, given by (1.1) and it follows from that for λ > and a, c R\Z 0 where and I λ = z + k=1 From (3), we deduce that and I λ = φ (+) (a, c; z) f (z), z U (c) k (λ + ) k a +k z +k (a) k (1) k = z 2F 1 (c, λ +, a; z) f (z) (z U). (3) φ (a, c; z) φ + (a, c; z) = φ (a, c; z) = z + k=1 z (1 z) λ+ (a) k (c) k z +k. z(i λ ) = (λ + )I λ+1 λi λ (4) z(i λ (a + 1, c) f (z)) = ai λ (a )I λ (a + 1, c) f (z). (5) We also note that I 0 ( + 1, 1) f (z) = z 0 f (t) dt, t I 0 (, 1) f (z) = I 1 ( + 1, 1) f (z) = f (z), I(, 1 1) f (z) = z f (z), I(, 2 1) f (z) = 2z f (z) + z 2 f (z), ( + 1) I( 2 + 1, 1) f (z) = f (z) + z f (z), + 1 I n (a, a) f (z) = D n+ 1 f (z), n N, n >.

3 R.Jayasankar et al. / Filomat 28:10 (2014), The Ruscheweyh derivative D n+ 1 f (z) was introduced by Y. C. Kim et al. [11] and the oerator I λ (a, c)(λ >, a, c R\Z ) was recently introduced by Cho et al. [6], who investigated (among other 0 things) some inclusion relationshis and roerties of various subclasses of multivalent functions in A, (see, [3, 17, 18]) which were defined by means of the oerator I λ (a, c). For λ = c = 1 and a = n +, the Cho-Kwon-Srivastava oerator I λ (a, c) yields the Noor integral oerator I(n 1 +, 1) = I n, (n > ) of (n + 1) the order, studied by Liu and Noor [9](see also the works of [4, 5, 15, 16]). The linear oerator I λ (µ + 2, 1) (λ > 1, µ > 2) was also recently introduced and studied by Choi et al. [8]. For relevant details 1 about further secial cases of the Choi-Saigo-Srivastava oerator I λ (µ + 2, 1), the interested reader may refer 1 to the works by Cho et al. [6] and Choi et al.[8] (see also [2, 7, 10]). In an earlier investigation, a sequence of results using differential subordination with convolution for the univalent case has been studied by Shanmugam [19] while shar coefficient estimates for a certain general class of sirallike functions by means of differential subordination was studied by Xu et al. [25]. A systematic study of the subordination and suerordination using certain oerators under the univalent case has also been studied by Shanmugam et al. [20 23] and by sun et al. [24]. We observe that for these results, many of the investigations have not yet been studied by using aroriate classes of admissible functions. Motivated by the aforementioned works, we obtain certain differential subordination and suerordination results for analytic functions in the oen unit disk using Cho-Kwon-Srivastava oerator by investigating aroriate classes of admissible functions. Sandwich-tye results are also obtained as a consequence of the main results. 2. Preliminaries To rove our results, we need the following definitions and theorems. Denote by L 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) 0 for ζ U\E(q). Further let the subclass of L for which q(0) = a be denoted by L (a), L (0) L 0 and L (1) L 1. Definition 2.1. [13, Definition 2.3a,.27] Let Ω be a set in C, q L and n be a ositive 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 whenever and r = q(ζ), s = kζq (ζ), { } { } t ζq R s + 1 (ζ) kr q + 1, (ζ) where z U, ζ U\E(q) and k n. We write Ψ 1 [Ω, q] as Ψ[Ω, q]. In articular when q(z) = M Mz + a, with M > 0 and a < M, M + az q(u) = U M := {w : w < M}, q(0) = a, E(q) = and q L. In this case, we set Ψ n [Ω, M, a] := Ψ n [Ω, q] and in the secial case when the set Ω = U M, the class is simly denoted by Ψ n [M, a].

