Differential subordination theorems for new classes of meromorphic multivalent Quasi-Convex functions and some applications

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1 Int. J. Adv. Appl. Math. and Mech. 2(3) (2015) (ISSN: ) Journal homepage: International Journal of Advances in Applied Mathematics and Mechanics Differential subordination theorems for new classes of meromorphic multivalent Quasi-Convex functions and some applications Abbas Kareem Wanas Department of Mathematics,College of Computer Science and Mathematics, University of Al-Qadisiya,Diwaniya, Iraq Research Article Received 15 December 2014; accepted (in revised version) 09 March 2015 Abstract: MSC: In the present paper, we study new classes of meromorphic multivalent quasi-convex functions, we obtain some subordination theorems for such classes in punctured unit disk. Also we give some applications of firstâăşorder differential subordination. 34G10 26A33 30C45 Keywords: Meromorphic multivalent quasi-convex functions Differential subordination Derivative operator. c 2015 IJAAMM all rights reserved. 1. Introduction Let L p (λ) denotes the class of all functions f of the form: f = p + a n nλ 0 < λ < 1, p N = {1, 2, }, (1) which are analytic in the punctured unit disk U = { C : 0 < < 1}. Also, let T p (λ) denotes the class of all functions f of the form: f = p a n nλ an > 0, 0 < λ < 1, p N = {1, 2, }, (2) which are analytic in the punctured unit disk U. For two functions f and g analytic in = { C : < 1}, we say f is subordinate to g in, denote by f g or f g ( ), if there exists a Schwar function w analytic in U with w (0) = 0 and w < 1( ) such that f = g (w ),( ). In particular, if the function g is univalent in, f g if and only if f (0) = g (0) and f ( ) g ( ). Let ψ : C 3 U C. and let h be univalent in. Assume that k, ψ are analytic and univalent in if k satisfies the differential subordination ψ(k, k, 2 k ; ) h, (3) k is called a solution of the differential subordination. The univalent function q is called a dominant of the solutions of the differential subordination, or more simply dominant if k q for all k satisfying (3). A dominant ˇq Corresponding author. address: abbas.alshareefi@yahoo.com 126

2 Abbas Kareem Wanas / Int. J. Adv. Appl. Math. and Mech. 2(3) (2015) that satisfies ˇq q for all dominants q of (3) is said to be the best dominant of (3). Let L p be the class of all functions Φ of the form: Φ = p + a n n p N = {1, 2, }, which are analytic in the punctured unit disk U. Also, let T p be the class of all functions Φ of the form: Φ = p a n n an > 0, p N = {1, 2, }, which are analytic in the punctured unit disk U. A function f L p (λ) T p (λ) is meromorphic multivalent starlike if f 0 and f R e > 0, U. f Similar,f L p (λ) T p (λ) is meromorphic multivalent convex if f 0 and R e 1 + f f > 0, U. A function f L p (λ) T p (λ) is called meromorphic multivalent Quasi-convex function if there exists a meromorphic multivalent convex function g such that g 0 and f R e g > 0, U. A function Φ L p Tp is meromorphic multivalent starlike if Φ 0 and Φ R e > 0, U. Φ Similar,a function Φ is meromorphic multivalent convex if Φ 0 and R e 1 + Φ > 0, U. Φ Moreover, a function Φ is called meromorphic multivalent Quasi-convex function if there exists a meromorphic multivalent convex function Ψ such that Ψ 0 and ( Φ ) R e Ψ > 0, U. 2. Preliminaries Definition 2.1 (Srivastava and Owa [11]). The fractional derivative of order λ,(0 < λ < 1) of a function f is defined by D λ f = 1 d Γ (1 λ) d 0 f (ɛ) d ɛ, (4) ( ɛ) λ where f is an analytic function in a simply-connected region of the -plane containing the origin, and the multiplicity of ( ɛ) λ is removed by requiring log( ɛ) to be real, when ( ɛ) > 0. Let a, b, c C with c 0,1,2,. The Gaussian hypergeometric function 2 F 1 ( see [12] ) is defined by 2F 1 = 2 F 1 (a, b, c ; ) = (a ) n (b ) n (c ) n n n! ( ), where (x ) n is the Pochhammer symbol defined, in terms of the Gamma function, by (x ) n = Γ (x + n) Γ (x ) = 1 (n = 0), x (x + 1) (x + n 1) (n N ). 127

