Some Results of Meromorphic Functions Related

Applied Mathematical Sciences, Vol. 5, 2011, no. 66, 3291 - 3301

Some Results of Meromorphic Functions Related to Cho-Kwon-Srivastava Operator 1

F. Ghanim and 2 M. Darus 1

Faculty of Management Multimedia University Cyberjaya 63100 Selangor D. Ehsan, Malaysia 2

School of Mathematical Sciences Faculty of Science and Technology Universiti Kebangsaan Malaysia Bangi 43600 Selangor D. Ehsan, Malaysia e-mail: 1 fi[email protected] e-mail: 2 [email protected] (Corresponding author) Abstract In the present paper, we obtain some interesting properties for meromorphic analytic functions related to Cho-Kwon-Srivastava Operator defined by means of an operator recently given by Ghanim and Darus.

Mathematics Subject Classification: 30C45 Keywords: Convex function; Meromorphic function; Cho-Kwon-Srivastava operator; Ghanim–Darus operator; Choi–Saigo–Srivastava operator; Linear operator; Hadamard product; Multiplier transformation

1

Introduction

Let Σ denote the class of meromorphic functions f (z) normalized by 1  an z n , f (z) = + z n=1 ∞

(1.1)

which are analytic in the punctured unit disk U = {z : 0 < |z| < 1}. For 0 ≤ β, we denote by S ∗ (β) and k(β), the subclasses of Σ consisting of all meromorphic functions which are, respectively, starlike of order β and convex of order β in U (cf. e.g., [[1] and [2]]).

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F. Ghanim and M. Darus

For functions fj (z)(j = 1; 2) defined by 1  an,j z n , fj (z) = + z n=1 ∞

(1.2)

we denote the Hadamard product (or convolution) of f1 (z) and f2 (z) by 1  an,1 an,2 z n . (f1 ∗ f2 ) = + z n=1 ∞

(1.3)

˜ β; z) by Let us define the function φ(α,  ∞    (α)n+1  n 1  z , φ˜ (α, β; z) = + z n=0  (β)n+1 

(1.4)

for β = 0, −1, −2, ..., and α ∈ C/ {0}, where (λ)n = λ(λ + 1)n+1 is the Pochhammer symbol. We note that 1 φ˜ (α, β; z) = 2 F1 (1, α, β; z) z where ∞  (b)n (α)n z n 2 F1 (b, α, β; z) = (β)n n! n=0

is the well-known Gaussian hypergeometric function. Let us set 1  q λ, μ (z) = + z n=1 ∞



λ n+1+λ

μ

zn,

(λ > 0, μ ≥ 0) .

˜ β; z) and qλ, μ (z), and using the Hadamard Corresponding to the functions φ(α, product for f (z) ∈ Σ, we define a new linear operator L(α, β, λ, μ) on Σ by   ˜ L (α, β, λ, μ) f (z) = f (z) ∗ φ (α, β; z) ∗ q λ, μ (z)  ∞  (1.5)   (α)n+1  λ μ 1 =z+  (β)  n+1+λ an z n . n=1

n+1

The meromorphic functions with the generalized hypergeometric functions were considered recently by Dziok and Srivastava ([6], [7]), Liu [16], Liu and Srivastava ( [19], [20],[21]) and, Cho and Kim [3] . For a function f ∈ L (α, β, λ, μ) f (z), we define μ, 0 Iα, β, λ = L (α, β, λ, μ) f (z)

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Some results of meromorphic functions

and for k = 1, 2, 3, ..., k−1

 μ, k Iα, L (α, β, λ, μ) f (z) + β, λ f (z) = z I  ∞   (α)  λ μ = z1 + nk  (β)n+1  n+1+λ an z n . n=1

2 z

(1.6)

n+1

μ, 0 Note that if n = β, k = 0, the operator Iα, n, λ have benn introduced by Cho μ, 0 et.al [4]. It was known that the definition of the operator Iα, n, λ was motivated essentially by the Choi-Saigo-Srivastava operator [5] for analytic functions, which includes a simpler integral operator studied earlier by Noor [22] and 0, k others (cf. [23], [17]and [18]). Note also the operator Iα, β have been recently introduced and studied by Ghanim and Darus [9] and Ghanim et.al [8] respecμ, 0 tively. To our best knowledge, the recent work regarding operator Iα, n, λ was μ, k charmingly studied by Piejko and Sok´ol [24]. Moreover, the operator Iα, β, λ was defined and studied by Ghanim and Darus [10]. In the same direction, we μ, k will study for the operator Iα, β, λ given in (1.6).

