DIFFERENTIAL SUBORDINATION FOR MEROMORPHIC MULTIVALENT QUASI-CONVEX FUNCTIONS. Maslina Darus and Imran Faisal. 1. Introduction and preliminaries

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1 FACTA UNIVERSITATIS (NIŠ) SER. MATH. INFORM. 26 (2011), 1 7 DIFFERENTIAL SUBORDINATION FOR MEROMORPHIC MULTIVALENT QUASI-CONVEX FUNCTIONS Maslina Darus and Imran Faisal Abstract. An attempt has been made to introduce certain new classes of meromorphic multivalent quasi-convex functions and discuss its differential subordination properties in the punctured unit disku. 1. Introduction and preliminaries LetE + p,α denotes the family of all functions F, of the form (1.1) F= 1 z p+ a n z n n/α α N\{1}, p=1, 2,... which are analytic in the punctured unit disku={z : z C z <1}. SimilarlyE p,α denotes the family of all functions F, of the form (1.2) F= 1 z p a n z n n/α α N\{1}, p=1, 2,... which are analytic in the punctured unit disku. For two functions f and analytic inu, we say that the function f is subordinate to inuand write f or simply f if there exists a Schwarz function w which is analytic inuwith w(0)=0 and w <1 such that f = (w) z U. Letφ : C 3 U C and let h analytic inu. Assume that p,φ are analytic and univalent inuand p satisfies the differential superordination (1.3) h φ(p, zp, z 2 p ; z). Then p is called a solution of the differential superordination. Received Mathematics Subject Classification. Primary 30C45 The authors were supported by UKM-ST-06-FRGS

2 2 M. Darus and I. Faisal An analytic function q is called a subordinant if q p,for all p satisfying equation (1.3). A univalent function q such that p qfor all subordinates p of equation (1.3) is said to be the best subordinant. LetE + p be the class of all functions of the form f = 1 z p+ a n z n p=1, 2,... which are analytic in the punctured unit disku. SimilarlyE p be the class of all functions of the form f = 1 z p a n z n p=1, 2,... which are analytic in the punctured unit disku. A function f E + p (E p ) is meromorphic multivalent starlike if f 0 and Re( z f )>0, z U. f Similarly, a function f is meromorphic multivalent convex if f 0 and Re(1+ z f f )>0, z U. Moreover, a function f is called meromorphic multivalent Quasi-convex function if there is a meromorphic multivalent convex function such that Re( (z f ) )>0 z U. A function F E + p,α (E p,α ) is meromorphic multivalent starlike if F 0and Re( zf )>0, z U. F Similar, F E + p,α(e p,α) is meromorphic multivalent convex if F 0and Re(1+ zf F )>0, z U. A function f E + p,α(e p,α) is called meromorphic multivalent Quasi-convex function if there exist a meromorphic multivalent convex function G such that G 0 and Re( (zf ) G )>0 z U.

3 Differential Subordination for Meromorphic... 3 In the present paper, we establish some sufficient conditions for the functions belong to the classese + p,α ande p,α to satisfy Re( (zp F ) G ) q, z U and q is the given univalent function in U. Moreover, we give applications for these results in fractional calculus. In order to prove our subordination results, we need to the following lemmas in the sequel. Lemma 1.1. [11] Let q be convex univalent in the unit diskuandψandγ Cwith If p is analytic inuand p q Re(1+ zq q + ψ γ )>0. ψp+γzp ψq+γzq, Lemma 1.2. [10] Let q be univalent in the unit disk U and θ be analytic in a domain D containing q(u). If zq θ is starlike inuand p q zp θ(p) zqθ(q) 2. Subordination Theorems In this section, we establish some sufficient conditions for subordination of analytic functions in the classese + p,α ande p,α. Note also similar work has been seen for different subclasses done by other authors (see for example [4-7]) Theorem 2.1. Let the function q be convex univalent inusuch that q 0 and (2.1) Re(1+ zq q + ψ )>0, γ 0. γ Suppose that (zp F ) G is analytic inu. If F E + p,α satisfies the subordination (zp F ) G (ψ+γ( z(zp F ) (z p F ) zg G )) ψq+γzq, (zp F ) G q,

