STRONG DIFFERENTIAL SUBORDINATIONS FOR HIGHER-ORDER DERIVATIVES OF MULTIVALENT ANALYTIC FUNCTIONS DEFINED BY LINEAR OPERATOR

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1 Khayyam J. Math , no. 2, DOI: /kjm STRONG DIFFERENTIA SUBORDINATIONS FOR HIGHER-ORDER DERIVATIVES OF MUTIVAENT ANAYTIC FUNCTIONS DEFINED BY INEAR OPERATOR ABBAS KAREEM WANAS 1 AND ABDURAHMAN H. MAJEED 2 Communicated by A.K. Mirmostafaee Abstract. In the present paper, we introduce and study a new class of higherorder derivatives multivalent analytic functions in the open unit disk and closed unit disk of the complex plane by using linear operator. Also we obtain some interesting properties of this class and discuss several strong differential subordinations for higher-order derivatives of multivalent analytic functions. 1. Introduction and preliminaries et U = { C : < 1 and U = { C : 1 denote the open unit disk and the closed unit disk of the complex plane, respectively. Denote by H U U the class of analytic functions in U U. For n a positive integer and a C, let H [a, n, ζ] = { f H U U : f, ζ = a + a n ζ n + a n+1 ζ n+1 +, U, ζ U, where a k ζ are holomorphic functions in U for k n. et A nζ = { f H U U : f, ζ = a n+1 ζ n+1 +, U, ζ U, with A 1ζ = A ζ, where a kζ are holomorphic functions in U for k n + 1. Denote by S ζ = { f H [a, n, ζ] : Re { f, ζ >, U forall ζ U f, ζ Date: Received: 13 June 217; Revised: 17 September 217; Accepted: 2 September 217. Corresponding author. 21 Mathematics Subject Classification. 3C45, 3A2, 34A4. Key words and phrases. Analytic functions, strong differential subordinations, convex function, higher-order derivatives, linear operator. 16

2 STRONG DIFFERENTIA SUBORDINATIONS FOR HIGHER-ORDER 161 the class of starlike functions in U U and by { { f Kζ = f H, ζ [a, n, ζ] : Re 2 f, ζ + 1 the class of convex functions in U U. >, U forall ζ U Definition 1.1. [8] et f, ζ, g, ζ be analytic in U U. The function f, ζ is said to be strongly subordinate to g, ζ, written f, ζ g, ζ, U, ζ U, if there exists an analytic function w in U with w = and w < 1, U such that f, ζ = g w, ζ for all ζ U. Remark 1.2. [8] 1 Since f, ζ is analytic in U U, for all ζ U and univalent in U, for all ζ U, Definition 1.1 is equivalent to f, ζ = g, ζ for all ζ U and f U U g U U. 2 If f, ζ = f and g, ζ = g, the strong subordination becomes the usual notion of subordination. et A ζ p denote the subclass of the functions f, ζ H U U of the form: f, ζ = p + a p+k ζ p+k, p N = {1, 2,, U, ζ U, 1.1 which are analytic and multivalent in U U. Upon differentiating both sides of 1.1 j-times with respect to, we obtain where f, ζ j = δp, j p j + δp, j = δp + k, ja p+k ζ p+k j, p N, j N = N {, p > j, p j! = 1 j = pp 1 p j + 1, j. For a R, c R \ Z, where Z = {, 1, 2,, λ < 1, p N, α > p, µ, ν R with µ ν p < 1 and A ζ p, the linear operator a, c : A ζ p A ζ p see [4] is defined by a, cf, ζ = p + It is easily verified from 1.2 that a, cf, ζ c k p + 1 µ k p + 1 λ + ν k α + p k a p+k ζ p+k. a k p + 1 k p + 1 µ + ν k k! 1.2 = α + p +1 a, cf, ζ α a, cf, ζ. 1.3