4 R.Jayasankar et al. / Filomat 28:10 (2014), Definition 2.2. [14, Definition 3,.817] Let Ω be a set in C, q(z) H [a, n] with q (z) 0. The class of admissible functions Ψ n[ω, q] consists of those functions ψ : C 3 U C that satisfy the admissibility condition whenever and where z U, ζ U and m n 1. In articular, we write Ψ 1 [Ω, q] as Ψ [Ω, q]. ψ(r, s, t; ζ) Ω r = q(z), s = zq (z) m, { } t R s { } zq m R (z) q (z) + 1, Theorem 2.3. [13, Theorem 2.3b,.28] Let ψ Ψ n [Ω, q] with q(0) = a. If the analytic function (z) = a + a n z n + a n+1 z n satisfies ψ ( (z), z (z), z 2 (z); z ) Ω, (z) q(z). Theorem 2.4. [14, Theorem 1,.818] Let ψ Ψ n[ω, q] with q(0) = a. If the analytic function (z) L (a) and ψ ( (z), z (z), z 2 (z); z ) is univalent in U, imlies Ω { ψ ( (z), z (z), z 2 (z); z ) : z U } q(z) (z). 3. Main Results Definition 3.1. Let Ω be a set in C and q(z) L 0 H [0, ]. The class of admissible functions Φ I [Ω, q] consists of those functions φ : C 3 U C that satisfy the admissibility condition whenever where z U, ζ U\E(q) and k. φ(u, v, w; z) Ω u = q(ζ), v = kζq (ζ) + λq(ζ), λ + { } (λ + )(λ + + 1)w 2λ(λ + )v + λ(λ 1)u R kr (λ + )v λu { } ζq (ζ) q + 1, (ζ)

5 R.Jayasankar et al. / Filomat 28:10 (2014), Theorem 3.2. Let φ Φ I [Ω, q]. If f (z) A satisfies { ( φ I λ, I λ+1, I λ+2 ; z ) : z U } Ω (6) I λ q(z). Proof. Define the analytic function (z) in U by (z) = I λ. (7) In view of the relation (λ + )I λ+1 = z(i λ ) + λi λ (8) this imlies from (7) we get, I λ+1 = z (z) + λ(z). (λ + ) (9) Further comutations show that, I λ+2 = z2 (z) + (2λ + 1)z (z) + λ(λ + 1)(z). (10) (λ + )(λ + + 1) Define the transformations from C 3 to C by u = r, v = s + λr t + (2λ + 1)s + λ(λ + 1)r, w =. (11) (λ + ) (λ + )(λ + + 1) Let ψ(r, s, t; z) = φ(u, v, w; z) = φ ( r, s + λr λ + ) t + (2λ + 1)s + λ(λ + 1)r, ; z. (12) (λ + )(λ + + 1) The roof shall make use of Theorem 2.3 Using equations (7), (9) and (10), from (12), we find ψ ( (z), z (z), z 2 (z); z ) = φ ( I λ, I λ+1, I λ+2 ; z ). (13) Hence (6) becomes ψ ( (z), z (z), z 2 (z); z ) Ω. The roof is comleted if it can be shown that the admissibility condition for φ Φ I [Ω, q] is equivalent to the admissibility condition for ψ as given in Definition 2.1. Note that and hence ψ Ψ [Ω, q]. By Theorem 2.3, (z) q(z) or t s (λ + )(λ + + 1)w 2λ(λ + )v + λ(λ 1)u + 1 =, (λ + )v λu I λ q(z). If Ω C is a simly connected domain, Ω = h(u) for some conformal maing h(z) of U onto Ω. In this case the class Φ I [h(u), q] can be written as Φ I [h, q]. The following result is an immediate consequence of Theorem 3.2.

