A NEW CLASS OF MEROMORPHIC FUNCTIONS RELATED TO CHO-KWON-SRIVASTAVA OPERATOR. F. Ghanim and M. Darus. 1. Introduction

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1 MATEMATIQKI VESNIK 66, 1 14, 9 18 March 14 originalni nauqni rad research paper A NEW CLASS OF MEROMORPHIC FUNCTIONS RELATED TO CHO-KWON-SRIVASTAVA OPERATOR F. Ghanim and M. Darus Abstract. In the present paper, we introduce a new class of meromorphic functions defined by means of the Hadamard product of Cho-Kwon-Srivastava operator and we define here a similar transformation by means of an operator introduced by Ghanim and Darus. We investigate a number of inclusion relationships of this class. We also derive some interesting properties of this class. 1. Introduction Let Σ denote the class of meromorphic functions f normalied by f = 1 + a n n, 1.1 which are analytic in the punctured unit disk U = { : < < 1}. For β, we denote by S β and kβ, the subclasses of Σ consisting of all meromorphic functions which are, respectively, starlike of order β and convex of order β in U. For functions f j j = 1; defined by f j = 1 + a n,j n, 1. we denote the Hadamard product or convolution of f 1 and f by f 1 f = 1 + a n,1 a n, n. 1.3 Let us define the function φα, β; by φ α, β; = 1 + n= α n+1 β n+1 n, AMS Subject Classification: 3C45, 3C5 Keywords and phrases: Subordination; meromorphic function; Cho-Kwon-Srivastava operator; Choi-Saigo-Srivastava operator; Hadamard product; integral operator. The work presented here was partially supported by MOHE: UKM-ST-6-FRGS

2 1 F. Ghanim, M. Darus for β, 1,,..., and α C/ {}, λn = λλ+1 n+1 is the Pochhammer symbol. We note that φ α, β; = 1 F 1 1, α, β; F 1 b, α, β; = b n α n n n= β n n! is the well-known Gaussian hypergeometric function. Let us put q λ,µ = 1 + µ λ n, λ >, µ. n λ Corresponding to the functions φα, β; and q λ,µ, and using the Hadamard product for f Σ, we define a new linear operator Lα, β, λ, µ on Σ by L α, β, λ, µ f = f φ α, β; q λ,µ = 1 + α n+1 µ λ a n n. 1.5 n λ β n+1 The meromorphic functions with the generalied hypergeometric functions were considered recently by Diok and Srivastava [4,5], Liu [1], Liu and Srivastava [13 15], Cho and Kim [1]. For a function f L α, β, λ, µ f we define and for k = 1,, 3,..., I µ,,λ = L α, β, λ, µ f,λ f = I k 1 L α, β, λ, µ f + = 1 + α n k n+1 µ λ a n n. 1.6 n λ β n+1 Note that if n = β, k = the operator I µ, α,n,λ have been introduced by N.E. Cho, O.S. Kwon and H.M. Srivastava [] for µ N = N. It was known that the definition of the operator I µ, α,n,λ was motivated essentially by the Choi-Saigo- Srivastava operator [3] for analytic functions, which includes a simpler integral operator studied earlier by Noor [17] and others cf. [11,1,18]. Note also the operator I,k have been recently introduced and studied by Ghanim and Darus [6 8]. To our best knowledge, the recent work regarding operator I µ, α,n,λ was charmingly studied by Piejko and Sokól [19]. Moreover, the operator,λ was defined and studied by Ghanim and Darus [9]. In the same direction, we will study for the operator,λ given in 1.6. Now, it follows from 1.5 and 1.6 that,λ f = αi µ,k α+1,β,λf α + 1 Iµ,k,λf. 1.7

3 Functions related to Cho-Kwon-Srivastava operator 11 Let Ω be the class of analytic functions h with h = 1, which are convex and univalent in the open unit disk U = U {}. For functions f and g analytic in U, we say that f is subordinate to g and write f g, if g is univalent in U, f = g and fu gu. Definition 1.1. A function f Σ is said to be in the class ρ; h, if it satisfies the subordination condition 1 + ρ,λ f + ρ,λ f h 1.8 ρ is a real or complex number and h Ω. Let A be class of functions of the form f = + n= a n n 1.9 which are analytic in U. A function h A is said to be in the class S a, if { f } Re > a U. f For some aa < 1. When < a < 1, S a is the class of starlike functions of order a in U. A function h A is said to be prestarlike of order a in U, if 1 1 a f S a a < 1 the symbol means the familiar Hadamard product or convolution of two analytic functions in U. We denote this class by Ra see [,4]. A function f A is in the class R, if and only if f is convex univalent in U and 1 1 R = S In this paper, we introduce and investigate various inclusion relationships and convolution properties of a certain class of meromorphic functions, which are defined in this paper by means of a linear operator.. Preliminary results In order to prove our main results, we need the following lemmas. Lemma.1. [16] Let g be analytic in U, and h be analytic and convex univalent in U with h = g. If, Re m and m, then g + 1 m g h.1 g h = m m t m 1 h t dt h and h is the best dominant of.1.

