Research Article | Open Access

# Fekete-Szegö Inequalities for Starlike Functions with respect to -Symmetric Points of Complex Order

**Academic Editor:**J. Dziok

#### Abstract

Sharp upper bounds of for the function belonging to certain subclass of starlike functions with respect to -symmetric points of complex order are obtained. Also, applications of our results to certain functions defined through convolution with a normalized analytic function are given. In particular, Fekete-Szegö inequalities for certain classes of functions defined through fractional derivatives are obtained.

#### 1. Introduction

Let denote the class of analytic functions of the following form: And let be the subclass of , which are univalent functions.

Let be given by the following: The Hadamard product (or convolution) of and is given by

If and are analytic functions in , we say that is subordinate to , written if there exists a Schwarz function , which is analytic in with and for all , such that . Furthermore, if the function is univalent in , then we have the following equivalence (see [1, 2]):

Sakaguchi [3] introduced a class of functions starlike with respect to symmetric points, which consists of functions satisfying the inequality

Chand and Singh [4] introduced a class of functions starlike with respect to -symmetric points, which consists of functions satisfying the inequality where

Al-Shaqsi and Darus [5] defined the linear operator as follows: and in general where

In this paper, we define the following class () as follows.

*Definition 1. *Let be univalent starlike function with respect to which maps the unit disk onto a region in the right half plane which is symmetric with respect to the real axis. Let be a complex number and let . Then functions are in the class if
where is defined by (9) and is defined by (7).

We note that for suitable choices of , , , , and we obtain the following subclasses:(i) (see Al-Shaqsi and Darus [6]),(ii) (see Al-Shaqsi and Darus [6]),(iii) (see Al-Shaqsi and Darus [6]),(iv) (see Shanmugam et al. [7]),(v) (see Sakaguchi [3]),(vi) (see Shanthi et al. [8] and Al-Shaqsi and Darus [9]),(vii) (see Ma and Minda [10]),(viii) (see Janowski [11]),(ix) and (see Ravichandran et al. [12]),(x) (see Nasr and Aouf [13]),(xi) (see Nasr and Aouf [14] and Aouf et al. [15]),(xii) (see Libera [16]),(xiii) (see Chichra [17]),(xiv) and (see Aouf and Silverman [18]),(xv) (see Keogh and Markes [19]).Also, we note the following: In this paper, we obtain the Fekete-Szegö inequalities for the functions in the class . We also give application of our results to certain functions defined through convolution and, in particular, we consider the class defined by fractional derivatives.

#### 2. Fekete-Szegö Problem

To prove our results, we need the following lemmas.

Lemma 2 (see [10]). *If is an analytic function with positive real part in and is a complex number, then
**
The result is sharp for the functions given by
*

Lemma 3 (see [10]). *If is an analytic function with positive real part in then
**
When or , the equality holds if and only if or one of its rotations. If , 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. If , the equality holds if and only if
**
Also the above upper bound is sharp and it can be improved as follows when :
*

Theorem 4. *Let . If given by (1) belongs to the class and , then
**
where
**
This result is sharp.*

*Proof. *Let ; then there is a Schwarz function in with and in such that
If the function is analytic and has positive real part in and , then
Since is a Schwarz function, define
From (7), we obtain
where is given by (21). In view of (22) and (23), we have
Since
therefore, we have
and, from this equation and (28), we obtain
Then, from (24), we see that
Now from (24), (28), and (30), we have
Therefore, we have
where
Our result now follows by an application of Lemma 2. The result is sharp for the functions
This completes the proof of Theorem 4.

*Remark 5. *(i) Putting , , , and in Theorem 4, we obtain the result obtained by Keogh and Markes [19, Theorem 1].

(ii) Putting and in Theorem 4, we obtain the result obtained by Shanthi et al. [8, Theorem 2.6].

(iii) Putting , , and in Theorem 4, we obtain the result obtained by Ma and Minda [10].

Putting and in Theorem 4, we obtain the following corollary.

Corollary 6. *Let . If given by (1) belongs to the class , then
**
This result is sharp.*

Putting , , and in Theorem 4, we obtain the following corollary.

Corollary 7. *Let . If given by (1) belongs to the class , then
**
This result is sharp.*

Putting in Theorem 4, we obtain the following corollary.