6 Theorem 3.3. Let φ Φ I [h, q]. If f (z) A satisfies R.Jayasankar et al. / Filomat 28:10 (2014), φ(i λ, I λ+1, I λ+2 ; z) h(z) (14) I λ q(z). Our next result is an extension of Theorem 3.3 to the class where the behavior of q(z) on U is not known. Corollary 3.4. Let Ω C and let q(z) be univalent in U, q(0) = 0. Let φ Φ I [Ω, q ρ ] for some ρ (0, 1) where q ρ (z) = q(ρz). If f (z) A and φ(i λ, I λ+1, I λ+2 ; z) Ω, I λ (a, c) q(z). Proof. Theorem 3.2 gives I λ (a, c) q ρ (z). The result is now deduced from q ρ (z) q(z). Theorem 3.5. Let h(z) and q(z) be univalent in U, with q(0) = 0 and set q ρ (z) = q(ρz) and h ρ (z) = h(ρz). Let φ : C 3 C satisfy one of the following conditions: 1. φ Φ I [h, q ρ ], for some ρ (0, 1) or 2. there exists ρ 0 (0, 1) such that φ Φ I [h ρ, q ρ ] for all ρ (ρ 0, 1). If f (z) A satisfies (14), I λ q(z). Proof. The roof is similar to the roof of [13, Theorem 2.3d,.30] and is therefore omitted. The next Theorem yields the best dominant of the differential subordination (14). Theorem 3.6. Let h(z) be univalent in U. Let φ : C 3 U C. Suose that the differential equation φ ( q(z), zq (z) + λq(z), z2 q (z) + (2λ + 1)zq ) (z) + λ(λ + 1)q(z) ; z = h(z) (15) λ + (λ + )(λ + + 1) has a solution q(z) with q(0) = 0 and satisfy one of the following conditions: 1. q(z) L 0 and φ Φ I [h, q], 2. q(z) is univalent in U and φ Φ I [h, q ρ ] for some ρ (0, 1) or 3. q(z) is univalent in U and there exists ρ 0 (0, 1) such that φ Φ I [h ρ, q ρ ], for all ρ (ρ 0, 1). If f (z) A satisfies (14) and q(z) is the best dominant. I λ q(z) Proof. Following the same argument in [13, Theorem 2.3e,.31]. We deduce that q(z) is a dominant from Theorems 3.3 and 3.5. Since q(z) satisfies (15) it is also a solution of (14) and therefore q(z) will be dominated by all dominants. Hence q(z) is the best dominant. In the articular case q(z) = Mz, M > 0 and in view of the Definition 3.1, the class of admissible functions Φ I [Ω, q], denoted by Φ I [Ω, M] is described below.

7 R.Jayasankar et al. / Filomat 28:10 (2014), Definition 3.7. Let Ω be a set in C and M > 0. The class of admissible functions Φ I [Ω, M] consists of those functions φ : C 3 U C such that ( φ Me iθ, k + λ ) L + ((2λ + 1)k + λ(λ + 1)) λ + Meiθ Meiθ, ; z Ω (16) (λ + )(λ + + 1) whenever z U, θ R, R ( Le iθ) (k 1)kM for all real θ and k. Corollary 3.8. Let φ Φ I [Ω, M]. If f (z) A satisfies φ ( I λ, I λ+1, I λ+2 ; z ) Ω, I λ < M. In the secial case Ω = q(u) = {w : w < M}, the class Φ I [Ω, M] is simly denoted by Φ I [M]. Corollary 3.9. Let φ Φ I [M]. If f (z) A satisfies φ ( I λ, I λ+1, I λ+2 ; z ) < M, I λ < M. Remark when Ω = U, λ = a 1(a > 0), = 1 and M = 1, Corollary 3.8 reduces to [6, Theorem2,.231]. When Ω = U, λ = 1, = 1 and M = 1, Corollary 3.8 reduces to [1, Theorem 1,. 477]. Corollary If M > 0 and f (z) A satisfies (λ + )(λ + + 1)I λ+2 (λ + )I λ+1 λ(λ + 1)I λ [ ] < (2 1)λ + ( 1) M I λ < M. Proof. This follows from Corollary 3.8 by taking φ(u, v, w; z) = (λ + )(λ + + 1)w (λ + )v λ(λ + 1)u and Ω = h(u) where h(z) = [ (2 1)λ + ( 1) ] Mz, M > 0. To use Corollary 3.8, we need to show that φ Φ I [Ω, M], that is the admissible condition (16) is satisfied. This follows since, ( φ Me iθ, k + λ λ + Meiθ, L + ((2λ + 1)k + λ(λ + 1))Meiθ ; z) (λ + )(λ + + 1) = L + ((2λ + 1)k + λ(λ + 1))Me iθ (k + λ)me iθ λ(λ + 1)Me iθ = L + (2k 1)λMe iθ (2k 1)λM + R(Le iθ ) (2k 1)λM + k(k 1)M [ (2 1)λ + ( 1) ] M, where z U, θ R, R ( Le ıθ) k(k 1)M and k. Hence by Corollary 3.8, we deduce the required result. (17)

