Abstract and Applied Analysis

Volume 2012 (2012), Article ID 279843, 11 pages

http://dx.doi.org/10.1155/2012/279843

## Some Properties of a Generalized Class of Analytic Functions Related with Janowski Functions

^{1}Department of Mathematics, Abdul Wali Khan University, 23200 Mardan, Pakistan^{2}Department of Mathematics, COMSATS Institute of Information Technology, 44000 Islamabad, Pakistan^{3}Department of Mathematics, GC University, 38000 Faisalabad, Pakistan

Received 9 February 2012; Accepted 12 March 2012

Academic Editor: Muhammad Aslam Noor

Copyright © 2012 M. Arif 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

We define a class [, , ] of analytic functions by using Janowski’s functions which generalizes a number of classes studied earlier such as the class of strongly close-to-convex functions. Some properties of this class, including arc length, coefficient problems, and a distortion result, are investigated. We also discuss the growth of Hankel determinant problem.

#### 1. Introduction

Let be the class of analytic functions satisfying the condition , in the open unit disc . Let and be analytic in . Then the function is said to be subordinate to , written as if there exists an analytic function in with and such that in . If is univalent in , then is equivalent to and .

A function , analytic in with is said to be in the class , if and only if It is noted that for , the class reduces to the class which was introduced by Janowski [1], and for , , and , we obtain the well-known class of functions with positive real part. Now, we consider the generalized class of Janowski functions which is defined as follows.

A function if and only if where , and . It is clear that and , the well-known class given and studied by Pinchuk [2].

We define the following classes as For ,, and , we obtain the well-known classes of bounded boundary rotation and bounded radius rotation , for details [3–8]. The classes and have been extensively studied by Noor in [9–11]. Also and , where and are the classes studied by Polatoğlu in [12].

Throughout in this paper, we assume that unless otherwise mentioned.

*Definition 1.1. *Let , then if and only if, for , there exists a function such that
For is the class of strongly close-to-convex functions of order in the sense of Pommerenke [13]. Also is the class of close-to-convex functions, see [14].

In [15], the th Hankel determinant , for a function is stated by Noonan and Thomas as follows.

*Definition 1.2. *Let , then the th Hankel determinant of is defined for by

The Hankel determinant plays an important role, for instance, in the study of the singularities by Hadamard, see [16, page 329], Edrei [17] and in the study of power series with integral coefficients by Pólya [18, page 323], Cantor [19], and many others.

In this paper, we will determine the rate of growth of the Hankel determinant for , as . This determinant has been considered by several authors. That is, Noor [20] determined the rate of growth of as for a function belongs to the class . Pommerenke in [21] studied the Hankel determinant for starlike functions. The Hankel determinant problem for other interesting classes of analytic functions was discussed by Noor [22–24].

Lemma 1.3. *Let . Let the th Hankel determinant of for , be defined by (1.5). Then, writting , we have
**
where with , one defines, for ,
*

Lemma 1.4. *With and any integer,
*

Lemmas 1.3 and 1.4 are due to Noonan and Thomas [15].

Lemma 1.5. *A function if and only if there exist two functions and such that
*

Using the definition of class and simple calculations yields the above result.

Lemma 1.6. *Let , then
**
with
*

This result follows easily by using Lemma 1.5 and a result for the class due to Polatoğlu et al. [12]. This result is best possible.

#### 2. Some Properties of the Class

Theorem 2.1. *The function if and only if there exist two functions such that
*

* Proof. *From (1.4), we have
where and . Using (1.10), we obtain
with and , which completes the required result.

Theorem 2.2. *Let then for , where is the root of
**
with and .*

*Proof. *From (1.4), we have
Since, therefore using (1.9), we have
Differentiating logarithmically (2.6) with respect to , we obtain
Using the well-known results for the classes and we have
where and . Let
then and for and therefore, there exists a root . This completes the proofs.

Theorem 2.3. *Let , then for , and ,
**
where is a constant depending upon , andonly.*

* Proof. * With ,
Since, therefore by using (1.9) with , we have
Using the well-known Holder’s inequality, with and such that and ,we can write
Also, it is known [13] that, for ,
Therefore,
Therefore, we have
Since , for , therefore
which is the required result.

Theorem 2.4. *Let , then for , and ,
*

*Proof. *By Cauchy’s theorem, we have
Now putting , we have
which is required.

Theorem 2.5. *Let , then
*

* Proof. *Since therefore
Using Lemma 1.5 and the well-known distortion result of class we obtain the required result.

Theorem 2.6. *Let , then for , and ,
**
where , and is a constant depending on , and only.*

* Proof. * From (1.4), we have
where . It follows easily from Alexander type relation that
Using (1.9) with , we have
Therefore,
Let , then for any nonzero complex and , consider as defined by (1.7). Then,
and by using (2.27), we have
where we have used the result proved in [25]. The well-known Holder’s inequality will give us
Using (2.14) in (2.30), we obtain
Therefore, we can write
Now, using a subordination result for starlike functions, we have
where is a constant depending on only and . Applying Lemma 1.4 and putting , we have for ,
where is a constant depending on , and only. We now estimate the rate of growth of . For and
For , we use similar argument due to Noonan and Thomas [15] together with Lemma 1.3 to have
and depends only on , and .