4 1 F. Ghanim, M. Darus Lemma.. [] Let a < 1, f S a and g Ra. For any analytic function F in U, then g ff g f U co F U, co F U denotes the convex hull of F U. 3. Main results Theorem 3.1. For some real ρ, let ρ 1 < ρ. Then ρ ; h ρ 1; h Proof. Let ρ 1 < ρ and suppose that g =,λ f 3.1 for f ρ ; h. Then the function g is analytic in U with g = 1. Differentiating both sides of 3.1 with respect to and using 1.7, we have 1 + ρ,λ f + ρ,λ f = g + ρ g h. 3. Hence an application of Lemma.1 with m = 1 ρ > yields Noting that ρ 1 ρ that 1 + ρ 1 = ρ 1 ρ,λ f [ 1 + ρ g h. 3.3 < 1 and that h is convex univalent in U, it follows from + ρ 1,λ f,λ f + ρ,λ f ] + 1 ρ 1 g h. ρ Thus, f ρ 1; h and the proof of Theorem 3.1 is complete. Theorem 3.. Let, { Re φ } α 1, α ; > 1 φ α 1, α ; is defined as in 1.4. Then, α,β U; α / {, 1,,...}, 3.4 ρ; h Σµ,k,λ α 1,β ρ; h. Proof. For f Σ it is easy to verify that α f 1,β,λ = φ α 1, α ; α f,β,λ 3.5

5 Functions related to Cho-Kwon-Srivastava operator 13 and α 1,β,λ f = φ α 1, α ; α,β,λ f. 3.6 Let f α,β ρ; h. Then from 3.5 and 3.6, we deduce that 1 + ρ α f 1,β,λ + ρ α f 1,β,λ = φ α 1, α ; Ψ 3.7 and Ψ = 1 + ρ α,β,λ f + ρ α,β,λ f h 3.8 In view of 3.4, the function φ α 1, α ; has the Herglot representation φ dm x α 1, α ; = U, x x =1 m x is a probability measure defined on the unit circle x = 1 and dm x = 1. x =1 Since h is convex univalent in U, it follows from that 1 + ρ α 1,β,λ f + ρ α 1,β,λ f = Ψ x dm x h. x =1 This shows that f α 1,β ρ; h and the theorem is proved. Theorem 3.3 Let < α 1 < α. Then α,β ρ; h Σµ,k,λ α 1,β ρ; h. Then, Proof. Define, g = + α 1 n+1 α n+1 U; < α 1 < α. n+1 φ α1, α ; = g A 3.1 φ α 1, α ; is defined as in 1.4, and 1 g = α α By 3.11, we see that 1 g α S 1 α 1 for < α 1 < α, which implies that g R 1 α S 1 α 3.1

6 14 F. Ghanim, M. Darus Let f α,β ρ; h. Then we deduce from 3.7, 3.8 and 3.1 that 1 + ρ α 1,β,λ f + ρ α 1,β,λ f g g Ψ = Ψ =, g 3.13 Ψ = 1 + ρ α,β,λ f + ρ α,β,λ f h Since belongs to S 1 α and h is convex univalent in U, it follows from and Lemma. that 1 + ρ α f 1,β,λ + ρ α f 1,β,λ h Thus, f α 1,β ρ; h and the proof is completed. As a special case of Theorem 3.3, we have α+1,β ρ; h Σµ,k,λ ρ; h α > In Theorem 3.4 below we give a generaliation of the above result. Theorem 3.4 Let Re α and α. Then, Proof. Let us define g g α+1,β h = α α g = 1 + ρ ρ; h Σµ,k,λ ρ; h, t α 1 h t dt h.,λ f + ρ,λ f 3.15 for f Σ. Then 1.7 and 3.15 lead to g = αρ α+1,β,λ f + 1 αρ Differentiating both sides of 3.16 and using 1.7, we obtain the following = αρ + 1 αρ By 3.16 and 3.17, we get that is, g αg g + g α α+1,β,λ f [ α = αρ,λ f ] α+1,β,λ f 1 + α α+1,β,λ f α+1,β,λ f + α 1 + ρ α+1,β,λ f, = 1 + ρ α+1,β,λ f + ρ α+1,β,λ f. 3.18