8 R.Jayasankar et al. / Filomat 28:10 (2014), Definition Let Ω be a set in C and q(z) L 0 H 0. The class of admissible functions Φ I,1 [Ω, q] consists of those functions φ : C 3 U C that satisfy the admissibility condition whenever R φ(u, v, w; z) Ω u = q(ζ), v = kζq (ζ) + (λ + 1)q(ζ) (λ + ) { [ ]} (λ + ) (λ + + 1)w (2λ + 2 1)v + 3(λ + 1)u where z U, ζ U\E(q) and k 1. (λ + )v (λ + 1)u kr { } ζq (ζ) q + 1, (ζ) Theorem Let φ Φ I,1 [Ω, q]. If f (z) A satisfies φ I λ, Iλ+1, Iλ+2 ; z : z U Ω (18) z 1 z 1 z 1 Proof. Define an analytic function (z) in U by I λ q(z). z 1 (z) := Iλ z 1. By making use of (8), we get, (19) I λ+1 z 1 Further comutations show that, I λ+2 z 1 = z (z) + (λ + 1)(z). (20) (λ + ) = z2 Define the transformations from C 3 to C by Let u = r, v = (z) + 2(λ + )z (z) + (λ + 1)(λ + + 1)(z). (21) (λ + )(λ + + 1) s + (λ + 1)r, w = (λ + ) ψ(r, s, t; z) = φ(u, v, w; z) = φ ( r, t + 2(λ + )s + (λ + 1)(λ + + 1)r. (22) (λ + )(λ + + 1) s + (λ + 1)r, (λ + ) ) t + 2(λ + )s + (λ + 1)(λ + + 1)r ; z (λ + )(λ + + 1). (23) The roof shall make use of Theorem 2.3 using equations (19), (20) and (21), from (23) we obtain ψ ( (z), z (z), z 2 (z); z ) I λ = φ, Iλ+1, Iλ+2 ; z. (24) z 1 z 1 z 1 Hence (18) becomes, ψ ( (z), z (z), z 2 (z); z ) Ω.

9 R.Jayasankar et al. / Filomat 28:10 (2014), The roof is comleted if it can be shown that the admissibility condition for φ Φ I,1 [Ω, q] is equivalent to the admissibility condition for ψ as given in Definition 2.1. Note that, t (λ + )[(λ + + 1)w (2λ + 2 1)v + 3(λ + 1)u] + 1 = s (λ + )v (λ + 1)u and hence, ψ Ψ[Ω, q]. By Theorem 2.3, (z) q(z) or I λ q(z). z 1 If Ω C is simly connected domain, Ω = h(u), for some conformal maing h(z) of U onto Ω. In this case the class Φ I,1 [h(u), q] can be written as Φ I,1 [h, q]. In the articular case q(z) = Mz, M > 0, the class of admissible functions Φ I,1 [Ω, q], denoted by Φ I,1 [Ω, M]. The following result is an immediate consequence of Theorem Theorem Let φ Φ I,1 [h, q]. If f (z) A satisfies I λ φ, Iλ+1, Iλ+2 ; z h(z) (25) z 1 z 1 z 1 I λ q(z). z 1 Definition Let Ω be a set in C and M > 0. The class of admissible functions Φ I,1 [Ω, M] consists of those functions φ : C 3 U C such that ( ) φ Me iθ k + (λ + 1), Me iθ L + [2(λ + )k + (λ + 1)(λ + + 1)]Meiθ, ; z Ω (26) (λ + ) (λ + )(λ + + 1) whenever z U, θ R, real θ and k 1. R ( Le iθ) (k 1)kM Corollary Let φ Φ I,1 [Ω, M]. If f (z) A satisfies I λ φ, Iλ+1 z 1 z 1, Iλ+2 I λ z 1 < M. ; z Ω, In the secial case Ω = q(u) = {w : w < M}, the class Φ I,1 [Ω, M] is simly denoted by Φ I,1 [M]. Corollary Let φ Φ I,1 [M]. If f (z) A satisfies φ I λ, Iλ+1 z 1 z 1, Iλ+2 I λ z 1 < M. z 1 ; z z 1 < M,

