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International Journal of Mathematics and Mathematical Sciences
Volume 2010 (2010), Article ID 343580, 11 pages
http://dx.doi.org/10.1155/2010/343580
Research Article

On Subclass of Analytic Univalent Functions Associated with Negative Coefficients

School of Mathematical Sciences, Faculty of Science and Technology, Universiti Kebangsaan Malaysia, Selangor D. Ehsan, Bangi 43600, Malaysia

Received 18 November 2009; Accepted 9 January 2010

Academic Editor: Stanisława R. Kanas

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.

Abstract

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 [1]. In [1] 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 [2] have recently introduced a new generalized derivative operator as follows.

Definition 1.1. For the generalized derivative operator is defined by where .
(1)Special cases of this operator include the Ruscheweyh derivative operator in the cases [3], the Salagean derivative operator [4], the generalized Ruscheweyh derivative operator [5], the generalized Salagean derivative operator introduced by Al-Oboudi [6], and the generalized Al-Shaqsi and Darus derivative operator where can be found in [7]. 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 where , 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 [8] (see also Srivastava et al. [9]), whereas the classes and were, respectively, studied by Altintaş [10] and Kamali and Akbulut [11]. When or , or in the class , we have the class introduced and studied by Ahuja [12]. Finally we note that when in the class we have the class introduced and studied by Al-Shaqsi and Darus [13].

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.

Theorem 2.2. Let be defined by (1.2) and (,). Then if and only if (2.1) is satisfied.

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

Theorem 2.3. Let the function given by (1.2) be in the class . Then where , and Equality holds for the function given by (2.9).

Proof. Since , then condition (2.1) gives for each where
Clearly the function given by (2.9) satisfies (2.10), and therefore, given by (2.9) is in for this function; the result is clearly sharp.

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

Proof. Since , by Theorem 2.2, we have that Now Hence Since then we have that and therefore, By using the inequality (3.11) in (3.14), we get Theorem 3.2. This completes the proof.

4. Extreme Points

The extreme points of the class are given by the following theorem.

Theorem 4.1. Let and where 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 .

Acknowledgments

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.

References

  1. H. Silverman, “Univalent functions with negative coefficients,” Proceedings of the American Mathematical Society, vol. 51, pp. 109–116, 1975. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  2. M. H. Al-Abbadi and M. Darus, “Differential subordination for new generalized derivative operator,” Acta Universitatis Apulensis, no. 20, pp. 265–280, 2009.
  3. St. Ruscheweyh, “New criteria for univalent functions,” Proceedings of the American Mathematical Society, vol. 49, pp. 109–115, 1975. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  4. G. Š. Sălăgean, “Subclasses of univalent functions,” in Complex Analysis—Fifth Romanian-Finnish Seminar, Part 1 (Bucharest, 1981), vol. 1013 of Lecture Notes in Mathematics, pp. 362–372, Springer, Berlin, Germany, 1983. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  5. K. Al-Shaqsi and M. Darus, “On univalent functions with respect to k-symmetric points defined by a generalized Ruscheweyh derivatives operator,” Journal of Analysis and Applications, vol. 7, no. 1, pp. 53–61, 2009. View at MathSciNet
  6. F. M. Al-Oboudi, “On univalent functions defined by a generalized Sălăgean operator,” International Journal of Mathematics and Mathematical Sciences, vol. 2004, no. 27, pp. 1429–1436, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  7. M. Darus and K. Al-Shaqsi, “Differential sandwich theorems with generalised derivative operator,” International Journal of Computational and Mathematical Sciences, vol. 2, no. 2, pp. 75–78, 2008. View at MathSciNet
  8. S. K. Chatterjea, “On starlike functions,” Journal of Pure Mathematics, vol. 1, pp. 23–26, 1981. View at Zentralblatt MATH · View at MathSciNet
  9. H. M. Srivastava, S. Owa, and S. K. Chatterjea, “A note on certain classes of starlike functions,” Rendiconti del Seminario Matematico della Università di Padova, vol. 77, pp. 115–124, 1987. View at Zentralblatt MATH · View at MathSciNet
  10. O. Altıntaş, “On a subclass of certain starlike functions with negative coefficients,” Mathematica Japonica, vol. 36, no. 3, pp. 489–495, 1991. View at Zentralblatt MATH · View at MathSciNet
  11. M. Kamali and S. Akbulut, “On a subclass of certain convex functions with negative coefficients,” Applied Mathematics and Computation, vol. 145, no. 2-3, pp. 341–350, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  12. O. P. Ahuja, “Integral operators of certain univalent functions,” International Journal of Mathematics and Mathematical Sciences, vol. 8, no. 4, pp. 653–662, 1985. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  13. K. Al Shaqsi and M. Darus, “On certain subclass of analytic univalent functions with negative coefficients,” Applied Mathematical Sciences, vol. 1, no. 21–24, pp. 1121–1128, 2007. View at MathSciNet