3 162 A.K. WANAS, A.H. MAJEED Differentiating 1.3 j-times with respect to, we get a, cf, ζ j+1 = α + p +1 a, cf, ζ j α + j a, cf, ζ j. 1.4 Note that the linear operator a, c unifies many other operators considered earlier. In particular: 1,p,α,ν a, c Jp α a, c see Cho et al. [1]. 2,p,α,ν a, a D α+p 1 see Goel and Sohi [2]. 3,p,1,ν p + 1 λ, 1 Ω λ,p see Srivastava and Aouf [1]. 4,1,α 1,ν a, c Jc a,α see Hohlov [3]. 5,1 α,α,ν a, c p a, c see Saitoh [9]. 6,p,1,ν p + α, 1 J α,p, α Z, α > p see iu and Noor [5]. The purpose of this paper is to apply a method based on the strong differential subordination in order to investigate various useful and interesting properties for higher-order derivatives of multivalent analytic functions involving linear operator. We need the following lemmas to study the strong differential subordinations: emma 1.3. [7] et h, ζ be convex function with h, ζ = a, for every ζ U and let γ C = C\{ with Re γ. If p H [a, n, ζ] and p, ζ + 1 γ p, ζ h, ζ, U, ζ U, 1.5 p, ζ q, ζ h, ζ, U, ζ U, where q, ζ = 1.5. γ n γ n t γ n 1 h t, ζ dt is convex and it is the best dominant of emma 1.4. [6] et q, ζ be convex function in U U for all ζ U and let h, ζ = q, ζ + nδq, ζ, U, ζ U, where δ > and n is a positive integer. If is analytic in U U and p, ζ = q, ζ + p n ζ n + p n+1 ζ n+1 +, p, ζ + δp, ζ h, ζ, U, ζ U, p, ζ q, ζ, U, ζ U, and this result is sharp.

4 STRONG DIFFERENTIA SUBORDINATIONS FOR HIGHER-ORDER Main results Definition 2.1. et η [, 1, a R, c R \ Z, λ < 1, p N, α > p, µ, ν R, µ ν p < 1, j N and p > j. A function f, ζ A ζ p is said to be in the class N a, c, λ, p, α, µ, ν, j, η, ζ if it satisfies the inequality { a, cf, ζ j+1 Re j+1 > η, U, ζ U. p j 1 In the first theorem, we demonstrate that the class N a, c, λ, p, α, µ, ν, j, η, ζ is convex set. Theorem 2.2. The set N a, c, λ, p, α, µ, ν, j, η, ζ is convex. Proof. et the functions f i, ζ = p + a p+k,i ζ p+k, i = 1, 2 be in the class N a, c, λ, p, α, µ, ν, j, η, ζ. It is sufficient to show that the function T, ζ = t 1 f 1, ζ + t 2 f 2, ζ is in the class N a, c, λ, p, α, µ, ν, j, η, ζ, where t 1 and t 2 non-negative and t 1 + t 2 = 1. Since T, ζ = p + t 1 a p+k,1 ζ + t 2 a p+k,2 ζ p+k, a, ct, ζ = p c k p + 1 µ k p + 1 λ + ν k α + p k + t 1 a p+k,1 ζ + t 2 a p+k,2 ζ p+k. a k p + 1 k p + 1 µ + ν k k! 2.1 Differentiating both sides of 2.1 j + 1-times with respect to, we obtain a, ct, ζ j+1 = p j 1! p j 1 + c kp + 1 µ k p + 1 λ + ν k α + p k a k p + 1 k p + 1 µ + ν k k! Thus, a, ct, ζ j+1 = p j 1 p j 1! + c kp + 1 µ k p + 1 λ + ν k α + p k a k p + 1 k p + 1 µ + ν k k! p + k! p + k j 1! t 1 a p+k,1 ζ + t 2 a p+k,2 ζ p+k j 1. p + k! p + k j 1! t 1 a p+k,1 ζ + t 2 a p+k,2 ζ k.

