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

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.

Geometrical meaning of inequality (2) is that ( )/ ( ) maps U onto the interior of the circle with center at 1 and radius 1 − . By S * ( ) and K * ( ), we mean the classes of starlike and convex functions of reciprocal order , 0 ≤ < 1 which are defined, respectively, by Recently in 2008, Nunokawa and his coauthors [2] improved inequality (2) for the class S * ( ) and they proved that, for ( ) ∈ S * ( ), 0 < < 1/2, 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.

2
The Scientific World Journal Definition 1. A function ( ) ∈ A is said to be in the class L( , ), with ∈ C \ {0} and ∈ [0, 1], if it satisfies the inequality where Example 2. Let us define the functions ( ) by This implies that and this further implies that The th Hankel determinant ( ), ≥ 1, ≥ 1, for a function ( ) ∈ A 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 th 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 | 3 − 2 2 | is 2 (1). Fekete-Szegő then further generalized the estimate | 3 − 2 2 | with real and ( ) ∈ S. Moreover, we also know that the functional | 2 4 − 2 3 | is equivalent to 2 (2). 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 ( ) ∈ S * and ( ) ∈ C, they obtained | 2 4 − 2 3 | ≤ 1 and 8| 2 4 − 2 3 | ≤ 1, respectively. In 2007, Babalola [9] considered the third Hankel determinant 3 (1) 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 3 (1) 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 L( , ).
For our results we will need the following Lemmas.
Proof. Let us define the function ( ) by where ( ) is given by (6) with and ( ) is analytic in U with (0) = 1, Re ( ) > 0. Now using (1) and (12), we have The Scientific World Journal Comparing coefficient of like power of , we obtain Using triangle inequality and Lemma 3, we get For = 2 and = 3 in (23), we easily obtain that Making = 4 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 = 0 and = 1 − , we get the following result.
Making = 1 and = 1 − , we get the following result.
Proof. Since