4 4 M. Darus and I. Faisal Proof. Let the function p be defined by It can easily observed that ψp+γzp = (zp F ) p= (zp F ) G, z U G (ψ+γ( z(zp F ) (z p F ) zg G )) ψq+γzq. Then using the assumption the theorem the assertion of the theorem follows by an application of Lemma 1.1. Corollary 2.1. Assume that eq. (2.1) holds. Let the function q be univalent in U. Let n=1, if q satisfies the subordination (zf ) G (ψ+γ( z(zf ) (zf ) zg G )) ψq+γzq, (zf ) G q, Theorem 2.2. Let the function q be univalent inusuch that q 0, z Uand zq q, is starlike univalent inu. If F E p,α satisfies the subordination a( z(zp F ) (z p F ) zg G ) azq q, (zp F ) G Proof. Let the functionψbe defined by By setting q, ψ= (zp F ) G, z U θ(ω)=a/ω,ω 0 it can be easily observed that θ is analytic in C {0}. Then by simple computation we have a zψ ψ = a( z(zp F ) (z p F ) zg G ) ψq+γzq. Then using the assumption the theorem the assertion of the theorem follows by an application of Lemma 1.2.

5 Differential Subordination for Meromorphic... 5 Corollary 2.2. Assume that q is convex univalent inu. Let p=1, if F E p,α and a( z(zf ) (zf ) zg G ) azq q, (zf ) G q, 3. Applications of Fractional Integral Operator In this section we introduce some applications of section (2) containing fractional integral operators. Assume that f = 0 φ nz n and let us begin with the following definition. Note also similar work has been seen for different subclasses done by other authors (see for example [1, 2, 3, 8, 9]). Definition 3.1. The fractional integral of orderα is defined for a function f by, Iz α f = 1 z f (z ζ) α 1 dζ, 0 α<1 Γ(α) 0 where, the function f is analytic in simply-connected region of the complex z- plane containing the origin and the multiplicity of (z ζ) α 1 is removed by requiring lo (z ζ) to be real when (z ζ)>0. Note that Iz α f = f zα 1 /Γ(α) for z>0 and 0. Let f = φ n z n n/β+1 α, this implies that 0 thus Iz α f = f z α 1 /Γ(α)=z α 1 /Γ(α) φ n z n n/β+1 α f orz>0 = a n z n n/β, where a n =φ n /Γ(α), o 1/z p ± I α z f E + p,α(e p,α) 0 Theorem 3.1. Let the function q be convex univalent inusuch that q 0 and (3.1) Re(1+ zq q + ψ )>0, γ 0. γ

6 6 M. Darus and I. Faisal Suppose that (zp (1/z p +Iz α f ) ) (1/z p +I is analytic inu. If F E + z ) α p,α satisfies the subordination (zp (1/z p + I α z f ) ) (1/z p + I α z ) (ψ+γ( z(zp (1/z p + I α z f ) ) (z p (1/z p + I α z f ) ) z(1/zp + I α z ) (1/z p + I α z ) )) ψq+γzq, (zp (1/z p + I α z f ) ) (1/z p + I α z ) q, Proof. Let the function p be defined by It can easily observed that p= (zp (1/z p + I α z f ) ) (1/z p + I α z ), z U ψp+γzp = (zp (1/z p + I α z f ) ) (1/z p + I α z ) (ψ+γ( z(zp (1/z p + I α z f ) ) (z p (1/z p + I α z f ) ) z(1/zp + I α z ) (1/z p + I α z ) )) ψq+γzq. Then using the assumption the theorem the assertion of the theorem follows by an application of lemma 1.1. Theorem 3.2. Let the assumptions of theorem 2.2. hold, (zp (1/z p I α z f ) ) (1/z p I α z ) q, z U where F=(1/z p I α z f, G=(1/z p I α z R E F E R E N C E S 1. H. M. SRIVASTAVA and S. OWA: Univalent Functions, Fractional Calculus and Their Applications. Halsted Press, John Wiley and Sons,New York, Chichester, Brisbane and Toronto (1989). 2. H. M. SRIVASTAVA and S. OWA: Current Topic in Analytic Function Theorey. World Scientific Publishing Company, Singapore, New Jersey, London and Hongkong (1992). 3. K. S. MILLER and B. ROSS: An Introduction to The Fractional Calculus and Fractional Differential Equations. 1st Edn. John Wiley and Sons (1993). 4. M. DARUS and R. W. IBRAHIM: Coefficient inequalities for a new class of univalent functions. Lobachevskii J. Math. 29(4) (2008),

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