5 164 A.K. WANAS, A.H. MAJEED Now, Re { + t 1 Re + t 2 Re { { a, ct, ζ j+1 = p j 1 p + k! p + k j 1! p + k! p + k j 1! p j 1! c k p + 1 µ k p + 1 λ + ν k α + p k a p+k,1 ζ k a k p + 1 k p + 1 µ + ν k k! c k p + 1 µ k p + 1 λ + ν k α + p k a p+k,2 ζ k. a k p + 1 k p + 1 µ + ν k k! 2.2 Since f 1, f 2 N a, c, λ, p, α, µ, ν, j, η, ζ, we get { p + k! c k p + 1 µ k p + 1 λ + ν k α + p k Re a p+k,i ζ k p + k j 1! a k p + 1 k p + 1 µ + ν k k! > η, i = 1, p j 1! From 2.2 and 2.3, we have { a, ct, ζ j+1 Re j+1 > p j 1 p j 1! + t 1 η + t 2 η p j 1! p j 1! = η. Therefore, T, ζ N a, c, λ, p, α, µ, ν, j, η, ζ and we obtain the desired result. Next, we establish the following inclusion relationship for the class N a, c, λ, p, α, µ, ν, j, η, ζ. Theorem 2.3. et h, ζ = ζ+2η ζ 1+, U, ζ U, η [, 1. Then N a, c, λ, p, α + 1, µ, ν, j, η, ζ N a, c, λ, p, α, µ, ν, j, η, ζ, where η = 2η ζ + 2 ζ η α + p Bα + p 1 with Bx = 1 Proof. Suppose that f, ζ N a, c, λ, p, α + 1, µ, ν, j, η, ζ. have α + p +1 a, cf, ζ j = a, cf, ζ j+1 Differentiating both sides of 2.4 with respect to, we get α + p +1 a, cf, ζ j+1 t x 1+t dt. Using 1.4, we + α + j a, cf, ζ j. 2.4 = a, cf, ζ j α + j + 1 a, cf, ζ j

6 STRONG DIFFERENTIA SUBORDINATIONS FOR HIGHER-ORDER 165 et the function F be defined by a, cf, ζ j+1 F, ζ = j+1, U, ζ U. p j 1 After some computations, we have p j 1 F, ζ + F a, cf, ζ j+2, ζ = j p j 2 It follows from 2.5 and 2.6 that +1 a, cf, ζ j+1 = F, ζ + 1 p j 1 α + p F, ζ. 2.7 Since f, ζ N a, c, λ, p, α + 1, µ, ν, j, η, ζ, from Definition 2.1 and the equation 2.7, we obtain { Re F, ζ + 1 α + p F, ζ > η, U, ζ U. We note that Hence h, ζ = { h, ζ Re 2 h, ζ + 1 = Re 2η ζ and h 4η ζ 2, ζ = { 1 = Re 1 + Therefore h, ζ is a convex function and = { 1 rcos θ + i sin θ 1 + rcos θ + i sin θ 1 r r cos θ + r 2 >. F, ζ + 1 α + p F ζ + 2η ζ, ζ. 1 + Since the image of the unit disk U through function h is the semi plane {w C : w > η. By using emma 1.3, we deduce that F, ζ q, ζ h, ζ, where q, ζ = α + p α+p ζ + 2η ζt t α+p 1 dt = 2η ζ +2 ζ η α + p 1 + t α+p t α+p t dt, and q, ζ is convex and which is the best dominant. Since q, ζ is convex and q U U is symmetric with respect to the real axis, we obtain Re { a, cf, ζ j+1 p j 1 min Re {q, ζ =1 = Re {q 1, ζ = 2η ζ + 2 ζ η α + p 1 t α+p t dt = η.