10 R.Jayasankar et al. / Filomat 28:10 (2014), Remark When Ω = U, λ = a 1(a > 0), = 1 and M = 1. Corollary 3.16 reduces to [12, Theorem 2,.231]. When Ω = U, λ = 1, = 1 and M = 1, Corollary 3.16 reduces to [1, Theorem 1,.477]. Corollary If f (z) A, I λ+1 z 1 < M I λ z 1 < M. This follows from Corollary 3.17 by taking φ(u, v, w; z) = v. Corollary If M > 0 and f (z) A satisfies (λ + )(λ + 1)Iλ+2 + (λ + ) Iλ+1 ( (λ + ) 2 1 ) Iλ z 1 z 1 z 1 < [3(λ + )]M I λ z 1. (27) Proof. This follows from Corollary 3.16 by taking φ(u, v, w; z) = (λ + )(λ + 1)w + (λ + )v (λ + 1)(λ + + 1)u and Ω = h(u) where h(z) = 3(λ + )Mz, M > 0. To use Corollary 3.16, we need to show that φ Φ I,1 [Ω, M], that is the admissible condition (26) is satisfied. This follows since ( φ Me iθ k + (λ + 1), Me iθ L + [2(λ + )k + (λ + 1)(λ + + 1)]Meiθ, ; z) (λ + ) (λ + )(λ + + 1) = L + [2(λ + )k + (λ + 1)(λ + + 1)]Me iθ + (k + λ + 1)Me iθ (λ + 1)(λ + + 1)Me iθ = L + { 2(λ + )k + (k + λ + 1) } Me iθ [ 2(λ + )k + (k + + λ 1) ] M + R [ Le iθ] [ 2(λ + )k + (k + + λ 1) ] M + k(k 1)M 3(λ + )M, where z U, θ R, R ( Le iθ) k(k 1)M and k 1. Hence by Corollary 3.16, we deduce the required result. Definition Let Ω be a set in C and q(z) L 1 H. The class of admissible functions Φ I,2 [Ω, q]consists of those functions φ : C 3 U C that satisfy the admissibility condition whenever kr { } ζq (ζ) q + 1 R (ζ) u = q(ζ), v = where U, ζ U\E(q) and k 1. φ(u, v, w; z) Ω [ ] (λ + )q(ζ) + kζq (ζ), ( q(ζ) 0 ), (λ + + 1) q(ζ) { (λ + + 1)(λ + + 2)u 2 w + (λ + + 1)uv (λ + )(λ + + 3)u 3 (λ + + 4)u 2 } + 4, (λ + + 1)uv (λ + )u 2 u