Now, it follows from (1.5) and (1.6) that   μ,k μ,k μ,k f (z) = αIα+1, z Iα, β,λ β,λ f (z) − (α + 1) Iα, β,λ f (z).

(1.7)

In the present paper, we shall derive some properties of univalent functions μ, k defined by the linear operator Iα, β, λ f (z). In order to prove our main results, we recall the following lemma due to Yang [29]. Lemma 1.1 Let q(z) = 1 + qn z n + qn+1 z n+1 + . . . be analytic functions in U with q(z) = 0 for z ∈ U. If  zq  (z)  1+a 2 0 and 11−

na , 2 log 2

2 (M − 1) log 2 na

The bound in (1.10) is the best possible.

(1.9)

(z ∈ U ) .

(1.10)

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F. Ghanim and M. Darus

Main Results

We begin with the following theorem:

μ, k μ, k Theorem 2.1 Let α + 1 > 0, Iα, f (z) Iα+1, β, λ β, λ f (z) = 0 for z ∈ U and suppose that     μ, k μ, k μ, k αIα+1, Iα+2, Iα+1, β, λ f (z) β, λ f (z) β, λ f (z) 1+ − μ, k < M,(2.1)  1+ μ, k μ, k (α + 1) Iα, Iα, Iα, β, λ f (z) β, λ f (z) β, λ f (z) where n . 2 (α + 1) log 2

11−

2 (α + 1) (M − 1) log 2 , n

(z ∈ U ) .

(2.3)

The bound in (2.3) is the best possible. Proof. Define the function q(z) by q(z) =

μ, k Iα, β, λ f (z) μ, k Iα+1, β, λ f (z)

.

(2.4)

Then, clearly q(z) = 1 + qn z n + qn+1 z n+1 + . . . analytic function in U with q(z) = 0 for z ∈ U. It follows from (2.4) and (1.7) that 

μ, k Iα, β, λ f

z (z) zq  (z) = μ, k q(z) Iα, β, λ f (z)

 −

z



μ, k Iα+1, β, λ f

(z)

 (2.5)

I k L∗p (a + 1, c) f (z)

by making use of the familiar identity (1.7) in (2.5), we obtain μ, k Iα+2, β, λ f (z) μ, k Iα+1, β, λ f (z)

=

1 zq  (z) 1 + − α + 1 (α + 1) q (z) (α + 1) q(z)

or, equivalent μ, k Iα+1, 1 zq  (z) β, λ f (z) = 1 + 1+ μ, k (α + 1) q 2 (z) (α + 1) Iα, β, λ f (z)

 1+

μ, k αIα+1, β, λ f (z) μ, k Iα, β, λ f (z)

 (2.6)

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Some results of meromorphic functions

−

μ, k Iα+2, β, λ f (z) μ, k Iα+1, β, λ f (z)

Applying Lemma 1.1, with a =

1 , 1+α

.

we get the required result.

Letting α = β = 1 in Theorem 2.1, we have    μ, k k Corollary 2.1 Let I λ f (z) z I μ, f (z) = 0 for z ∈ U and suppose that λ ⎧ ⎫  ⎛  ⎞     μ, k μ, k μ, k μ, k+2 ⎪ 1 ⎨ ⎬ f (z) ⎪ z Iλ f (z) ⎜ z Iλ f (z) ⎟ z Iλ f (z) + 2 Iλ  1+ 1 + − < M, ⎝ ⎠ ⎪ ⎪ 2Iλμ, k f (z) Iλμ, k f (z) Iλμ, k f (z) ⎩ ⎭ where 11−

n . 4 log 2

4 (M − 1) log 2 , n

(2.8)

(z ∈ U ) .

