Abstract and Applied Analysis

Volume 2011 (2011), Article ID 759175, 13 pages

http://dx.doi.org/10.1155/2011/759175

## A Study on Becker's Univalence Criteria

School of Mathematical Sciences, Faculty of Science and Technology, Universiti Kebangsaan Malaysia, Bangi, Selangor D. Ehsan 43600, Malaysia

Received 26 January 2011; Accepted 11 May 2011

Academic Editor: Allan C. Peterson

Copyright © 2011 Maslina Darus and Imran Faisal. 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 study univalence properties for certain subclasses of univalent functions , , , and , respectively. These subclasses are associated with a generalized integral operator. The extended Becker-typed univalence criteria will be studied for these subclasses.

#### 1. Introduction and Preliminaries

Let denote the class of analytic functions in the open unit disk normalized by . Thus, each has a Taylor series representation Let be the subclass of consisting of functions of the form Let be the univalent subclass of which satisfies Let be the subclass of for which . Let be the subclass of consisting of functions of the form (1.2) which satisfy Next, we define a subclass of consisting of all functions that satisfy For functions and , the Hadamard product (or convolution) is defined as usual by Define the function by where is the famous Pochhammer symbol defined in terms of Gamma function. It is easily seen that is a convex function, since

Using the fractional derivative of order , [1], Owa and Srivastava [2] introduced the operator which is known as an extension of fractional derivative and fractional integral, as follows: Note that

For a function in , we define , the linear fractional differential operator, as follows: If is given by (1.1), then by (1.8) and (1.9), we see that From (1.8) and (1.9), can be written in terms of convolution as where which generalizes many operators. Indeed, if we choose suitably values of , , , and in (1.12), we have the following.(i), , and , we obtain given by Aouf et al. [3].(ii), , , and , we obtain given by Al-Oboudi [4]. (iii), , , , and , we obtain given by Sălăgean [5].(iv), , , , and , we obtain given by Uralegaddi and Somanatha [6].(v), , , and , we obtain given by Cho and Srivastava [7] and Cho and Kim [8].(vi), , , , and , we obtain Owa and Srivastava differential operator [2].(vii), , and , we obtain given by Al-Oboudi and Al-Amoudi [9, 10].(viii), , and , we obtain given by Catas [11].(ix), , , and , we obtain given by Kumar et al. and Srivastava et al., respectively [12, 13].

Next, we introduce a new family of integral operator by using generalized differential operator already defined above.

For and , , we define a family of integral operators by which generalize many integral operators. In fact, if we choose suitable values of parameters in this type of operator, we get the following interesting operators.(i), , , , , and , we obtain given by Bulut [14].(ii), , , , , , and , we obtain given by Breaz et al. [15].(iii), , , , , , and , we obtain given by D. Breaz and N. Breaz [16].

For our main result, we need the following lemmas.

Lemma 1.1 (see [17, 18]). *Let be a complex number, , . If is a regular function in and
**
then the function is regular and univalent in .*

Lemma 1.2 (Schwarz Lemma). *Let the function be regular in the disk with . If has one zero with multiply for , then
**
and equality holds only if , where is constant.*

Lemma 1.3 (see [19]). *Let be a complex number with such that . If satisfies the condition
**
then the function
**
is analytic and univalent in .*

Lemma 1.4 (see [20]). *If a function , then
*

#### 2. Univalence Properties

In this section, we will discuss the univalence properties of the new family of integral operators mentioned above.

Theorem 2.1. *Let be a complex number, , for all and for such that
**
where , are complex numbers. If
**
then the family is univalent.*

*Proof. *Since , so by Lemma 1.4, we have
Now, by using hypothesis, we have
so by Lemma 1.3, we get
Let
so
Let
which implies that
This implies that
or
Using (2.5), we get
This implies that
By using (2.3), we get
which implies that
because implies that
Now, we calculate
This implies that
By using (2.46), we conclude that
Hence, by Lemma 1.3, the family of integral operators is univalent.

Corollary 2.2. *Let be a complex number, , for all and , , for all such that
**
where , are complex numbers. If
**
then the family is univalent.*

Corollary 2.3. *Let be a complex number, , , for all and the family , , , for all such that
**
where , are complex numbers. If
**
then the family is univalent. *

Using the method given in the proof of Theorem 2.1, one can prove the following results.

Theorem 2.4. *Let be a complex number, , for all and the family for and such that
**
where , are complex numbers. If
**
then the family is univalent.*

Theorem 2.5. * Let be a complex number, , for all and for such that
**
where , are complex numbers. If
**
then the family is univalent.*

Theorem 2.6. *Let be a complex number, , for all and for such that
**
where , are complex numbers. If
**
then the family is univalent.*

Theorem 2.7. * Let be a complex number, , for all and , for such that
**
where , are complex numbers. If
**
then the family is univalent.*

*Proof. * Using the proof of Theorem 2.1, we have
Since , so by using (1.4), we get
So from (2.33), we get
or
Now, we evaluate the expression
Using (2.45) and (2.46), we conclude that
Hence by using Lemma 1.3, the family is univalent.

Corollary 2.8. *Let be a complex number, , for all and , for such that
**
where , are complex numbers. If
**
then the family is univalent.*

Corollary 2.9. *Let be a complex number, , , for all and , for such that
**
where , are complex numbers. If
**
then the family is univalent.*

Using a similar method as in the proof of Theorem 2.7, one can prove the following results.

