Filomat 28:10 (2014), 2009 2026 DOI 10.2298/FIL1410009J Published by Faculty of Sciences and Mathematics, University of Niš, Serbia Available at: htt://www.mf.ni.ac.rs/filomat 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 - 631 209, India. b School of Mathematical Sciences, Faculty of Science and Technology Universiti Kebangsaan Malaysia, Bangi 43600. c Deartment of Mathematics, University College of Engineering Tindivanam, Anna University Chennai, Melakkam -604 001, 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+1 +..., 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=+1 2010 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 Email addresses: jaymaths@gmail.com (R.Jayasankar), maslina@krisc.cc.ukm.my (Maslina Darus), sivasaisastha@rediffmail.com (S. Sivasubramanian)
R.Jayasankar et al. / Filomat 28:10 (2014), 2009 2026 2010 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 >.
R.Jayasankar et al. / Filomat 28:10 (2014), 2009 2026 2011 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].
R.Jayasankar et al. / Filomat 28:10 (2014), 2009 2026 2012 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 + 1 1 { } 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+1 +... 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)
R.Jayasankar et al. / Filomat 28:10 (2014), 2009 2026 2013 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.
Theorem 3.3. Let φ Φ I [h, q]. If f (z) A satisfies R.Jayasankar et al. / Filomat 28:10 (2014), 2009 2026 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.
R.Jayasankar et al. / Filomat 28:10 (2014), 2009 2026 2015 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 3.10. 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 3.11. 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)
R.Jayasankar et al. / Filomat 28:10 (2014), 2009 2026 2016 Definition 3.12. 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 3.13. 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.
R.Jayasankar et al. / Filomat 28:10 (2014), 2009 2026 2017 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 3.13. Theorem 3.14. 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 3.15. 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 3.16. 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 3.17. 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,
R.Jayasankar et al. / Filomat 28:10 (2014), 2009 2026 2018 Remark 3.18. 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 3.19. 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 3.20. 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 3.21. 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 [ ] 1 1 + (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
R.Jayasankar et al. / Filomat 28:10 (2014), 2009 2026 2019 Theorem 3.22. 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) + 2 + ]. (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 + 2 + (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 + + 1 [ ( 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 + 2 + (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.
Note that, R.Jayasankar et al. / Filomat 28:10 (2014), 2009 2026 2020 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 3.22. Theorem 3.23. 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 3.24. 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 3.25. 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.
Proof. From (13) and (36), we have R.Jayasankar et al. / Filomat 28:10 (2014), 2009 2026 2021 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 3.25. Theorem 3.26. 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 3.27. 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 3.28. 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,
R.Jayasankar et al. / Filomat 28:10 (2014), 2009 2026 2022 Now we will give the dual result of Theorem 3.13 for differential suerordination. Theorem 3.29. 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 3.29. Theorem 3.30. 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.
R.Jayasankar et al. / Filomat 28:10 (2014), 2009 2026 2023 Definition 3.31. 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 = [ ] 1 1 + (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 2 + 4 where z U, ζ U and m 1. (λ + + 1)uv (λ + )u 2 u Theorem 3.32. 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 3.32. Theorem 3.33. 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
R.Jayasankar et al. / Filomat 28:10 (2014), 2009 2026 2024 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