3 Differential Subordination Theorems for New Classes of Meromorphic Multivalent Quasi-Convex... Definition 2.2 (Goyal and Goyal [4]). Let 0 λ < 1 and µ,ν R. Then, in terms of familiar (Gauss) hypergeometric function 2 F 1, the generalied fractional derivative operator J λ,µ,ν J λ,µ,ν f = of a function f is defined by: 1 d Γ (1λ) d λµ ( 0 ɛ)λ f (ɛ). 2 F 1 (µ λ,ν; 1 λ; 1 ɛ )d ɛ,(0 λ < 1) d n d J λn,µ,ν n f (n λ < n + 1, n N ). where the function f is analytic in a simply-connected region of the -plane containing the origin, with the order f = O ( ε ), ( 0) (5) for ε > ma x {0,µ ν} 1, and the multiplicity of ( ε) λ is removed by requiring l o g ( ε) > 0 to be real, when ( ε) > 0. By comparing (4) with (5), we find J λ,λ,ν f = D λ f,(0 λ < 1). In terms of gamma function, we have J λ,µ,ν n = Γ (n + 1)Γ (n µ + ν + 1) Γ (n µ + 1)Γ (n λ + ν + 1) nµ,(0 λ < 1,µ,ν R, n > ma x 0,µ ν 1). (6) Lemma 2.1 (Miller and Mocanu [8]). Let q be univalent in the unit disk and θ and φ be analytic in a domain D containing q ( ) with φ (w ) 0 when w q ( ). Set Q = q φ q and h = θ q +Q. Suppose that 1- Q is starlike univalent in, and 2- R e h Q > 0 for. If θ (k ) + k φ (k ) θ q + q φ q, k q Lemma 2.2 (Shanmugam and et al. [9]). Let q be convex univalent in the unit disk and ψ and γ C with R e 1 + q + ψ q γ > 0. If k is analytic in and ψk + γ k ψq + γ q, k q Such type of study was carried out by various authors for another classes, like, Ibrahim and Darus [5 7], Darus and Ibrahim [3], Singh et al. [10], Billing [2] and Atshan and Wanas [1]. 3. Subordination results In this section, we obtain some sufficient conditions for subordination of analytic functions in the classes L p (λ) and T p (λ). Theorem 3.1. Let the function q be univalent in U, q 0 and assume that R e 1 + p r + s (1 t ) s r q q + (r 1) + q > 0, (7) t q q where r, s C, t C \{0}. Suppose that q r 1 q is starlike univalent in U. If f L p (λ) satisfies the subordination p f αs p f αr p f (1 t ) p + α g g g p f g (1 t ) q s + t q r p f α q,( U,α C \{0}) g p + q, (8) q 128

4 Abbas Kareem Wanas / Int. J. Adv. Appl. Math. and Mech. 2(3) (2015) Define the function k by p f α, U. (9) k = g Note that (1 t )(k ) s + t (k ) r p + k p f αs p f αr = (1 t ) k g g p f p + α g. (10) p f g From (8) and (10), we have (1 t )(k ) s + t (k ) r By setting p + k (1 t ) q s r + t q k θ (w ) = (1 t ) w s + t p w r a nd φ (w ) = t w r 1, w 0, p + q q we see that θ (w ) is analytic in C, φ (w ) is analytic in C \{0} and that φ (w ) 0, w C \{0}. Also, we get and Q = q φ q = t q r 1 q h = θ q +Q = (1 t ) q s + t q r p + q. q. (11) It is clear that Q is starlike univalent in U, h R e = R e 1 + p r + s (1 t ) s r q q + (r 1) + q > 0. (12) Q t q q From (7) and (12), we have h R e > 0. Q Therefore, by Lemma 2.1, we get k q. By using (9), we obtain the result. By fixing α = p = 1 in Theorem 3.1, we obtain the following corollary: Corollary 3.1. Let the function q be univalent in U, q 0 and assume that R e 1 + p r + s (1 t ) s r q q + (r 1) + q > 0, t q q where r, s C, t C \{0}. Suppose that q r 1 q is starlike univalent in U. If f L p (λ) satisfies the subordination f s f r f (1 t ) 1 + α g g g f g (1 t ) q s + t q r f g q 1 + q, q 129

