Certain Bound for -Starlike and -Convex Functions with respect to Symmetric Points

Abstract

The aim of this paper is to establish the coefficient estimates for the subclasses of -starlike and -convex functions with respect to symmetric points involving -difference operator. Also certain applications based on these results for subclasses of univalent functions defined by convolution are given.

1. Introduction

The -difference calculus or quantum calculus was initiated at the beginning of 19th century that was initially developed by Jackson [1, 2]. Basic definitions and properties of -deference calculus can be found in the book mentioned in [3]. The fractional -difference calculus had its origin in the works by Al-Salam [4] and Agarwal [5]. Recently, the area of -calculus has attracted the serious attention of researchers. The great interest is due to its application in various branches of mathematics and physics. Mohammed and Darus [6] studied approximation and geometric properties of these -operators for some subclasses of analytic functions in compact disk.

Let denote the class of all functions of the formwhich are analytic in the open unit disk .

Let be the subclass of consisting of all univalent functions in .

If and are analytic in , then we say that the function is subordinate to , if there exists a Schwarz function , analytic in with such thatWe denote this subordination by In particular, if the function is univalent in , the above subordination is equivalent to and (see [7, 8]).

A -analog of the class of starlike functions was first introduced in [9] by means of the -difference operator acting on functions given by (1) and , and the -derivative of a function is defined by (see [1, 2]) and . From (5), we deduce thatwhereAs , . For a function , we observe that where is the ordinary derivative.

As a right inverse, Jackson [2] introduced the -integral provided that the series converges. For a function , we have where is the ordinary integral. Note that the -difference operator plays an important role in the theory of hypergeometric series and quantum physics (see, e.g., [10–14]).

Ma and Minda [15] unified various subclasses of starlike and convex functions for which either quantity or quantity is subordinate to a more general superordinate function. For this purpose, they considered an analytic function with positive real part in the unit disc , with , , and maps onto a region starlike, with respect to the real axis.

The class of Ma-Minda starlike functions consists of functions satisfying the subordination . Similarly, the class of Ma-Minda convex functions consists of functions satisfying the subordination .

Sakaguchi [16] introduced and studied class of starlike functions with respect to symmetric points. Class defined below is the generalization of class .

Definition 1 (see [17]). Let be a univalent starlike function with respect to 1 which maps the unit disk onto a region in the right half plane which is symmetric with respect to the real axis and let . The function is in classes and :

Definition 2. A function is said to belong to classes and if respectively.

As , we denote subclasses and by and which are the class of starlike and convex functions with respect to symmetric points, respectively (see [18]).

The main objective of the present paper is to derive the Fekete-Szegö inequality for functions in classes and . Also, by using convolution product, we give the applications of our results to the defined functions and in particular we consider classes and defined by fractional derivatives.

In order to prove the main result, we need the following lemma.

Lemma 3 (see [15]). If is an analytic function with positive real part in , thenWhen or , the equality holds if and only if or one of its rotations. When , then the equality holds if and only if or one of its rotations. If , the equality holds if and only if , or one of its rotations. While , equality holds if and only if is the reciprocal of one of the functions for which the equality holds in the case of . Also the above upper bound can be improved as follows: when

We also need the following results in our investigation.

Lemma 4 (see [19]). If is an analytic function with positive real part in , then The result is sharp for the function and .

2. Main Results

Theorem 5. Let . Also let , where the coefficients are real with and .
If given by (1) belongs to , thenwhereThe result is sharp.

Proof. For , letFrom (18), we obtain Since is univalent and , the function is analytic and has positive real part in . Thus we haveand from (18) and (21), where Our result now follows by an application of Lemma 3. To show that these bounds are sharp, we define the functions by and the functions and by Clearly the functions . Also we write . If or , the equality holds if and only if is or one of its rotations. When , the equality holds if and only if is or one of its rotations. If then the equality holds if and only if is or one of its rotations. If then the equality holds if and only if is or one of its rotations.
If , then, in view of Lemma 3, Theorem 5 can be improved.

Theorem 6. Let . Also let , where the coefficients are real with and .
Let given by (1) belongs to . Let be given by .
If , then If , then

Corollary 7 (see [18]). Let . Also let , where the coefficients are real with and .
If given by (1) belongs to and (i.e., reduced to ) thenwhere The result is sharp.

Theorem 8. Let . Also let , where the coefficients are real with and .
If given by (1) belongs to , thenwhereThe result is sharp.

Proof. LetSince is univalent and , the function is analytic and has positive real part in . Thus we haveand from (34), where Our result now follows by an application of Lemma 3. To show that these bounds are sharp, we define the functions by and the functions and by Clearly the functions . Also we write . If or , the equality holds if and only if is or one of its rotations. When , the equality holds if and only if is or one of its rotations. If then the equality holds if and only if is or one of its rotations. If then the equality holds if and only if is or one of its rotations.
If , then, in view of Lemma 3, Theorem 8 can be improved.

Corollary 9. Let . Also let , where the coefficients are real with and .
Let given by (1) belong to . Let be given by .
If , thenIf , then

Corollary 10 (see [18]). Let . Also let , where the coefficients are real with and .
If given by (1) belongs to and (i.e., is reduced to ) thenwhereThe result is sharp.

3. Applications to Functions Defined by Fractional Derivative

In order to introduce classes and we need the following.

Definition 11 (see [20]). Let 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 that is real for . Using the above Definition 11 and its known extensions involving fractional derivatives and fractional integrals, Owa and Srivastava [20] introduced the operator defined by Classes and consist of functions for which and , respectively.

For a fixed , let be the class of functions for which and let be the class of functions for which .

Classes and are the special case of classes and , respectively, whenLet

Since if and only if , we obtain the coefficient estimates for functions in classes and , from the corresponding estimates for functions in classes and .

Applying Theorem 5 for the function , we get the following theorem for an obvious change of .

Theorem 12. Let the function be given by and let , . If given by (1) belongs to , thenwhereThe result is sharp.

Sincewe have

Theorem 13. Let the function , . If given by (1) belongs to , thenwhere The result is sharp.

For and given by (50), Theorems 12 and 13 reduce to the following.

Corollary 14. Let . If given by (1) belongs to , thenwhere

Corollary 15. Let . If given by (1) belongs to , thenwhere

Conflict of Interests

The authors declare that there is no conflict of interest regarding the publication of this paper.