R.Jayasankar et al. / Filomat 28:10 (2014), 2009 2026 2025 is univalent in U, imlies I λ+1 h 1 (z) φ I λ, Iλ+2 I λ+1, Iλ+3 I λ+2 ; z h 2(z), q 1 (z) Iλ+1 q I λ 2 (z). Acknowledgement : The authors would like to record their sincere thanks to the referees for their valuable suggestions. The work here is artially suorted by AP-2013-009 and DIP-2013-001 of the second author. References [1] M. K. Aouf, H. M. Hossen and A. Y. Lashin, An alication of certain integral oerators, J. Math. Anal. Al. 248 (2000), no. 2, 475 481. [2] M. K. Aouf and H. M. Srivastava, A linear oerator and associated families of meromorhically multivalent functions of order α, Adv. Stud. Contem. Math. (Kyungshang) 13 (2006), no. 1, 27 56. [3] M. K. Aouf, H. Silverman and H. M. Srivastava, Some subclasses of multivalent functions involving a certain linear oerator, Adv. Stud. Contem. Math. (Kyungshang) 14 (2007), no. 2, 215 232. [4] M. K. Aouf and B. A. Frasin, Proerties of some families of meromorhic multivalent functions involving certain linear oerator, Filomat 24(3) (2010), 35 54. [5] M. Çağlar, H. Orhan and E. Deniz, Majorization for certain subclasses of analytic functions involving the generalized Noor integral oerator, Filomat 27(1) (2013), 143 148. [6] N. E. Cho, O. S. Kwon and H. M. Srivastava, Inclusion relationshis and argument roerties for certain subclasses of multivalent functions associated with a family of linear oerators, J. Math. Anal. Al. 292 (2004), no. 2, 470 483. [7] N. E. Cho, O. S. Kwon and H. M. Srivastava, Inclusion and argument roerties for certain subclasses of meromorhic functions associated with a family of multilier transformations, J. Math. Anal. Al. 300 (2004), no. 2, 505 520. [8] J. H. Choi, M. Saigo and H. M. Srivastava, Some inclusion roerties of a certain family of integral oerators, J. Math. Anal. Al. 276 (2002), no. 1, 432 445. [9] J.-L. Liu and K. I. Noor, Some roerties of Noor integral oerator, J. Nat. Geom. 21 (2002), no. 1-2, 81 90. [10] J.-L. Liu and H. M. Srivastava, A linear oerator and associated families of meromorhically multivalent functions, J. Math. Anal. Al. 259 (2001), no. 2, 566 581. [11] Y. C. Kim, K. S. Lee and H. M. Srivastava, Some alications of fractional integral oerators and Ruscheweyh derivatives, J. Math. Anal. Al. 197 (1996), no. 2, 505 517. [12] Y. C. Kim and H. M. Srivastava, Inequalities involving certain families of integral and convolution oerators, Math. Inequal. Al. 7 (2004), no. 2, 227 234. [13] S. S. Miller and P. T. Mocanu, Differential subordinations, Monograhs and Textbooks in Pure and Alied Mathematics, 225, Dekker, New York, 2000. [14] S. S. Miller and P. T. Mocanu, Subordinants of differential suerordinations, Comlex Var. Theory Al. 48 (2003), no. 10, 815 826. [15] K. I. Noor, Some classes of -valent analytic functions defined by certain integral oerator, Al. Math. Comut. 157 (2004), no. 3, 835 840. [16] K. I. Noor and M. A. Noor, On certain classes of analytic functions defined by Noor integral oerator, J. Math. Anal. Al. 281 (2003), no. 1, 244 252. [17] J. K. Prajaat, R. K. Raina and H. M. Srivastava, Inclusion and neighborhood roerties for certain classes of multivalently analytic functions associated with the convolution structure, JIPAM. J. Inequal. Pure Al. Math. 8 (2007), no. 1, Article 7, 8. (electronic). [18] R. K. Raina and H. M. Srivastava, Inclusion and neighborhood roerties of some analytic and multivalent functions, JIPAM. J. Inequal. Pure Al. Math. 7 (2006), no. 1, Article 5, 6. (electronic). [19] T. N. Shanmugam, Convolution and differential subordination, Internat.J Math. Math. Sci. Vol.12 (2) (1989), 333 340. [20] T. N. Shanmugam, V. Ravichandran and S.Sivasubramanian, Differential sandwich theorems for some subclasses of analytic functions, Austral. Math. Anal. Al., 3(1), Art. 8, (2006), : 1 11. [21] T. N. Shanmugam, S. Sivasubramanian and H. M. Srivastava, Differential sandwich theorems for certain subclasses of analytic functions involving multilier transformations, Int. Transforms Sec. Functions., (17)(12) (2006), 889 899. [22] T. N. Shanmugam, S. Sivasubramanian and M.Darus, On certain subclasses of functions involving a linear Oerator, Far East J. Math. Sci. 23(3), (2006), 329 339. [23] T. N. Shanmugam, S. Sivasubramanian and H. Silverman, On sandwich theorems for some classes of analytic functions, Int. J. Math. Math. Sci, 2006, Article ID: 29864, :1 13. [24] Y. Sun, W.-P. Kuang and Z.-H. Liu, Subordination and suerordination results for the family of Jung-Kim-Srivastava integral oerators, Filomat 24(1) (2010), 69 85.
R.Jayasankar et al. / Filomat 28:10 (2014), 2009 2026 2026 [25] Q.-H. Xu, C.-B. Lv, N.-C. Luo, and H. M. Srivastava, Shar coefficient estimates for a certain general class of sirallike functions by means of differential subordination, Filomat 27 (2013), 1351 1356.