Table of Contents Author Guidelines Submit a Manuscript
Abstract and Applied Analysis
Volume 2013 (2013), Article ID 621810, 6 pages
http://dx.doi.org/10.1155/2013/621810
Research Article

On General Integral Operator of Analytic Functions

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

Received 3 August 2013; Accepted 27 October 2013

Academic Editor: Mohamed Kamal Aouf

Copyright © 2013 Nasser Alkasbi and Maslina Darus. 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

Let be the integral operator defined by , where each of the functions and is, respectively, analytic functions and functions with positive real part defined in the open unit disk for all . The object of this paper is to obtain several univalence conditions for this integral operator. Our main results contain some interesting corollaries as special cases.

1. Introduction and Definitions

Let denote the class of functions of the form which are analytic in the open unit disk is univalent in . Also, let be the class of all functions which are analytic in and satisfy , . Frasin and Darus [1] defined the family , , so that it consists of functions satisfying the condition In this paper, we obtain new sufficient conditions for the univalence of the general integral operator defined by where , , , and for all .

Here and throughout in the sequel, every multivalued functions is taken with the principal branch.

Remark 1. Note that the integral operator generalizes the following operators introduced and studied by several authors as follows.(i) For , where , we obtain the integral operator introduced and studied by Frasin [2].(ii) For , , we obtain the integral operator introduced and studied by Frasin [3].(iii) For , , we obtain the integral operator introduced and studied by Frasin [4].(iv) For , , and , we obtain the integral operator introduced and studied by Frasin [5].(v) For and , we obtain the integral operator introduced and studied by D. Breaz and N. Breaz [6].(vi) For , , and , we obtain the integral operator introduced and studied by D. Breaz and N. Breaz [6].(vii) For , , and , we obtain the integral operator introduced and studied by Breaz et al. [7].(viii) For , , , and , we obtain the integral operator studied in [8].(ix) For , , , and , we obtain the integral operator studied in [9]. In particular, for , we obtain Alexander integral operator which was introduced in [10] as follows (x) For , , , and , we obtain the integral operatorstudied in [11].In order to derive our main results, we have to recall here the following lemmas.

Lemma 2 (see [12]). Let with . If satisfies for all , then the integral operator is in the class .

Lemma 3 (see [13]). Let with , with , . If satisfies for all , then the integral operator defined by (16) is in the class .

Lemma 4 (see [14]). If , then

Lemma 5 (see [9]). If , then When , so .

Lemma 6 (see [9]). If , then Also we need the following general Schwarz lemma.

Lemma 7 (see [15]). Let the function be regular in the disk , with for fixed . If has one zero with multiplicity order bigger than for , then The equality holds only if where is constant.

Lemma 8 (see [16]). If , then

2. Univalence Conditions for the Operator

We first prove the following theorem.

Theorem 9. Let , , for all and with . If then the integral operator defined by (3) is in the class .

Proof. Define the regular function by Then it is easy to see that and . Differentiating both sides of (26) logarithmically, we obtain Thus, we have Since , for all , from (28), (18), (19), and (20), we obtain Multiplying both sides of (29) by , we get for all .
Let us denote , , , and . It is easy to prove that From (31), (30), and the hypotheses (24), we have for all . Applying Lemma 2 for the function , we prove that .

Letting , , , , and in Theorem 9, we obtain the following corollary.

Corollary 10. Let , , , and all with . If and then the integral operator defined by is in the class .

If we set in Corollary 10, we have the following.

Corollary 11. Let , and all with . If then the integral operator defined by (34) is in the class .

Next, we prove the following theorem.

Theorem 12. Let for all and each satisfies condition (15) with and and then, for any complex number with , the integral operator defined by (4) is in the class .

Proof. Suppose that for all . Thus we have where for all . Differentiating both sides of (37) logarithmically, we obtain Define the regular function as in (25). Thus from (27) we have and so From Lemma 8, it follows that Multiplying both sides of (41) by , from Lemma 5 with , we get Suppose that . Define a function by Then is an increasing function and consequently, for ; , we obtain We thus find from (42) and (44) that Using the hypotheses (36) for , we readily get Now if , we define a function by We observe that the function is decreasing and consequently, for , we have for all . It follows from (40) and (42) that Using once again the hypotheses (36) when , we easily get Finally by applying Lemma 2, we conclude that the integral operator defined by (4) is in the class .

