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Journal of Mathematics

Volume 2013 (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].

#### References

- L. Spacek, “Prispěvek k teorii funkei prostych,”
*Čapopis pro pěstováni matematiky a fysiky*, vol. 62, pp. 12–19, 1933. - M. S. Robertson, “Univalent functions $f(z)$ for which $z{f}^{\text{'}}(z)$ is spirallike,”
*The Michigan Mathematical Journal*, vol. 16, pp. 97–101, 1969. View at Zentralblatt MATH · View at MathSciNet - K. S. Padmanabhan and R. Parvatham, “Properties of a class of functions with bounded boundary rotation,”
*Annales Polonici Mathematici*, vol. 31, no. 3, pp. 311–323, 1975. View at Zentralblatt MATH · View at MathSciNet - B. Pinchuk, “Functions of bounded boundary rotation,”
*Israel Journal of Mathematics*, vol. 10, pp. 7–16, 1971. View at Zentralblatt MATH · View at MathSciNet - S. Owa, “On the distortion theorems. I,”
*Kyungpook Mathematical Journal*, vol. 18, no. 1, pp. 53–59, 1978. View at Zentralblatt MATH · View at MathSciNet - H. M. Srivastava and S. Owa, Eds.,
*Univalent Functions, Fractional Calculus, and Their Applications*, Ellis Horwood Series: Mathematics and its Applications, Ellis Horwood, Chichester, UK, John Wiley & Sons, New York, NY, USA, 1989. View at MathSciNet - S. Owa and H. M. Srivastava, “Univalent and starlike generalized hypergeometric functions,”
*Canadian Journal of Mathematics*, vol. 39, no. 5, pp. 1057–1077, 1987. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - F. M. Al-Oboudi and K. A. Al-Amoudi, “On classes of analytic functions related to conic domains,”
*Journal of Mathematical Analysis and Applications*, vol. 339, no. 1, pp. 655–667, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - F. M. Al-Oboudi, “On univalent functions defined by a generalized Sălăgean operator,”
*International Journal of Mathematics and Mathematical Sciences*, no. 25–28, pp. 1429–1436, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - G. Ş. Sălăgean, “Subclasses of univalent functions,” in
*Complex Analysis-Fifth Romanian-Finnish seminar. Part 1*, vol. 1013 of*Lecture Notes in Mathematics*, pp. 362–372, Springer, Berlin, Germany, 1983. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - L. Dileep and S. Latha, “A generalized integral operator associated with functions of bounded boundary rotation,”
*General Mathematics*, vol. 19, no. 3, pp. 25–30, 2011. View at MathSciNet - K. I. Noor, M. Arif, and A. Muhammad, “Mapping properties of some classes of analytic functions under an integral operator,”
*Journal of Mathematical Inequalities*, vol. 4, no. 4, pp. 593–600, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - E. J. Moulis Jr., “Generalizations of the Robertson functions,”
*Pacific Journal of Mathematics*, vol. 81, no. 1, pp. 167–174, 1979. View at Zentralblatt MATH · View at MathSciNet - K. I. Noor, M. Arif, and W. U. Haq, “Some properties of certain integral operators,”
*Acta Universitatis Apulensis. Mathematics. Informatics*, no. 21, pp. 89–95, 2010. View at Zentralblatt MATH · View at MathSciNet - B. A. Frasin, “Family of analytic functions of complex order,”
*Acta Mathematica Academiae Paedagogiace Nyíregyháziensis (N. S.)*, vol. 22, no. 2, pp. 179–191, 2006. View at Zentralblatt MATH · View at MathSciNet - S. Bulut, “A new general integral operator defined by Al-Oboudi differential operator,”
*Journal of Inequalities and Applications*, vol. 2009, Article ID 158408, 13 pages, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - S. Bulut, “Some properties for an integral operator defined by Al-Oboudi differential operator,”
*Journal of Inequalities in Pure and Applied Mathematics*, vol. 9, no. 4, article 115, 5 pages, 2008. View at MathSciNet - D. Breaz, H. Ö. Güney, and G. Ş. Sălăgean, “A new general integral operator,”
*Tamsui Oxford Journal of Mathematical Sciences*, vol. 25, no. 4, pp. 407–414, 2009. View at Zentralblatt MATH · View at MathSciNet - D. Breaz and N. Breaz, “Two integral operators,”
*Studia Universitatis Babes-Bolyai Mathematica*, vol. 47, no. 3, pp. 13–19, 2002. View at Zentralblatt MATH · View at MathSciNet - S. S. Miller, P. T. Mocanu, and M. O. Reade, “Starlike integral operators,”
*Pacific Journal of Mathematics*, vol. 79, no. 1, pp. 157–168, 1978. View at MathSciNet - I. W. Alexander, “Functions which map the interior of the unit circle upon simple regions,”
*Annals of Mathematics*, vol. 17, no. 1, pp. 12–22, 1915. View at Publisher · View at Google Scholar · View at MathSciNet - D. Breaz, S. Owa, and N. Breaz, “A new integral univalent operator,”
*Acta Universitatis Apulensis. Mathematics. Informatics*, no. 16, pp. 11–16, 2008. View at MathSciNet - J. A. Pfaltzgraff, “Univalence of the integral of ${f}^{\text{'}}{(z)}^{\lambda}$,”
*The Bulletin of the London Mathematical Society*, vol. 7, no. 3, pp. 254–256, 1975. View at Zentralblatt MATH · View at MathSciNet - N. N. Pascu and V. Pescar, “On the integral operators of Kim-Merkes and Pfaltzgraff,”
*Mathematica (Cluj)*, vol. 32 (55), no. 2, pp. 185–192, 1990. View at MathSciNet - S. Bulut, “A note on the paper of Breaz and Güney,”
*Journal of Mathematical Inequalities*, vol. 2, no. 4, pp. 549–553, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet