- About this Journal ·
- Abstracting and Indexing ·
- Aims and Scope ·
- Annual Issues ·
- Article Processing Charges ·
- Articles in Press ·
- Author Guidelines ·
- Bibliographic Information ·
- Citations to this Journal ·
- Contact Information ·
- Editorial Board ·
- Editorial Workflow ·
- Free eTOC Alerts ·
- Publication Ethics ·
- Reviewers Acknowledgment ·
- Submit a Manuscript ·
- Subscription Information ·
- Table of Contents
Abstract and Applied Analysis
Volume 2012 (2012), Article ID 279843, 11 pages
Some Properties of a Generalized Class of Analytic Functions Related with Janowski Functions
1Department of Mathematics, Abdul Wali Khan University, 23200 Mardan, Pakistan
2Department of Mathematics, COMSATS Institute of Information Technology, 44000 Islamabad, Pakistan
3Department 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.
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.
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 , 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 .
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 .
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 . Also is the class of close-to-convex functions, see .
In , 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  and in the study of power series with integral coefficients by Pólya [18, page 323], Cantor , 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  determined the rate of growth of as for a function belongs to the class . Pommerenke in  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,
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
2. Some Properties of the Class
Theorem 2.1. The function if and only if there exist two functions such that
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  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 . 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  together with Lemma 1.3 to have and depends only on , and .
- W. Janowski, “Some extremal problems for certain families of analytic functions. I,” Polska Akademia Nauk. Annales Polonici Mathematici, vol. 28, pp. 297–326, 1973.
- B. Pinchuk, “Functions of bounded boundary rotation,” Israel Journal of Mathematics, vol. 10, pp. 6–16, 1971.
- 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.
- 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.
- 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.
- V. Paatero, “Uber gebiete von beschrankter randdrehung,” Annales Academiae Scientiarum Fennicae. Series A, vol. 37, no. 9, 20 pages, 1933.
- 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.
- 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.
- 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.
- 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.
- 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.
- 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.
- Ch. Pommerenke, “On close-to-convex analytic functions,” Transactions of the American Mathematical Society, vol. 114, pp. 176–186, 1965.
- K. I. Noor, “On strongly close-to-convex functions,” Mathematica, vol. 44, no. 1, pp. 25–28, 2002.
- 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.
- 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.
- 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.
- D. G. Cantor, “Power series with integral coefficients,” Bulletin of the American Mathematical Society, vol. 69, pp. 362–366, 1963.
- 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.
- C. Pommerenke, “On starlike and close-to-convex functions,” Proceedings of the London Mathematical Society. Third Series, vol. 13, pp. 290–304, 1963.
- 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.
- 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.
- 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.
- G. M. Golusin, “On distortion theorem and coefficients cients of univalent functions,” Mathematicheskii Sbornik, vol. 19, pp. 183–2023, 1946.