7 Functions related to Cho-Kwon-Srivastava operator 15 If f α+1,β ρ; h, then it follows from 3.18 that g + g h Re α, α. α Hence an application of Lemma.1 yields which shows that g h = α α t α 1 h t dt h, f ρ; h ρ; h Theorem 3.5 Let ρ >, δ > and f ρ; δh + 1 δ. If δ δ, then f ρ; h. Proof. Let us define for f g + ρg = 1 + ρ δ = ρ 1 g =,λ f u 1 1 ρ u du ρ; δh + 1 δ. with ρ >, and δ >. Then we have,λ f + ρ,λ f δ h Hence an application of Lemma.1 yields g δ ρ 1 ρ Ψ = δ ρ 1 ρ t 1 ρ 1 h t dt + 1 δ = h Ψ, 3.1 t 1 ρ 1 dt + 1 δ 3. 1 t If < δ δ, δ > 1 is given by 3.19, then it follows from 3. that Re Ψ = δ 1 u 1 ρ 1 1 Re du + 1 δ > δ 1 u 1 ρ 1 ρ 1 u ρ 1 + u du + 1 δ 1 U. Now, by using the Herglot representation for Ψ, from 3. and 3.1 we get,λ f h Ψ h because h is convex univalent in U. This shows that f Σ α, β, k, ρ; h. For h = 1 1 and f Σ defined by,λ f = δ 1 t 1 ρ 1 ρ ρ dt + 1 δ, 1 t

8 16 F. Ghanim, M. Darus it is easy to verify that 1 + ρ,λ f + ρ,λ f = δ h Thus, f ρ; δh + 1 δ. Also, for δ > δ, we have Re,λ f δ ρ 1 which implies that f / ρ; h. Then, Theorem 4.1. Let f Proof. For f 1 + ρ,λ f g = 1 + ρ g u 1 ρ u du + 1 δ < 1 1, 4. Convolution properties ρ; h, g Σ and Re g > 1 U. f g ρ; h ρ; h and g Σ. we have + ρ,λ f g,λ f + ρg,λ f = g Ψ 4.1 Ψ = 1 + ρ,λ f + ρ,λ f h 4. The remaining part of the proof of Theorem 4.1 is similar to that of Theorem 3. and hence we omit it. Corollary 4.1. Let f ρ; h be given by 1.1 and let Then the function ω m = 1 + m 1 is also in the class ρ; h. Proof. We have σ m = 1 + m 1 σ m = a n n 1 m N\ {1}. 1 tω m t dt a n n + 1 n 1 = f g m m N \ {1}, 4.3 f = 1 + a n n 1 Σ α, β, k, ρ; h

9 and Functions related to Cho-Kwon-Srivastava operator 17 g m = 1 + m 1 n 1 n + 1 Σ. Also, for m N\ {1}, it is known from [1] that { Re {g m } = Re 1 + m 1 n } > 1 n + 1 U. 4.4 In view of 4.3 and 4.4, an application of Theorem 4.1 leads to σ m ρ; h. Then, Theorem 4.. Let f ρ; h, g Σ and g R a a < 1. f g ρ; h. Proof. For f ρ; h andg Σ, from 4.1 used in the proof of Theorem 4.1, we can write 1 + ρ,λ f g + ρ,λ f g Ψ is defined as in 4.. = g Ψ g U, 4.5 Since h is convex univalent in U, Ψ h, g R a and S a a < 1, the desired result follows from 4.5 and Lemma. Taking a = and a = 1, Theorem 4. reduces to the following. Corollary 4.. Let f Σ α, β, k, ρ; h and let g Σ satisfy either of the following conditions i g is convex univalent in U or ii g S 1. Then, f g REFERENCES ρ; h. [1] N.E. Cho, I.H. Kim, Inclusion properties of certain classes of meromorphic functions associated with the generalied hypergeometric function, Appl. Math. Comput , [] N.E. Cho, O.S. Kwon, H.M. Srivastava, Inclusion and argument properties for certain subclasses of meromorphic functions associated with a family of multiplier transformations, J. Math. Anal. Appl. 3 4, [3] J.H. Choi, M. Saigo, H.M. Srivastava, Some inclusion properties of a certain family of integral operators, J. Math. Anal. Appl. 76, [4] J. Diok, H.M. Srivastava, Some subclasses of analytic functions with fixed argument of coefficients associated with the generalied hypergeometric function, Adv. Stud. Contemp. Math. Kyungshang 5,

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