Table of Contents Author Guidelines Submit a Manuscript
The Scientific World Journal
Volume 2014, Article ID 989640, 6 pages
http://dx.doi.org/10.1155/2014/989640
Research Article

Coefficient Bounds for Some Families of Starlike and Convex Functions of Reciprocal Order

1Department of Mathematics, Abdul Wali Khan University Mardan, Mardan, Khyber Pakhtunkhwa 23200, Pakistan
2School of Mathematical Sciences, Faculty of Science and Technology, Universiti Kebangsaan Malaysia, 43600 Bangi, Selangor, Malaysia
3Department of Mathematics, Government College University, Faisalabad 38000, Pakistan

Received 31 May 2014; Revised 7 September 2014; Accepted 12 October 2014; Published 24 November 2014

Academic Editor: Minghe Sun

Copyright © 2014 Muhammad 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

The aim of the present paper is to investigate coefficient estimates, Fekete-Szegő inequality, and upper bound of third Hankel determinant for some families of starlike and convex functions of reciprocal order.

1. Introduction

Let denote the class of functions which are analytic in the open unit disk and normalized by Also let and denote the usual classes of starlike and convex functions of order , , respectively. In 1975, Silverman [1] proved that if it satisfies the condition Geometrical meaning of inequality (2) is that maps onto the interior of the circle with center at 1 and radius .

By and , we mean the classes of starlike and convex functions of reciprocal order , which are defined, respectively, by Recently in 2008, Nunokawa and his coauthors [2] improved inequality (2) for the class and they proved that, for , , if and only if the following inequality holds: In view of these results we now define the following subclass of analytic functions of reciprocal order and investigate its various properties.

Definition 1. A function is said to be in the class , with and , if it satisfies the inequality where

Example 2. Let us define the functions by This implies that Hence and this further implies that The th Hankel determinant , , , for a function is studied by Noonan and Thomas [3] as In literature many authors have studied the determinant . For example, Arif et al. [4, 5] studied the Hankel determinant for some subclasses of analytic functions. Hankel determinant of exponential polynomials is obtained by Ehrenborg in [6]. The Hankel transform of an integer sequence and some of its properties were discussed by Layman [7]. It is well known that the Fekete-Szegő functional is . Fekete-Szegő then further generalized the estimate with real and . Moreover, we also know that the functional is equivalent to . The sharp upper bounds of the second Hankel determinant for the familiar classes of starlike and convex functions were studied by Janteng et al. [8]; that is, for and , they obtained and , respectively. In 2007, Babalola [9] considered the third Hankel determinant and obtained the upper bound of the well-known classes of bounded-turning, starlike and convex functions. In 2013 Raza and Malik [10] studied the Hankel third determinant related with lemniscate of Bernoulli. In the present investigation, we study the upper bound of for a subclass of analytic functions of reciprocal order by using Toeplitz determinants.

In this paper we study some useful results including coefficient estimates, Fekete-Szegő inequality, and upper bound of third Hankel determinant for the functions belonging to the class .

Throughout in this paper we assume that and unless otherwise stated.

For our results we will need the following Lemmas.

Lemma 3 (see [11]). If is a function with and is of the form then

Lemma 4 (see [12]). If is of the form (12) with positive real part, then the following sharp estimate holds:

Lemma 5 (see [13]). If is of the form (12) with positive real part, then for some , with and .

2. Some Properties of the Class

Theorem 6. Let . Then and for all

Proof. Let us define the function by where is given by (6) with and is analytic in with , .
Now using (1) and (12), we have where Comparing coefficient of like power of , we obtain Using triangle inequality and Lemma 3, we get For and in (23), we easily obtain that Making in (23), we see that equivalently, we have Using the principal of mathematical induction, we obtain Now from the use of relation (21), we obtain the required result.

If we take and , we get the following result.

Corollary 7 (see [14]). Let . Then, for , one has with .

Making and , we get the following result.

Corollary 8 (see [14]). Let . Then, for , one has with .