Theorem 2.10. *Let be a complex number, , for all and , for such that
**
where , are complex numbers. If
**
then the family is univalent.*

Theorem 2.11. *Let be a complex number, , for all and , for such that
**
where , are complex numbers. If
**
then the family is univalent. *

Note that some other related work involving integral operators regarding univalence criteria can also be found in [21–23].

#### Acknowledgment

The work presented here was partially supported by UKM-ST-06-FRGS0244-2010.

#### References

- S. Owa, “On the distortion theorems. I,”
*Kyungpook Mathematical Journal*, vol. 18, no. 1, pp. 53–59, 1978. View at Google Scholar · View at Zentralblatt MATH - 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 - M. K. Aouf, R. M. El-Ashwah, and S. M. El-Deeb, “Some inequalities for certain
*p*-valent functions involving extended multiplier transformations,”*Proceedings of the Pakistan Academy of Sciences*, vol. 46, no. 4, pp. 217–221, 2009. View at Google Scholar - F. M. Al-Oboudi, “On univalent functions defined by a generalized Salagean 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 - G. Ş. Sălăgean, “Subclasses of univalent functions,” in
*Complex Analysis—5th Romanian-Finnish Seminar, Part 1 (Bucharest, 1981)*, 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 - B. A. Uralegaddi and C. Somanatha, “Certain classes of univalent functions,” in
*Current Topics in Analytic Function Theory*, pp. 371–374, World Scientific, Singapore, 1992. View at Google Scholar · View at Zentralblatt MATH - N. E. Cho and H. M. Srivastava, “Argument estimates of certain analytic functions defined by a class of multiplier transformations,”
*Mathematical and Computer Modelling*, vol. 37, no. 1-2, pp. 39–49, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - N. E. Cho and T. H. Kim, “Multiplier transformations and strongly close-to-convex functions,”
*Bulletin of the Korean Mathematical Society*, vol. 40, no. 3, pp. 399–410, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - 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 - F. M. Al-Oboudi, “On classes of functions related to starlike functions with respect to symmetric conjugate points defined by a fractional differential operator,”
*Complex Analysis and Operator Theory*. In press. - A. Catas, “On certain classes of
*p*-valent functions defined by multiplier transformations,” in*Proceedings of the International Symposium on Geometric Function Theory and Applications (GFTA '07)*, S. Owa and Y. Polatoglu, Eds., vol. 91, TC Istanbul Kultur University Publications, Istanbul, Turkey, August 2007. - S. S. Kumar, H. C. Taneja, and V. Ravichandran, “Classes multivalent functions defined by Dziok—Srivastava linear operaor and multiplier transformations,”
*Kyungpook Mathematical Journal*, vol. 46, pp. 97–109, 2006. View at Google Scholar - H. M. Srivastava, K. Suchithra, B. A. Stephen, and S. Sivasubramanian, “Inclusion and neighborhood properties of certain subclasses of analytic and multivalent functions of complex order,”
*Journal of Inequalities in Pure and Applied Mathematics*, vol. 7, no. 5, pp. 1–8, 2006. View at Google Scholar · View at Zentralblatt MATH - S. Bulut, “Sufficient conditions for univalence of an integral operator defined by Al-Oboudi differential operator,”
*Journal of Inequalities and Applications*, Article ID 957042, 5 pages, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - D. Breaz, N. Breaz, and H. M. Srivastava, “An extension of the univalent condition for a family of integral operators,”
*Applied Mathematics Letters*, vol. 22, no. 1, pp. 41–44, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - D. Breaz and N. Breaz, “Two integral operators,”
*Universitatis Babeş-Bolyai*, vol. 47, no. 3, pp. 13–19, 2002. View at Google Scholar · View at Zentralblatt MATH - L. V. Ahlfors, “Sufficient conditions for quasiconformal extension,” in
*Discontinuous Groups and Riemann Surfaces (Proc. Conf., Univ. Maryland, College Park, Md., 1973)*, pp. 23–29, Princeton University Press, Princeton, NJ, USA, 1974. View at Google Scholar - J. Becker, “Löwnersche differentialgleichung und Schlichtheitskriterien,”
*Mathematische Annalen*, vol. 202, pp. 321–335, 1973. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - V. Pescar, “A new generalization of Ahlfors's and Becker's criterion of univalence,”
*Malaysian Mathematical Society. Bulletin. Second Series*, vol. 19, no. 2, pp. 53–54, 1996. View at Google Scholar · View at Zentralblatt MATH - V. Singh, “On a class of univalent functions,”
*International Journal of Mathematics and Mathematical Sciences*, vol. 23, no. 12, pp. 855–857, 2000. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - N. Breaz, D. Braez, and M. Darus, “Convexity properties for some general integral operators on uniformly analytic functions classes,”
*Computers and Mathematics with Applications*, vol. 60, pp. 3105–3017, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - A. Mohammed and M. Darus, “A new integral operator for meromorphic functions,”
*Acta Universitatis Apulensis*, no. 24, pp. 231–238, 2010. View at Google Scholar - A. Mohammed, M. Darus, and D. Breaz, “Some properties for certain integral operators,”
*Acta Universitatis Apulensis*, no. 23, pp. 79–89, 2010. View at Google Scholar