#### References

- W. Janowski, “Some extremal problems for certain families of analytic functions. I,”
*Polska Akademia Nauk. Annales Polonici Mathematici*, vol. 28, pp. 297–326, 1973. View at Google Scholar - B. Pinchuk, “Functions of bounded boundary rotation,”
*Israel Journal of Mathematics*, vol. 10, pp. 6–16, 1971. View at Google Scholar - K. I. Noor, M. A. Noor, and E. Al-Said, “On analytic functions of bounded boundary rotation of complex order,”
*Computers & Mathematics with Applications*, vol. 62, no. 4, pp. 2112–2125, 2011. View at Publisher · View at Google Scholar - K. I. Noor, M. Arif, and W. Ul Haq, “Some properties of certain integral operators,”
*Acta Universitatis Apulensis. Mathematics. Informatics*, no. 21, pp. 89–95, 2010. View at Google Scholar - K. I. Noor, W. Ul-Haq, M. Arif, and S. Mustafa, “On bounded boundary and bounded radius rotations,”
*Journal of Inequalities and Applications*, vol. 2009, Article ID 813687, 12 pages, 2009. View at Publisher · View at Google Scholar - V. Paatero, “Uber gebiete von beschrankter randdrehung,”
*Annales Academiae Scientiarum Fennicae. Series A*, vol. 37, no. 9, 20 pages, 1933. View at Google Scholar - K. S. Padmanabhan and R. Parvatham, “Properties of a class of functions with bounded boundary rotation,”
*Annales Polonici Mathematici*, vol. 31, no. 3, pp. 311–323, 1975/76. View at Google Scholar - O. Tammi, “On the maximalization of the coefficients o schlicht and related functions,”
*Annales Academiae Scientiarum Fennicae. Series A*, vol. 1952, no. 114, 51 pages, 1952. View at Google Scholar - K. I. Noor, “Some radius of convexity problems for analytic functions related with functions of bounded boundary rotation,”
*The Punjab University. Journal of Mathematics*, vol. 21, pp. 71–81, 1988. View at Google Scholar - K. I. Noor, “On radii of convexity and starlikeness of some classes of analytic functions,”
*International Journal of Mathematics and Mathematical Sciences*, vol. 14, no. 4, pp. 741–746, 1991. View at Publisher · View at Google Scholar - K. I. Noor, “On some integral operators for certain families of analytic functions,”
*Tamkang Journal of Mathematics*, vol. 22, no. 4, pp. 113–117, 1991. View at Google Scholar - Y. Polatoğlu, M. Bolcal, A. Şen, and E. Yavuz, “A study on the generalization of Janowski functions in the unit disc,”
*Acta Mathematica. Academiae Paedagogicae Nyíregyháziensis*, vol. 22, no. 1, pp. 27–31, 2006. View at Google Scholar - Ch. Pommerenke, “On close-to-convex analytic functions,”
*Transactions of the American Mathematical Society*, vol. 114, pp. 176–186, 1965. View at Google Scholar - K. I. Noor, “On strongly close-to-convex functions,”
*Mathematica*, vol. 44, no. 1, pp. 25–28, 2002. View at Google Scholar - J. W. Noonan and D. K. Thomas, “On the Hankel determinants of areally mean
*p*-valent functions,”*Proceedings of the London Mathematical Society. Third Series*, vol. 25, pp. 503–524, 1972. View at Google Scholar - P. Dienes,
*The Taylor Series: An Introduction to the Theory of Functions of a Complex Variable*, Dover, New York, NY, USA, 1957. - A. Edrei, “Sur les déterminants récurrents et les singularités d'une fonction donnée par son développement de Taylor,”
*Compositio Mathematica*, vol. 7, pp. 20–88, 1940. View at Google Scholar - G. Pólya and I. J. Schoenberg, “Remarks on de la Vallée Poussin means and convex conformal maps of the circle,”
*Pacific Journal of Mathematics*, vol. 8, pp. 295–334, 1958. View at Google Scholar - D. G. Cantor, “Power series with integral coefficients,”
*Bulletin of the American Mathematical Society*, vol. 69, pp. 362–366, 1963. View at Google Scholar - K. I. Noor, “Hankel determinant problem for the class of functions with bounded boundary rotation,”
*Académie de la République Populaire Roumaine. Revue Roumaine de Mathématiques Pures et Appliquées*, vol. 28, no. 8, pp. 731–739, 1983. View at Google Scholar - C. Pommerenke, “On starlike and close-to-convex functions,”
*Proceedings of the London Mathematical Society. Third Series*, vol. 13, pp. 290–304, 1963. View at Google Scholar - K. I. Noor, “On the Hankel determinants of close-to-convex univalent functions,”
*International Journal of Mathematics and Mathematical Sciences*, vol. 3, no. 3, pp. 447–481, 1980. View at Publisher · View at Google Scholar - K. I. Noor, “On the Hankel determinant problem for strongly close-to-convex functions,”
*Journal of Natural Geometry*, vol. 11, no. 1, pp. 29–34, 1997. View at Google Scholar - K. I. Noor and I. M. A. Al-Naggar, “On the Hankel determinant problem,”
*Journal of Natural Geometry*, vol. 14, no. 2, pp. 133–140, 1998. View at Google Scholar - G. M. Golusin, “On distortion theorem and coefficients cients of univalent functions,”
*Mathematicheskii Sbornik*, vol. 19, pp. 183–2023, 1946. View at Google Scholar