Journal of Mathematics

Volume 2013, Article ID 541964, 7 pages

http://dx.doi.org/10.1155/2013/541964

## Mapping Properties of Some Classes of Analytic Functions under Certain Integral Operators

Civil Aviation College, Kocaeli University, Arslanbey Campus, 41285 Kocaeli, Turkey

Received 15 November 2012; Accepted 3 January 2013

Academic Editor: Abdellatif Agouzal

Copyright © 2013 Serap Bulut. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

We consider certain subclasses of analytic functions with bounded radius and bounded boundary rotation and study the mapping properties of these classes under certain integral operators.

#### 1. Introduction

Let be the class of all functions of the following form: which are analytic in the open unit disc

A function is said to be spiral-like if there exists a real number such that The class of all spiral-like functions was introduced by Spacek [1] in 1933 and we denote it by . Later in 1969, Robertson [2] considered the class of analytic functions in for which .

Let be the class of functions analytic in with and where is real with .

For , this class was introduced in [3] and for , see [4]. For and , the class reduces to the class of functions analytic in with and whose real part is positive.

The following definition of fractional derivative by Owa [5] (also by Srivastava and Owa [6]) will be required in our investigation.

The fractional derivative of order is defined, for a function , by where the function is analytic in a simply connected region of the complex -plane containing the origin, and the multiplicity of is removed by requiring to be real when .

It readily follows from (5) that

Using , Owa and Srivastava [7] introduced the operator , which is known as an extension of fractional derivative and fractional integral, as follows: Note that

In [8], Al-Oboudi and Al-Amoudi defined the linear multiplier fractional differential operator (namely, generalized Al-Oboudi differential operator) as follows:

If is given by (1), then by (7) and (9), we see that where

*Remark 1. *(i) When , we get Al-Oboudi differential operator [9].

(ii) When and , we get Sălăgean differential operator [10].

(iii) When and , we get Owa-Srivastava fractional differential operator [7].

*Definition 2. *A function is said to belong to the class if and only if
where is real with and is the generalized Al-Oboudi differential operator.

*Definition 3. *A function is said to belong to the class if and only if
where is real with and is the generalized Al-Oboudi differential operator.

*Remark 4. *(i) Letting and in Definition 2, we have the class introduced by Dileep and Latha [11].

(ii) For and , we obtain the classes and , respectively, introduced and studied by Noor et al. [12] and Moulis [13].

(iii) For and , we have the classes and , respectively, introduced and studied by Noor et al. [14].

(iv) For , and , we have the classes and , respectively, introduced by Frasin [15].

*Definition 5. *Let , , and . One defines the integral operator as
where and is the generalized Al-Oboudi differential operator.

*Remark 6. *The integral operator generalizes many operators which were introduced and studied recently.

(i) For , we have the integral operator introduced by Bulut [16]. Here is the Al-Oboudi differential operator.

(ii) For , and , we have the integral operator introduced by Bulut [17]. Here is the Al-Oboudi differential operator.

(iii) For and , we have the integral operator introduced by Breaz et al. [18]. Here is the Sălăgean differential operator.

(iv) For and , we have the integral operator introduced by D. Breaz and N. Breaz [19].

(v) For , , , and (consists of functions that are analytic, univalent and starlike), we have the integral operator studied by Miller et al. [20].

(vi) For , , , and , we have the integral operator of Alexander introduced by Alexander [21].

*Definition 7. *Let , , and . One defines the integral operator as
where and is the generalized Al-Oboudi differential operator.

*Remark 8. *The integral operator generalizes many operators which were introduced and studied recently.(i)For and , we have the integral operator
introduced by Breaz et al. [22].(ii)For , , , and , we have the integral operator
introduced by Pfaltzgraff [23] (see also Pascu and Pescar [24]).

In this paper, we investigate some propeties of the above integral operators and for the classes

#### 2. Main Results

Theorem 9. *Let for with . Also let be real with . If
**
then the integral operator defined by (15) is in the class with
*

*Proof. *Since , by (10), we have
for all . By (15), we get
This equality implies that
or equivalently
By differentiating the above equality, we get
Hence, we obtain from this equality that
Then by multiplying the above relation with , we have
or equivalently
Subtracting and adding on the left hand side and then taking real part, we have
where is given by (28). Integrating (37) and then using (28), we have
Since , we get
for . Using (39) in (38), we obtain
Hence, we obtain with is given by (28).

By setting , , , in Theorem 9, we obtain the following.

Corollary 10 (see [11, Theorem 1]). *Let for with . Also let be real with . If
**
then the integral operator defined by (17) is in the class with
*

*Remark 11. *Letting in Corollary 10, then we have [12, Theorem 3.1].

By setting , and in Theorem 9, we obtain the following.

Corollary 12. *Let for with . Also let . If
**
then the integral operator defined by (19) is in the class with
*

*Remark 13. *In Corollary 12, letting(i), we have [14, Theorem 2.1],(ii), we have [25, Theorem 1].

Theorem 14. *Let for with . Also let be real with . If
**
then the integral operator defined by (23) is in the class with
*

*Proof. *By (23), we get
This equality implies that
Thus by using (47) and (48), we obtain
Then by multiplying the above relation with , we have
or equivalently
Subtracting and adding on the left hand side and then taking real part, we have
where is given by (46). Integrating (52) and then using (46), we have
Since , we get
for . Using (54) in (53), we obtain
Hence, we obtain with given by (46).

By setting , and in Theorem 14, we obtain the following.

Corollary 15. *Let for with . Also let . If
**
then the integral operator defined by (24) is in the class with
*

*Remark 16. *In Corollary 15, letting(i), we have [14, Theorem 2.5],(ii), we have [25, Theorem 3].

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