Letting , , , , and in Theorem 12, we obtain the following corollary.

Corollary 13. Let , and satisfies condition (15). If then, for any complex number with , the integral operator defined by (4) is in the class .

Using Lemma 3, we derive the following theorem.

Theorem 14. Let for all , , and each satisfies condition (15). If then, for any complex number with , the integral operator defined by (4) is in the class .

Proof. From (40), we have Suppose that . Define a function by Then is an increasing function and consequently for ; , we obtain We thus find from (53) that Using the hypotheses (52) for , we readily get Now if , we define a function by We observe that the function is decreasing and consequently for ; , and using once again the hypotheses (36) when , we easily get Finally, by applying Lemma 3, we conclude that .

Conflict of Interests

The authors declare that they have no conflict interests.

Authors’ Contribution

The first author is currently a Ph.D. student under supervision of the second author and jointly worked on deriving the results. All authors read and approved the paper.

Acknowledgment

The work presented here was partially supported by ERGS/1/2013/STG06/UKM/01/2 and DIP-2013-1.

References

  1. B. A. Frasin and M. Darus, “On certain analytic univalent functions,” International Journal of Mathematics and Mathematical Sciences, vol. 25, no. 5, pp. 305–310, 2001. View at Google Scholar
  2. B. A. Frasin, “General integral operator of analytic functions involving functions with positive real part,” Journal of Mathematics, vol. 2013, Article ID 260127, 4 pages, 2013. View at Publisher · View at Google Scholar
  3. B. A. Frasin, “Integral operator of analytic functions with positive real part,” Kyungpook Mathematical Journal, vol. 51, no. 1, pp. 77–85, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
  4. B. A. Frasin, “Order of convexity and univalency of general integral operator,” Journal of the Franklin Institute, vol. 348, no. 6, pp. 1013–1019, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
  5. B. A. Frasin, “New general integral operator,” Computers and Mathematics with Applications, vol. 62, no. 11, pp. 4272–4276, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
  6. D. Breaz and N. Breaz, “Two integral operators, Studia Universitatis Babes-Bolyai,” Mathematica, vol. 3, pp. 13–21, 2002. View at Google Scholar
  7. D. Breaz, S. Owa, and N. Breaz, “A new integral univalent operator,” Acta Universitatis Apulensis, no. 16, pp. 11–16, 2008. View at Google Scholar
  8. M. Dorf and J. Szynal, “Linear invariance and integral operators of univalent functions,” Demonstratio Mathematica, vol. 38, no. 1, pp. 47–57, 2005. View at Google Scholar
  9. E. Deniz and H. Orhan, “An extension of the univalence criterion for a family of integral operators,” Annales Universitatis Mariae Curie-Sklodowska A, vol. 64, no. 2, pp. 29–35, 2010. View at Google Scholar
  10. 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 Google Scholar
  11. N. Pascu and V. Pescar, “On the integral operators of Kim-Merkes and Pfaltzgraff,” Mathematica, vol. 32, no. 2, pp. 185–192, 1990. View at Google Scholar
  12. N. Pascu and V. Pescar, “An improvement of Becker’s univalence criterion,” in Proceedings of the Commemorative Session: Simion Stoilow (Brasov 1987) (Brasov), pp. 43–48, University of Brasov, 1987.
  13. V. Pescar, “Univalence of certain integral operators,” Acta Universitatis Apulensis, vol. 12, no. 2006, pp. 43–48, 1989. View at Google Scholar
  14. T. H. MacGregor, “The radius of univalence of certain analytic functions,” Proceedings of the American Mathematical Society, vol. 14, pp. 514–520, 1963. View at Google Scholar
  15. Z. Nehari, Conformal Mapping, Dover, New York, NY, USA, 1975.
  16. M. Obradovic and S. Owa, “A criterion for starlikeness,” Mathematische Nachrichten, vol. 140, pp. 97–102, 1978. View at Google Scholar