Abstract

By making use of the fractional differential operator due to Owa and Srivastava, a class of analytic functions is introduced. The sharp bound for the nonlinear functional is found. Several basic properties such as inclusion, subordination, integral transform, Hadamard product are also studied.

1. Introduction

Let denote the class of functions analytic in the open unit disc and let be the class of functions in given by the normalized power series

Also let and denote, respectively, the subclasses of consisting of functions which are univalent, starlike of order , convex of order (cf. [1]), and close-to-convex (cf. [2]) in . In particular, and are the familiar classes of starlike and convex functions in (cf. [2]).

Given and in , the function is said to be subordinate to in if there exits a function satisfying the conditions of the Schwarz Lemma such that . We denote the subordination by It is well known [2] that if is univalent in then in is equivalent to and .

For the functions and given by the power series their Hadamard product (or convolution), denoted by , is defined by Note that .

By making use of the Hadamard product, Carlson-Shaffer [3] defined the linear operator by where and is the Pochhammer symbol (or shifted factorial) defined in terms of the gamma function by

It can be readily verified that is the identity operator; the operators commute, where , that is, and the transitive property, that is, holds. Each of the following definitions will also be required in our present investigation.

Definition 1.1 (cf. [4, 5], see also [6]). Let the function be analytic in a simply connected region of the -plane containing the origin. The fractional derivative of   of order  is defined by where the multiplicity of is removed by requiring to be real when .

Using Definition 1.1 and its known extensions involving fractional derivatives and fractional integrals, Owa and Srivastava [5] introduced the fractional differintegral operator  defined by Note that , and

Definition 1.2 (cf. [7]). For the function given by (1.2) and , the th Hankel determinant of is defined by

We now introduce the following class of functions.

Definition 1.3. The function is said to be in the class if it satisfies the inequality

Write Let be the family of functions satisfying and .

It follows from (1.15) that where is real, .

We note that and the class has been studied in [8].

It is well known (cf. [2]) that for and given by (1.2), the sharp inequality holds. This corresponds to the Hankel determinant with and . For a given family of functions in , the more general problem of finding sharp estimates for is popularly known as the Fekete-Szegö problem for . The Fekete-Szegö problem for the families has been completely solved by many authors including [9–12].

In the present paper, we consider the Hankel determinant for and and we find the sharp bound for the functional . We also obtain some basic properties of the class . Our investigation includes a recent result of Janteng et al. [13]. We also generalize some results of Ling and Ding [8].

2. Preliminaries

To establish our results, we recall the following.

Lemma 2.1 (See [2]). Let the function and be given by the series Then, the sharp estimate holds.

Lemma 2.2 (cf. [14, Page 254], See Also [15]). Let the function be given by the power series (2.1). Then, for some , and for some .

Lemma 2.3 (See [16]). Let  and  be univalent convex functions in  Then, the Hadamard product  is also a univalent convex function in .

Lemma 2.4 (See [17]). Let  and  be univalent convex functions in  Also let  and  in  Then,  in .

Lemma 2.5 (See [16], Also See [8]). Let  and  be starlike of order  Then, for each function  satisfying  one has 

Lemma 2.6 (See [8]). Let the function  be univalent convex in  For  if  then 

3. Main Results

We prove the following.

Theorem 3.1. Let the functiongiven by (1.2) be in the class Then, The estimate (3.1) is sharp.

Proof. Let . Then, by (1.17), where and is given by (2.1). Using (1.6), (1.7), and (1.13), we write Comparing the coefficients, we get Therefore, (3.4) yields Since the functions and are members of the class simultaneously, we assume without loss of generality that . For convenience of notation, we take .
Using (2.3) along with (2.4), we get An application of triangle inequality and replacement of by give where We next maximize the function on the closed square . Since and , we have for . Thus cannot have a maximum in the interior of the closed square . Moreover, for fixed Next, so that for and has real critical point at . Also . Therefore, occurs at . Therefore, the upper bound of (3.7) corresponds to and . Hence, which is the assertion (3.1). Equality holds for the function The proof of Theorem 3.1 is complete.

The choice of yields what follows.

Corollary 3.2. Let the functiongiven by (1.2) be a member of the class  Then, Equality holds for the function

Remark 3.3. Taking , and , we get a recent result due to Janteng et al. [13].

Theorem 3.4. Suppose and  Then,

Proof. Let Using the associative and commutative properties of the operator , we write where the function is defined by (1.7). Therefore, where . We note that , and . Moreover, it is well known (cf. [18]) that . Therefore, by Lemma 2.5, Hence, , and the proof of Theorem 3.4 is complete.

Theorem 3.5. Letand Then the Hadamard product