11 R.Jayasankar et al. / Filomat 28:10 (2014), Theorem Let φ Φ I,2 [Ω, q] and I λ 0. If f (z) A satisfies φ I λ+1 I λ, Iλ+2 I λ+1, Iλ+3 I λ+2 ; z : z U Ω, (28) I λ+1 I λ q(z). Proof. Define an analytic function (z) in U by (z) := Iλ+1 I λ. (29) By making use of (8) and (29) we get, I λ+2 I λ+1 = 1 (λ + + 1) Further comutations show that I λ+3 I λ+2 = 1 (λ + + 2) [ 1 + (λ + )(z) + z (z) (z) z (z) (z) Define the transformations from C 3 to C by Let u = r, v = + (λ + )(z) ]. (30) (λ + )z (z) + z2 (z) (z) [ ( 1 s 1 + (λ + )r + (λ + + 1) r)] + z (z) (z) 1 + (λ + )(z) + z (z) (z) ( z (z) (z) ) 2. (31) ) 1 s w = (λ + + 2) ( + (λ + )r (λ + )s + ( r) ( r) ( ) s 2 s + t r r 1 + (λ + )r + ( ) s. (32) r ψ(r, s, t; z) = φ(u, v, w; z) (33) = φ r, 1 λ [ ( s 1 + (λ + )r +, r)] The roof shall make use of Theorem 2.3. Using equations (29), (30) and (31), from (33) we obtain ) 1 s (λ + + 2) ( + (λ + )r (λ + )s + ( s r 1 + (λ + )r + ( s r ) r) ( r) ( 2 s + t ) r ; z ψ ( (z), z (z), z 2 (z); z ) I λ+1 = φ I λ, Iλ+2 I λ+1, Iλ+3 I λ+2 ; z (34) Hence (28) becomes, ψ ( (z), z (z), z 2 (z); z ) Ω. The roof is comleted if it can be shown that the admissibility condition for φ Φ I,2 [Ω, q] is equivalent to the admissibility condition for ψ as given in Definition 2.1.

12 Note that, R.Jayasankar et al. / Filomat 28:10 (2014), t s + 1 = (λ + + 1)(λ + + 2)u2 w + (λ + + 1)uv (λ + )(λ + + 3)u 3 (λ + + 4)u 2 + u (λ + + 1)uv (λ + )u 2 u and hence ψ Ψ[Ω, q]. By Theorem 2.3, (z) q(z) or I λ+1 I λ q(z). If Ω C is a simly connected domain, Ω = h(u), for some conformal maing h(z) of U onto Ω. In this case the class Φ I,2 [h(u), q] is written as Φ I,2 [h, q]. In the articular case q(z) = 1 + Mz, M > 0, the class of admissible functions Φ I,2 [Ω, q] becomes the class Φ I,2 [Ω, M]. Proceeding similarly as in the revious section, the following result is an immediate consequence of Theorem Theorem Let φ Φ I,2 [h, q]. If f (z) A satisfies I λ+1 φ I λ, Iλ+2 I λ+1, Iλ+3 I λ+2 ; z h(z) (35) I λ+1 I λ q(z). The dual roblem of differential subordination, that is, differential suerordination of Cho-Kwon- Srivastava oerator is investigated in this section. For this urose the class of admissible functions is given in the following definition. Definition Let Ω be a set in C and q(z) H [0, ] with zq (z) 0. The class of admissible functions Φ [Ω, q] I consists of those functions φ : C 3 U C that satisfy the admissibility condition whenever z U, ζ U and m. φ(u, v, w; ζ) Ω zq (z) m u = q(z), v = + λq(z), (λ + ) { } (λ + )(λ + + 1)w 2λ(λ + )v + λ(λ 1)u R 1 (λ + )v λu m Theorem Let φ Φ I [Ω, q]. If f (z) A, I λ L 0 and is univalent in U, imlies φ ( I λ, I λ+1, I λ+2 ; z ) { } zq (z) q (z) + 1, Ω { φ ( I λ, I λ+1, I λ+2 ; z ) : z U } (36) q(z) I λ.

