1. Introduction

If and are analytic in we say that subordinate to written

(1)

if there exist a Schwarz function which is analytic in with and for all such that

(2)

Furthermore, if the function is univalent in then we have the following equivalence (see [7]):

and

The pioneering work of [9], we say that is majorized by in and write

(3)

If there exists a function analytic in such that

Let denote the class of function of the form

(4)

which are analytic and multivalent in the punctured unit disc

For and we have (see [10])

In general, we have

(5)

Note that for we have the operator studied by Aouf and Hossen [5], Aouf and Al-Ashwah [4], Liu and Srivastava [8] and Srivastava and patel [13]. Also when and we have the operator studied by Urafegaddi and Somaatha [14].

The following relation for the operator can be obtained by Simple Calculation (see [10])

(6)

By making use of the operator we define the following class.

Let then if and only if

(7)

where

The majorization (3) is closely related to the concept of quasi-subordination between analytic functions in which was considered by Altinatas and Owa [2]. Recently, Ali [1] results investigated some majorization results for some classes of meromorphic multivalent functions defined by integral operator.

In the present paper, as a sequel to the work of [1] and [6], we investigate majorization problem for the class

2. Majorization Problem for the Class ∑_p,m

We shall assume throughout the paper that where

Theorem 2.1. Let the function and suppose that If

in then

where is the smallest positive root of the equation

(8)

Proof. Since from (7), we get

(9)

where is analytic function in with and then we have

(10)

Since therefore

(11)

where differentiating it with respect to and multiplying by we get

(12)

Thus, by noting that satisfies the inequality (see [1])

(13)

Using equations (11) and (12) and inequality (13), we obtain

(14)

which upon setting and in (14) leads to the inequality

(15)

where

(16)

Takes its maximum value at with is the smallest positive root of (8). Therefore, the function defined by

(17)

is an increasing function for so that

where

Hence, upon setting in (17), we conclude that (8) holds true for which complete the proof.

Putting and in Theorem 2.1, we obtain the following corollary:

Corollary 2.2. Let the function and suppose that If

in then

where

(18)

and

Putting in corollary 2.2, we obtain the following corollary:

Corollary 2.3. Let the function and suppose that If

in then

Where given by (18) with

Putting in corollary 2.3, we obtain the following corollary:

Corollary 2.4. Let the function and suppose that If in then

Where given by (18) with

Remark 2.5. Putting in corollary 2.4, we obtain the result of Ali ([1], Corollary 3).

ACKNOWLEDGEMENTS

The author would like to thank the referee(s) for his/her helpful comments.