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On Subclass of Analytic Univalent Functions Associated with Negative Coefficients
M. H. Al-Abbadi and M. Darus (2009) recently introduced a new generalized derivative operator , which generalized many well-known operators studied earlier by many different authors. In this present paper, we shall investigate a new subclass of analytic functions in the open unit disk which is defined by new generalized derivative operator. Some results on coefficient inequalities, growth and distortion theorems, closure theorems, and extreme points of analytic functions belonging to the subclass are obtained.
1. Introduction and Definitions
Let denote the class of functions of the form
and , which are analytic in the open unit disc on the complex plane ; note that and . Suppose that denote the subclass of consisting of functions that are univalent in . Further, let and be the classes of consisting of functions, respectively, starlike of order and convex of order in , for . Let denote the subclass of consisting of functions of the form
defined on the open unit disk . A function is called a function with negative coefficient and the class was introduced and studied by Silverman . In  Silverman investigated the subclasses of denoted by and for . That are, respectively, starlike of order and convex of order . Now denotes the Pochhammer symbol (or the shifted factorial) defined by
The authors in  have recently introduced a new generalized derivative operator as follows.
Definition 1.1. For the generalized derivative operator is defined by
(1)Special cases of this operator include the Ruscheweyh derivative operator in the cases , the Salagean derivative operator , the generalized Ruscheweyh derivative operator , the generalized Salagean derivative operator introduced by Al-Oboudi , and the generalized Al-Shaqsi and Darus derivative operator where can be found in . It is easily seen that , and also where
By making use of the generalized derivative operator the authors introduce a new subclass as follows.
Definition 1.2. For , let be the subclass of consisting of functions satisfying
, and .
Further, we define the class by for , and .
Also note that various subclasses of and have been studied by many authors by suitable choices of , and . For example,
starlike of order with negative coefficients. And
class of convex function of order with negative coefficients. Also
The classes and were studied by Chatterjea  (see also Srivastava et al. ), whereas the classes and were, respectively, studied by Altintaş  and Kamali and Akbulut . When or , or in the class , we have the class introduced and studied by Ahuja . Finally we note that when in the class we have the class introduced and studied by Al-Shaqsi and Darus .
2. Coefficient Inequalities
In this section, we provide a necessary and sufficient condition for a function analytic in to be in and in .
Theorem 2.1. For and , let be defined by (1.1). If then , where and .
Proof. Assume that (2.1) holds true. Then we shall prove condition (1.5). It is sufficient to show that
So, we have that
and expression (2.3) is bounded by .
Hence (2.2) holds if which is equivalent to by (2.1). Thus . Note that the denominator in (2.3) is positive provided that (2.1) holds.
Proof. We only prove the right-hand side, since the other side can be justified using similar arguments in proof of Theorem 2.1. Since by condition (1.5), we have that Choose values of on real axis so that is real. Letting through real values, we have that Thus we obtain which is (2.1). Hence the proof is complete.
The result is sharp with the extremal function given by
3. Growth and Distortion Theorems
In this section, growth and distortion theorems will be considered and covering property for function in the class will also be given.
Theorem 3.1. Let the function given by (1.2) be in the class . Then for , where .
Proof. We only prove the right-hand side inequality in (3.1), since the other inequality can be justified using similar arguments. Since by Theorem 2.2, we have that Now And therefore, Since then we have that After that, By aid of inequality (3.4), it yields the right-hand side inequality of (3.1). Thus, this completes the proof.
Theorem 3.2. Let the function given by (1.2) be in the class . Then for , where
4. Extreme Points
The extreme points of the class are given by the following theorem.
Theorem 4.1. Let and
Then if and only if it can be expressed in the form where and
Proof. Suppose that can be expressed as in (4.2). Our goal is to show that .
By (4.2), we have that Now so that Now, we have that Setting we arrive to And therefore, It follows from Theorem 2.2 that .
Conversely, let us suppose that ; our goal is, to get (4.2). From (4.2) and using similar last arguments, it is easily seen that which suffices to show that Now, we have that , then by previous Theorem 2.3, That is Since , we see , for each and
We can set that Thus, the desired result is that This completes the proof of the theorem.
Corollary 4.2. The extreme points of are the functions where , and .
This work is fully supported by UKM-GUP-TMK-07-02-107, Malaysia. The authors are also grateful to the referee for his/her suggestions which helped us to improve the contents of this article.
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Copyright © 2010 Ma'moun Harayzeh Al-Abbadi and Maslina Darus. 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.