Corollary 8. *Let If given by (1) belongs to the class and , then
**
This result is sharp.*

Putting in Corollary 8, we obtain the following corollary.

Corollary 9. *Let . If given by (1) belongs to the class , then
**
This result is sharp.*

Putting in Theorem 4, we obtain the following corollary.

Corollary 10. *Let . If given by (1) belongs to the class and , then
**
This result is sharp.*

Putting in Corollary 10, we obtain the following corollary.

Corollary 11. *Let . If given by (1) belongs to the class , then
**
This result is sharp.*

By using Lemma 3, we can obtain the following theorem.

Theorem 12. *Let If given by (1) belongs to the class and , then
**
where
**
and is given by (21). The result is sharp.*

*Proof. *To show that the bounds are sharp, we define the functions by
and the functions and by
Cleary the functions , , and Also we write . If or , then the equality holds if and only if is or one of its rotations. When then the equality holds 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 , in view of Lemma 3. This completes the proof of Theorem 12.

*Remark 13. *(i) Putting , , and in Theorem 12, we obtain the result obtained by Al-Shaqsi and Darus [6, Theorem 2.1].

(ii) Putting , , , and in Theorem 12, we obtain the result obtained by Shanmugam et al. [7, Theorem 2.1].

(iii) Putting and in Theorem 12, we obtain the result obtained by Al-Shaqsi and Darus [9, Theorem 2.1].

(iv) Putting , , and in Theorem 12, we obtain the result obtained by Ma and Minda [10].

Putting in Theorem 12, we obtain the following corollary.

Corollary 14. *Let . If given by (1) belongs to the class and if , then
**
where
**
The result is sharp.*

The proof of Theorem 15 is similar to the proof of Theorem 12, so the details are omitted.

Theorem 15. *Let Let given by (1) belong to the class and . Let be given by
**
where is given by (21). If , then
**
If , then
**
where and are given in Theorem 12.*

*Remark 16. *(i) Putting , , and in Theorem 15, we obtain the result obtained by Al-Shaqsi and Darus [6, Remark 2.1].

(ii) Putting , , and in Theorem 15, we obtain the result obtained by Shanmugam et al. [7, Theorem 2.2].

(iii) Putting and in Theorem 15, we obtain the result obtained by Al-Shaqsi and Darus [9, Remark 2.2].

Putting in Theorem 15, we obtain the following corollary.

Corollary 17. *Let Let given by (1) belong to the class and . Let be given by
**
If , then
**
If , then
**
where and are given in Corollary 14.*

#### 3. Applications to Functions Defined by Fractional Derivatives

*Definition 18 (see [20]). *Let be analytic in a simply connected region of the -plane containing origin. The fractional derivative of of order is defined by
where the multiplicity of is removed by requiring that be real for .

Using Definition 18, Owa and Srivastava (see [21–23]) introduced a fractional derivative operator defined by

*Definition 19. *A fixed given by (2) and given by (1) belong to the class if
where is defined in Definition 1 and is defined in (9).

The class consists of the functions for which . The class is a special case of the class when Now applying Theorem 4 for the function , we get the following theorem.

Theorem 20. *Let . If given by (2) and given by (1) belong to the class and if , then
**
where is given by (21). This result is sharp.*

*Proof. *The proof of Theorem 20 is similar to the proof of Theorem 4, so the details are omitted.

Putting and in Theorem 20, we obtain the following corollary.

Corollary 21. *Let . If given by (2) and given by (1) belong to the class , then
**
where is given by (21). This result is sharp.*

Putting , , and in Theorem 20, we obtain the following corollary.

Corollary 22. *Let . If given by (2) and given by (1) belong to the class , then
**
This result is sharp.*

Now applying Theorem 12 for the function , we get the following theorem.

Theorem 23. *Let . If given by (1) and given by (2), then belongs to the class and , then**
where
**
and is given by (21). The result is sharp.*

*Remark 24. *(i) Putting , , and in Theorem 23, we obtain the result obtained by Al-Shaqsi and Darus [6, Theorem 3.1].

(ii) Putting , , and in Theorem 23, we obtain the result obtained by Shanmugam et al. [7, Theorem 3.2].

(iii) Putting in Theorem 23, we obtain the result obtained by Al-Shaqsi and Darus [9, Theorem 3.2].

Since
we have