13 Proof. From (13) and (36), we have R.Jayasankar et al. / Filomat 28:10 (2014), Ω { ψ ( (z), z (z), z 2 (z); z ) : z U }. From (11), we see that the admissibility condition for φ Φ I [Ω, q] is equivalent to the admissibility condition for ψ as given in Definition 2.2. Hence ψ Ψ [Ω, q] and by Theorem 2.4, q(z) (z) or q(z) I λ. If Ω C is a simly connected domain, Ω = h(u) for some conformal maing h(z) of U onto Ω. In this case the class Φ I [h(u), q] can be written as Φ [h, q]. Proceeding similarly as in the revious section, the I following result is an immediate consequence of Theorem Theorem Let q(z) H [0, ], h(z) is analytic on U and φ Φ I [h, q].if f (z) A, I λ (a, c), f (z) L 0 and φ ( I λ, I λ+1, I λ+2 ; z ) is univalent in U, imlies h(z) φ ( I λ, I λ+1, I λ+2 ; z ) (37) q(z) I λ. Theorem 3.25 and Theorem 3.26 can only be used to obtain subordinants of differential suerordination of the form (36) and (37). The following theorem roves the existence of the best subordinant of (37) for certain φ. Theorem Let h(z) be analytic in U and φ : C 3 U C. Suose that the differential equation ( φ q(z), zq (z) + λq(z), z2 q (z) + (2λ + 1)zq ) (z) + λ(λ + 1)q(z) ; z = h(z) (38) (λ + ) (λ + )(λ + + 1) has a solution q(z) L 0. If φ Φ I [h, q], f (z) A, I λ L 0 and φ ( I λ, I λ+1, I λ+2 ; z ) is univalent in U, imlies and q(z) is the best subordinant. h(z) φ ( I λ, I λ+1, I λ+2 ; z ) q(z) I λ Proof. The roof is similar to the roof of Theorem 3.6 and is therefore omitted. Definition Let Ω be a set in C and q(z) H 0 with zq (z) 0. The class of admissible functions Φ I,1 [Ω, q] consists of those functions φ : C 3 U C that satisfy the admissibility condition whenever R { (λ + ) where z U, ζ U and m 1. φ(u, v, w; ζ) Ω zq (z) m + (λ + 1)q(z) u = q(z), v =, (λ + ) [ ] (λ + + 1)w (2λ + 2 1)v + 3(λ + 1)u (λ + )v (λ + 1)u } 1m { } zq R (z) q (z) + 1,

14 R.Jayasankar et al. / Filomat 28:10 (2014), Now we will give the dual result of Theorem 3.13 for differential suerordination. Theorem Let φ Φ I,1 [Ω, q]. If f (z) A, Iλ L 0 and is univalent in U, Ω φ I λ z 1, Iλ z 1 I λ φ z 1, Iλ+2 z 1, Iλ+1 z 1, Iλ+2 z 1 ; z : z U (39) z 1 imlies q(z) Iλ z 1. Proof. From (24) and (40), we have Ω { ψ((z), z (z), z 2 (z); z) : z U }. From (22), we see that the admissibility condition for φ Φ I,1 [Ω, q] is equivalent to the admissibility condition for ψ as given in Definition 2.2. Hence ψ Ψ [Ω, q] and by Theorem 2.4, q(z) (z) or q(z) Iλ z 1. If Ω C is a simly connected domain, and Ω = h(u) for some conformal maing h(z) of U onto Ω and the class Φ I,1 [h(u), q] cam be written as Φ [h, q]. Proceeding similarly as in the revious section, the following I,1 result is an immediate consequence of Theorem Theorem Let q(z) H 0, h(z) is analytic on U and φ Φ I,1 [h, q]. If f (z) A, I λ L 0 and I λ φ, Iλ+1, Iλ+2 ; z is univalent in U, I λ h(z) φ z 1, Iλ+1 z 1 z 1, Iλ+2 z 1 z 1 ; z (40) z 1 imlies q(z) Iλ z 1. Now, we will give the dual result of Theorem 3.22 for the differential suerordination.