Theorem 9. Let and be of the form (1). Then where

Proof. Let . Then from (22) we have We now consider Using Lemma 4, we obtain where is given by (31).

Putting , we obtain the following result.

Corollary 10. Let . Then

Theorem 11. Let and be of the form (1). Then

Proof. Let . Then, from (22), we have Consider Now using values of and from Lemma 5, we obtain Applying triangle inequality and replacing by , by , and by , we get Differentiating with respect to , we get Now since for and , maximum of will exist at and let . Then Now by differentiating with respect to , we obtain Since for , has a maximum value at and hence

Theorem 12. Let and be of the form (1). Then

Proof. From (37), we can write Using Lemma 5 for the values of and , we have Applying triangle inequality and then putting , , and , we have Now by using the same procedure as we did in the proof of Theorem 11, we obtain the required result.

Theorem 13. If and is of the form (1), then

Proof. Since using Theorem 6, Corollary 10, and Theorems 11 and 12, we have This completes the proof of this result.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgment

The work here is supported by LRGS/TD/2011/UKM/ICT/03/02.

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. Nunokawa, S. Owa, J. Nishiwaki, K. Kuroki, and T. Hayami, “Differential subordination and argumental property,” Computers & Mathematics with Applications, vol. 56, no. 10, pp. 193–217, 2008. View at Google Scholar
  3. J. W. Noonan and D. K. Thomas, “On the second Hankel determinant of areally mean p-valent functions,” Transactions of the American Mathematical Society, vol. 223, pp. 337–346, 1976. View at Google Scholar · View at MathSciNet
  4. M. Arif, K. I. Noor, and M. Raza, “Hankel determinant problem of a subclass of analytic functions,” Journal of Inequalities and Applications, vol. 2012, article 22, 2012. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  5. M. Arif, K. I. Noor, M. Raza, and W. Haq, “Some properties of a generalized class of analytic functions related with Janowski functions,” Abstract and Applied Analysis, vol. 2012, Article ID 279843, 11 pages, 2012. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  6. R. Ehrenborg, “The Hankel determinant of exponential polynomials,” The American Mathematical Monthly, vol. 107, no. 6, pp. 557–560, 2000. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  7. J. W. Layman, “The Hankel transform and some of its properties,” Journal of Integer Sequences, vol. 4, no. 1, pp. 1–11, 2001. View at Google Scholar
  8. A. Janteng, S. A. Halim, and M. Darus, “Hankel determinant for starlike and convex functions,” International Journal of Mathematical Analysis, vol. 1, no. 13, pp. 619–625, 2007. View at Google Scholar · View at MathSciNet
  9. K. O. Babalola, “On H3(1) Hankel determinant for some classes of univalent functions,” Inequality Theory and Applications, vol. 6, pp. 1–7, 2007. View at Google Scholar
  10. M. Raza and S. N. Malik, “Upper bound of the third Hankel determinant for a class of analytic functions related with lemniscate of Bernoulli,” Journal of Inequalities and Applications, vol. 2013, article 412, 2013. View at Publisher · View at Google Scholar · View at Scopus
  11. C. Pommerenke, Univalent Functions, Vandenhoeck & Ruprecht, Göttingen, Germany, 1975.
  12. F. R. Keogh and E. P. Merkes, “A coefficient inequality for certain classes of analytic functions,” Proceedings of the American Mathematical Society, vol. 20, pp. 8–12, 1969. View at Publisher · View at Google Scholar · View at MathSciNet
  13. U. Grenander and G. Szegö, Toeplitz Forms and Their Applications, California Monographs in Mathematical Sciences, University of California, Berkeley, Calif, USA, 1958. View at MathSciNet
  14. J. Nishiwaki and S. Owa, “Coefficient inequalities for starlike and convex functions of reciprocal order,” Electronic Journal of Mathematical Analysis and Applications, vol. 1, no. 2, pp. 212–216, 2013. View at Google Scholar