7 166 A.K. WANAS, A.H. MAJEED Thus, f, ζ N a, c, λ, p, α, µ, ν, j, η, ζ and we obtain the desired result. In the next result, we discuss strong subordination property of the integral operator. Theorem 2.4. et q, ζ be a convex function such that q, ζ = 1 and let h be the function h, ζ = q, ζ + 1 α+2p q, ζ, where α > p. Suppose that G, ζ = α + 2p α+p If f A ζ p satisfies the strong differential subordination p j 1! p j 1! and this result is sharp. Proof. Suppose that t α+p 1 f t, ζ dt, U, ζ U. 2.8 a, cf, ζ j+1 h, ζ, 2.9 p j 1 a, cg, ζ j+1 q, ζ p j 1 F, ζ = p j 1! a, cg, ζ j+1, U, ζ U. 2.1 p j 1 Then the function F, ζ is analytic in U U and F, ζ = 1. From 2.8, we have α+p G, ζ = α + 2p Differentiating both sides of 2.11 with respect to, we get and α + 2p t α+p 1 f t, ζ dt α + 2p f, ζ = α + p G, ζ + G, ζ a, cf, ζ = α + p a, cg, ζ+ a, cg, ζ. Differentiating the last relation j + 1-times with respect to, we have a, cf, ζ j+1 = α + p + j + 1 α + 2p + α + 2p a, cg, ζ j+1 a, cg, ζ j+2. +2

8 So STRONG DIFFERENTIA SUBORDINATIONS FOR HIGHER-ORDER 167 p j 1! = + a, cf, ζ j+1 p j 1 α + p + j + 1p j 1! α + 2p p j 1! α + 2p From 2.1 and 2.12, we obtain F, ζ + 1 α + 2p F, ζ = Using 2.13, 2.9 becomes a, cg, ζ j+1 p j 1 a, cg, ζ j p j 2 p j 1! a, cf, ζ j p j 1 F, ζ + 1 α + 2p F, ζ q, ζ + 1 α + 2p q, ζ. An application of emma 1.4 yields F, ζ q, ζ. By using 2.9, we obtain p j 1! a, cg, ζ j+1 q, ζ. p j 1 Next we will give particular case of Theorem 2.4 obtained for appropriate choices of the functions f, ζ = ζe and q, ζ = 1 + ζ with j = λ = µ = 1, 2 p = α = c = 1 and a = 2. Example 2.5. et U, ζ U. If 3ζ ζ + 1e ζ, [ ] e e ζ 3 2. Theorem 2.6. et q, ζ be a convex function such that q, ζ = 1 and let h be the function h, ζ = q, ζ + q, ζ. If f A ζ p satisfies the strong differential subordination +1 a, cf, ζ j j j h, ζ, 2.14 a, cf, ζ +1 a, cf, ζ j j j q, ζ. a, cf, ζ

9 168 A.K. WANAS, A.H. MAJEED Proof. Suppose that F, ζ = + +1 a, cf, ζ j j j, U, ζ U a, cf, ζ Then the function F, ζ is analytic in U U and F, ζ = 1. Differentiating both sides of 2.15 with respect to and using 2.14, we have +1 F, ζ + F a, cf, ζ j, ζ = j j a, cf, ζ a, cf, ζ j +1 a, cf, ζ j+1 +1 = +1 [ a, cf, ζ a, cf, ζ j [ j a, cf, ζ a, cf, ζ j ] j 2 a, cf, ζ j+1 ] j 2 +1 a, cf, ζ j [ a, cf, ζ a, cf, ζ j [ j a, cf, ζ = +1 a, cf, ζ j j j a, cf, ζ ] j 2 a, cf, ζ j+1 ] j 2 h, ζ. An application of emma 1.4, we obtain +1 a, cf, ζ j j j q, ζ. a, cf, ζ Next we will give particular case of Theorem 2.6 obtained for appropriate choices of the functions f, ζ = ζ 2η 1ζ+ζ and q, ζ =, η < with j = λ = µ = 1, p = α = c = 1 and a = 2. Example 2.7. et U, ζ U. If η 1ζ2 + 2ηζ + ζ 1 + 2,

