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