5 Differential Subordination Theorems for New Classes of Meromorphic Multivalent Quasi-Convex... By taking q = 1+A 1+B (1 B < A 1) in Corollary 3.1, we obtain the following corollary: Corollary 3.2. Let the function q be convex univalent in U, and assume that R e 1 + p r + s (1 t ) 1 + A s r r (A B ) AB + > 0, t 1 + B (1 + A )(1 + B ) where r, s C, t C \{0}. If f L p (λ) satisfies the subordination f s f r f (1 t ) 1 + α g g g f g 1 + A s 1 + A r 1 + 2A + AB 2 (1 t ), 1 + B 1 + B (1 + A )(1 + B ) f g 1 + A 1 + B,(1 B < A 1) and q = 1+A 1+B is the best dominant. Theorem 3.2. Let the function q be convex univalent in U, q 0 and assume that R e 1 + q + 1 > 0, (13) q γ where γ C \{0}. Suppose that p f g α is analytic in U. If f T p (λ) satisfies the subordination p f α + αγ p f α p f g q + γ q, (14) g g p f g p f α q,( U,α C \{0}) g Define the function k by k = p f α, U. (15) g Note that p f α k + γ k = + αγ p f α p f g. (16) g g p f g From (14) and (16), we have k + γ k q + γ q. (17) By setting ψ = 1 in Lemma 2.2, we get k q. By using (15), we obtain the result. By fixing α = p = 1 in Theorem 3.2, we obtain the following corollary: 130

6 Abbas Kareem Wanas / Int. J. Adv. Appl. Math. and Mech. 2(3) (2015) Corollary 3.3. Let the function q be convex univalent in U, q p f 0 and assume that (3.2). Suppose that is analytic in U. g If f T p (λ) satisfies the subordination f g f g f + γ g q, ( U ). f f g q + γ q, g By taking q = 1+ 1 in Corollary 3.3, we obtain the following corollary: Corollary 3.4. Let the function q be convex univalent in U and assume that R e (1 )(1 + ) + 1 > 0. γ If f T p (λ) satisfies the subordination f g f + γ g f f g g γ (1 )(1 + ), f g 1 +, ( U ) 1 4. Applications of fractional derivative operator In this section, we introduce some applications of section 3 containing fractional derivative operators. Assume that Φ = σ n n. By Definition 2.1, we have where D λ Φ = Γ (n + 1) Γ (n + 1 λ) σ n nλ = a n nλ, a n = Γ (n + 1) Γ (n + 1 λ) σ n, n = 0, 1, 2,.... Thus p + D λ Φ L p (λ) and p D λ Φ T p (λ)(σ n 0), we have the following results: Theorem 4.1. Let the assumptions of Theorem 3.1 hold. Then α p p λ + D Φ q, U p + D λ Ψ 131

7 Differential Subordination Theorems for New Classes of Meromorphic Multivalent Quasi-Convex... Define the function f by f = p + D λ Φ ( U ), it can easily observed thatf L p (λ). Thus by using Theorem 3.1, we obtain the result. Theorem 4.2. Let the assumptions of Theorem 3.2 hold. Then α p p λ D Φ q, U p D λ Ψ Define the function f by f = p D λ Φ ( U ), it can easily observed thatf T p (λ). Thus by using Theorem 3.2, we obtain the result. By using (6), we have where J λ,µ,ν Φ = Γ (n + 1)Γ n µ + ν + 1 Γ n µ + 1 Γ (n λ + ν + 1) σ n nµ = a n nµ, a n = Γ (n + 1)Γ n µ + ν + 1 Γ n µ + 1 Γ (n λ + ν + 1) σ n, n = 0, 1, 2,.... Let µ = λ. Then p + J λ,µ,ν Φ L p (λ) and p J λ,µ,ν Φ T p (λ) (σ n 0), we have the following results: Theorem 4.3. Let the assumptions of Theorem 3.1 hold. Then p p + J λ,µ,ν α Φ p + J λ,µ,ν Ψ q, U Define the function f by f = p + J λ,µ,ν Φ ( U ), it can easily observed thatf L p (λ). Thus by using Theorem 3.1, we obtain the result. Theorem 4.4. Let the assumptions of Theorem 3.2 hold. Then p p J λ,µ,ν α Φ p J λ,µ,ν Ψ q, U Define the function f by f = p J λ,µ,ν Φ ( U ), it can easily observed thatf T p (λ). Thus by using Theorem 3.2, we obtain the result. 132

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