The Scientific World Journal
Volume 2014 (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

Muhammad Arif,¹ Maslina Darus,² Mohsan Raza,³ and Qaiser Khan¹

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.

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