Advances in Decision Sciences

VolumeΒ 2011, Article IDΒ 167672, 9 pages

http://dx.doi.org/10.1155/2011/167672

## Majorization for A Subclass of -Spiral Functions of Order Involving a Generalized Linear Operator

^{1}School of Mathematical Sciences, Faculty of Science and Technology, Universiti Kebangsaan Malaysia, 43600 Bangi, Malaysia^{2}Department of Mathematics-Informatics, Faculty of Sciences, 1 Decembrie 1918 University of Alba Iulia, 510009 Alba Iulia, Romania

Received 22 June 2011; Accepted 18 August 2011

Academic Editor: SheltonΒ Peiris

Copyright Β© 2011 Afaf A. Ali Abubaker et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

Motivated by Carlson-Shaffer linear operator, we define here a new generalized linear operator. Using this operator, we define a class of analytic functions in the unit disk . For this class, a majorization problem of analytic functions is discussed.

#### 1. Introduction

Let denote the class of functions of the form which are analytic in the unit disk .

Let and be analytic in . Then, we say that function is subordinate to if there exists a Schwarz function , analytic in with and , such that , (see [1]). We denote this subordination by

Further, is said to be quasi subordinate to if there exists an analytic function such that is analytic in , and . Note that the quasi subordination (1.3) is equivalent to where and (see [2]). If , then (1.3) becomes (1.2).

Let functions and be analytic functions in . If , then there exists a function analytic in such that in , for which In this case, we say that is majorized by in (see [3]), and we write If we take in (1.4), then the quasi subordination (1.3) becomes the majorization (1.6).

Also, let denote the subclass of consisting of all functions which are univalent in .

In [4], Robertson introduced star-like functions of order on .

*Definition 1.1. *Let and ; then, is a star-like function of order on if and only if
Let denote the whole star-like functions of order in .

SpaΔek [5] extended the class of and obtained the class of -spiral-like functions. In the same article, the author gave an analytical characterization of spirallikeness of type on .

*Definition 1.2. *Let and ; then, is -spiral-like function on if and only if
We denote the whole -spiral-like functions in by .

Finally, Libera [6] introduced and studied the class of -spiral-like functions of order .

*Definition 1.3. *Let , and ; then, is -spiral function of order if and only if
We denote the whole -spiral-like functions of order in by .

In particular, we consider the convolution with function defined by where , , and is the Pochhammer symbol defined by Function is an incomplete beta-function related to the Gauss hypergeometric function by It has an analytic continuation to the -plane cut along the positive real line from 1 to . We note that and are the Koebe functions.

Carlson and Shaffer [7] defined a convolution operator on involving an incomplete beta-function as

*Definition 1.4. *Let function be given by
where and . The generalized linear operator is given as

We note here some special cases.(1) is the Carlson and Shaffer operator [7]. (2), is the Ruscheweyh derivative [8]. (3), is the Al-Oboudi operator [9]. (4) is the linear operator introduced by Al-Refai and Darus [10]. (5), is the generalized multiplier transformation which was introduced and studied by CΓ‘tΓ‘Ε [11].(6), is the multiplier transformation which was introduced and studied by Cho and Srivastava [12] and Cho and Kim [13].

*Remark 1.5. *It follows from the above definition that

We introduce the class as follows.

*Definition 1.6. *Let , , , , , , and ; then, one has if and only if
Obviously, when and we obtain , when and , we obtain that is a starl-like function of order on , and also when and , we obtain that is spiral-like function of type on .

Biernacki [14] in 1936 obtained the first results of majorization-subordination theory. He showed that, if and in , then in . Goluzin [15] improved the result and Shah [16] obtained the complete solution for by showing that in and that the result is the best possible. A majorization problem for star-like functions has been given by MacGregor [3]. Also, majorization problem for star-like functions of complex order has recently been investigated by AltintaΕ et al. [17].

The main object of this paper is to investigate the problem of majorization of the class defined by a generalized linear operator.

In order to prove our main theorem we need the following lemma.

Lemma 1.7 (see [18]). *Let be analytic in satisfying for . Then,
*

#### 2. Main Results

Theorem 2.1. *Let function and suppose that . If is majorized by in , then
**
where
**
for , and .*

*Proof. *Since , we have
where is analytic in , with and
By using (1.16) in (2.4), we get
Hence,
which, in view of (2.5), immediately yields the inequality
Next, since is majorized by in , from (1.5) we have
Also, by using (1.16) in (2.11), we get
then, we have
Thus, by Lemma 1.7, since the Schwarz function satisfies the inequality in (1.18) and using (2.8) in (2.11), we get
Hence,
which, upon setting
yields
where function defined by
takes on its maximum value at with
where
then, we have
A simple calculus in (2.19) is equivalent to
while the inequality in (2.19) takes its minimum value at , that is,
for all , where given in (2.2) holds true for , which proves the conclusion (2.1).

Putting in Theorem 2.1, we obtain the following result.

Corollary 2.2. *Let function and suppose that . If is majorized by in , then
**
where
*

Further, putting and in Theorem 2.1, we obtain the result of AltintaΕ et al. [17].

Corollary 2.3. *Let function and suppose that , where and . If is majorized by in , then
**
where
*

Putting in Corollary 2.3, we obtain the result as follows.

Corollary 2.4. *Let function and suppose that , where . If is majorized by in , then
**
where
*

Also, putting in Corollary 2.3, we obtain the result of MacGregor [3].

Corollary 2.5. *Let function and suppose that . If is majorized by in , then
*

#### Acknowledgment

The work presented here was partially supported by UKM-ST-06-FRGS0244-2010.

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