The Scientific World Journal

Volume 2014 (2014), Article ID 989640, 6 pages

http://dx.doi.org/10.1155/2014/989640

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

^{1}Department of Mathematics, Abdul Wali Khan University Mardan, Mardan, Khyber Pakhtunkhwa 23200, Pakistan^{2}School of Mathematical Sciences, Faculty of Science and Technology, Universiti Kebangsaan Malaysia, 43600 Bangi, Selangor, Malaysia^{3}Department 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 *

*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*

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

*Acknowledgment*

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

*References*

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