(2.9)

The bound in (2.9) is the best possible. Letting M = 1 + n/4 log 2 in Corollary 2.1, we have Corollary 2.2 Let

k I μ, λ f (z)

   μ, k z I λ f (z) = 0 and k = μ = 0 for z ∈ U and

suppose that    zf  (z) + 12 z 2 f  (z) zf  (z) n zf  (z) 1+ − 0, zIα+1, β, λ f (z) = 0 for z ∈ U and suppose that    δ I μ, k  f (z) α+2, β, λ μ, k  zIα+1, 0, Iα+1, β, λ f (z) = 0 for z ∈ U and suppose that ⎧ ⎛  ⎞⎫  μ, k μ, k+2 ⎪ 1 ⎨ ⎬ δ z I λ f (z) + 2 I λ f (z) ⎟⎪ ⎜ μ, k  zI λ f (z) ⎝  (2.18) ⎠ < M, k ⎪ ⎪ I μ, ⎩ ⎭ λ f (z)

where 11−  zI μ, λ n

(z ∈ U ) .

(2.20)

The bound in (2.20) is the best possible. Letting δ = 1, M = 1 + n/8 log 2 and k = μ = 0 in Corollary 2.3, we have μ, k Corollary 2.4 Let Iα+1, β, λ f (z) = 0 for z ∈ U and suppose that

   zf  (z) n  0,

(z ∈ U ) .

(2.22)

The result is sharp. Theorem 2.3 Let ξ > 0, z



μ, k Iα+1, β, λ f (z)

  μ, k Iα, β, λ f (z) = 0 for z ∈ U and

suppose that ⎧ ξ    μ, k μ, k ⎨ (α + 1) Iα+2, f (z) Iα, β, λ f (z) β, λ  1+ μ, k μ, k ⎩ Iα+1, β, λ f (z) Iα+1, β, λ f (z)  −α

μ, k Iα, β, λ f (z)

ξ −1

μ, k Iα+1, β, λ f (z)

⎫ ⎬ ⎭

(2.23)

< M,

where 1 1−

2ξ (M − 1) log 2 n

The bound in (2.25) is the best possible.

(z ∈ U ) .

(2.25)

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F. Ghanim and M. Darus

Proof. Define the function q(z) by  q(z) =

μ, k Iα+1, β, λ f (z) μ, k Iα, β, λ f (z)

ξ .

(2.26)

Then, clearly q(z) = 1 + qn z n + qn+1 z n+1 + . . . analytic function in U with q(z) = 0 for z ∈ U . Also by a simple computation and by making use of the familiar identity (??), we find from (2.26)that 1 zq  (z) 1+ =1+ ξ q 2 (z) ⎛ ×⎝

μ, k (α + 1) Iα+2, β, λ f (z) μ, k Iα+1, β, λ f (z)



μ, k Iα, β, λ f (z)

ξ (2.27)

μ, k Iα+1, β, λ f (z)

 −α

μ, k Iα, β, λ f (z) μ, k Iα+1, β, λ f (z)

⎞

ξ

− 1⎠

Applying Lemma 1.1, with a = 1ξ , we get the required result. Letting α = β = 1 in Theorem 2.3, we have    μ, k k Corollary 2.5 Let ξ > 0, z I λ f (z) I μ, λ f (z) = 0 for z ∈ U and suppose that ⎧  ξ ⎛  ξ ⎞⎫ ⎨ μ, k μ, k+2 μ, k ⎬ I λ f (z) f (z) I λ f (z) ⎝1 + zI λ ⎠ < M,(2.28)  1+ − k  k k  ⎩ ⎭ zI μ, I μ, zI μ, λ f (z) λ (z) λ f (z) where 1+

n 1 zq  (z) 11−

2ξ (M − 1) log 2, n

(z ∈ U).

(2.30)

The bound in (2.30) is the best possible. Letting ξ = 1, M = 1 + n/2 log 2 and k = μ = 0 in Corollary 2.5, we have

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Some results of meromorphic functions

Corollary 2.6 Let zf  (z)/f (z) = 0 for z ∈ U and suppose that    ξ  ξ  f (z) f (z) zI 2 f (z) n − .  1+ 1+ 0,

(z ∈ U).

(2.32)

The result is sharp. Some other work related to meromorphic functions can be seen in ([11], [12], [13], [25]-[28]) and many elsewhere. ACKNOWLEDGEMENT: The work presented here was fully supported by UKM-ST-06-FRGS0244-2010.

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