10 STRONG DIFFERENTIA SUBORDINATIONS FOR HIGHER-ORDER η 1ζ + ζ. 1 + Theorem 2.8. et h, ζ be a convex function such that h, ζ = 1. If f A ζ p satisfies the strong differential subordination p j 1! p j! a, cf, ζ j+1 h, ζ, 2.16 p j 1 a, cf, ζ j q, ζ h, ζ, p j where q, ζ = p j p j tp j 1 h t, ζ dt is convex and it is the best dominant. Proof. Suppose that F, ζ = p j! a, cf, ζ j, U, ζ U p j Then the function F, ζ is analytic in U U and F, ζ = 1. Simple computations from 2.17, we get F, ζ + 1 p j F, ζ = Using 2.18, 2.16 becomes p j 1! F, ζ + 1 p j F, ζ h, ζ. a, cf, ζ j p j 1 An application of emma 1.3 with n = 1, γ = p j yields p j! a, cf, ζ j q, ζ = p j t p j 1 h t, ζ dt h, ζ. p j p j By taking p = 1, j = and h, ζ = ζ+2η ζ 1+, η < 1 in Theorem 2.8, we obtain the following corollary: Corollary 2.9. If f A ζ p satisfies the strong differential subordination λ,1,α a, cf, ζ λ,1,α a, cf, ζ 1 ζ + 2η ζ, 1 + ζ + 2η ζt 2ζ η dt = 2η ζ + ln t

11 17 A.K. WANAS, A.H. MAJEED Theorem 2.1. et h, ζ be a convex function such that h, ζ = 1. f A ζ p satisfies the strong differential subordination 1 βp j! a, cf, ζ j σ σp j! p j + βp j! σp j! a, cf, ζ j+1 σ p j p j 1 If h, ζ, 2.19 p j! σp j! a, cf, ζ j σ q, ζ h, ζ, p j where q, ζ = p j β Proof. Suppose that F, ζ = p j β t p j β p j! σp j! 1 h t, ζ dt is convex and it is the best dominant. a, cf, ζ j σ, U, ζ U. 2.2 p j Then the function F, ζ is analytic in U U and F, ζ = 1. Differentiating both sides of 2.2 with respect to, we have F, ζ + β p j F, ζ = + 1 βp j! σp j! βp j! σp j! a, cf, ζ j p j σ a, cf, ζ j+1 σ p j p j From 2.19 and 2.21, we get F, ζ + β p j F, ζ h, ζ. An application of emma 1.3 with n = 1, γ = p j β p j! σp j! q, ζ = p j p j β β a, cf, ζ j p j yields σ t p j β 1 h t, ζ dt h, ζ. Acknowledgement. The authors would like to thank the referees for their valuable suggestions.

12 STRONG DIFFERENTIA SUBORDINATIONS FOR HIGHER-ORDER 171 References 1. N.E. Cho, O.H. Kwon and H.M. Srivastava, Inclusion and argument properties for certain subclass of multivalent functions associated with a family of linear operators, J. Math. Anal. Appl., , R.M. Goel and N.S. Sohi, A new criterion for p-valent functions, Proc. Amer. Math. Soc., Yu. E. Hohlov, Operators and operations in the class of univalent functions, Iv, Vvssh. Ucebn. Zaved. Math R. Kargar, A. Bilavi, S. Abdolahi and S. Maroufi, A class of multivalent analytic functions defined by a new linear operator, J. Math. Comp. Sci., J.. iu and K.I. Noor, Some properties of Noor integral operator, J. Natur. Geom., S.S. Miller and P.T. Mocanu, On some classes of first-order differential subordinations, Michigan Math. J., , no. 2, S.S. Miller, P.T. Mocanu, Differential Subordinations: Theory and Applications, Series on Monographs and Textbooks in Pure and Applied Mathematics, Vol. 225, Marcel Dekker Inc., New York, Basel, G.I. Oros and Gh. Oros, Strong differential subordination, Turk. J. Math., H. Saitoh, A liner operator and its applications of first order differential subordinations, Math. Japon., H.M. Srivastava and M.K. Aouf, A certain fractional derivative operator and its applications to a new class of analytic and multivalent functions with negative coefficients, Int. J. Math. Anal. Appl., Department of Mathematics, College of Science, Baghdad University, Iraq. address: abbas.kareem.w@qu.edu.iq 2 Department of Mathematics, College of Science, Baghdad University, Iraq. address: ahmajeed6@yahoo.com