15 R.Jayasankar et al. / Filomat 28:10 (2014), Definition Let Ω be a set in C, q(z) 0, zq (z) 0 and q(z) H. The class of admissible functions Φ I,2 [Ω, q] consists of those functions φ : C 3 U C that satisfy the admissibility condition φ(u, v, w; ζ) Ω whenever u = q(ζ), v = [ ] (λ + )q(z) + zq (z), (λ + + 1) mq(z) R { (λ + + 1)(λ + + 2)u 2 w + (λ + + 1)uv (λ + )(λ + + 3)u 3 (λ + + 4)u where z U, ζ U and m 1. (λ + + 1)uv (λ + )u 2 u Theorem Let φ Φ I,2 [Ω, q]. If f (z) A, Iλ+1 L I λ 1 and I λ+1 φ I λ, Iλ+2 I λ+1, Iλ+3 I λ+2 ; z } 1m { } zq R (z) q (z) + 1, is univalent in U, Ω φ I λ+1 I λ, Iλ+2 I λ+1, Iλ+3 I λ+2 ; z : z U (41) imlies q(z) Iλ+1 I λ. Proof. From (34) and (42), we have Ω { ψ((z), z (z), z 2 (z); z) : z U }. From (33), we see that the admissibility condition for φ Φ I,2 [Ω, q] is equivalent to the admissibility condition for ψ as given in Definition 2.2. Hence ψ Ψ [Ω, q], and by Theorem 2.4, q(z) (z) or q(z) Iλ+1 I λ. If Ω C is a simly connected domain, Ω = h(u) for some conformal maing h(z) of U onto Ω. In this case the class Φ I,2 [h(u), q] can be written as Φ I,2 [h, q]. The following result is an immediate consequence of Theorem Theorem Let h(z) be analytic in U and φ Φ I,2 [h, q]. If f (z) A, Iλ+1 L I λ 1 and I λ+1 φ I λ, Iλ+2 I λ+1, Iλ+3 I λ+2 ; z

16 R.Jayasankar et al. / Filomat 28:10 (2014), is univalent in U, I λ+1 h(z) φ I λ, Iλ+2 I λ+1, Iλ+3 I λ+2 ; z (42) imlies q(z) Iλ+1 I λ. 4. Further Corollaries and Observations Combining Theorem 3.3 and Theorem 3.26, we obtain the following sandwich theorem. Corollary 4.1. Let h 1 (z) and q 1 (z) be analytic functions in U, h 2 (z) be univalent in U.q 2 (z) L 0 with q 1 (0) = q 2 (0) = 0 and φ Φ I [h 2, q 2 ] Φ I [h 1, q 1 ]. If f (z) A, I λ H [0, ] L 0 and φ ( I λ, I λ+1, I λ+2 ; z ) h 2 (z), imlies q 1 (z) I λ q 2 (z). Combining Theorem 3.14 and Theorem 3.30 we obtain the following sandwich theorem. Corollary 4.2. Let h 1 (z) and q 1 (z) be analytic functions in U, h 2 (z) be univalent function in U, q 2 (z) L 0 with q 1 (0) = q 2 (0) = 0 and φ Φ I,1 [h 2, q 2 ] Φ I,1 [h 1, q 1 ]. If f (z) A, Iλ H 0 L 0 and I λ φ z 1, Iλ+1 z 1, Iλ+2 z 1 ; z z 1 is univalent in U, imlies I λ h 1 (z) φ z 1, Iλ+1 z 1 q 1 (z) Iλ z 1, Iλ+2 ; z h 2(z), z 1 q 2 (z). Combining Theorem 3.23 and Theorem 3.33, we obtain the following sandwich theorem. Corollary 4.3. Let h 1 (z) and q 1 (z) be analytic functions in U, h 2 (z) be univalent function in U, q 2 (z) L 1 with q 1 (0) = q 2 (0) = 1 and φ Φ I,2 [h 2, q 2 ] Φ I,2 [h 1, q 1 ]. If f (z) A, Iλ+1 H L I λ 1, I λ 0 and I λ+1 φ I λ, Iλ+2 I λ+1, Iλ+3 I λ+2 ; z

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