- About this Journal ·
- Abstracting and Indexing ·
- Advance Access ·
- Aims and Scope ·
- Article Processing Charges ·
- Articles in Press ·
- Author Guidelines ·
- Bibliographic Information ·
- Citations to this Journal ·
- Contact Information ·
- Editorial Board ·
- Editorial Workflow ·
- Free eTOC Alerts ·
- Publication Ethics ·
- Reviewers Acknowledgment ·
- Submit a Manuscript ·
- Subscription Information ·
- Table of Contents
International Journal of Mathematics and Mathematical Sciences
Volume 2013 (2013), Article ID 769537, 5 pages
Univalence of a New General Integral Operator Associated with the -Hypergeometric Function
School of Mathematical Sciences, Faculty of Science and Technology, Universiti Kebangsaan Malaysia, 43600 Bangi, Selangor, Malaysia
Received 11 December 2012; Revised 3 February 2013; Accepted 17 February 2013
Academic Editor: Shyam Kalla
Copyright © 2013 Huda Aldweby 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.
Motivated by the familiar -hypergeometric functions, we introduce a new family of integral operators and obtain new sufficient conditions of univalence criteria. Several corollaries and consequences of the main results are also pointed out.
Let denote the class of functions of the form which are analytic in the open unit disk , and the class of functions which are univalent in .
Let , where is defined by (1) and is given by Then the Hadamard product (or convolution) of the functions and is defined by
For complex parameters and, we define the -hypergeometric function by , where denotes the set of positive integers and is the -shifted factorial defined by
Corresponding to the function defined by (4), consider
Recently, the authors  defined the linear operator by where
It should be remarked that the linear operator (7) is a generalization of many operators considered earlier. For , and , we obtain the Dziok-Srivastava linear operator  (for), so that it includes (as its special cases) various other linear operators introduced and studied by Ruscheweyh , Carlson and Shaffer  and the Bernardi-Libera-Livingston operators [6–8].
Definition 2. A function is said to be in the class if it is satisfying the condition where is the operator defined by (7).
Note that , where the class of analytic and univalent functions was introduced and studied by Frasin and Darus .
Using the operator , we now introduce the following new general integral operator.
For , , and , we define the integral operator by where .
Remark 3. It is interesting to note that the integral operator generalizes many operators introduced and studied by several authors, for example,
(2) For , and , we obtain the integral operator studied recently by Breaz et al. .
(3) For , and , we obtain the integral operator introduced and studied by D. Breaz and N. Breaz .
(4) For , and , we obtain the integral operator introduced by Selvaraj and Karthikeyan .
(5) For , and , we obtain the integral operator recently introduced and studied by Breaz and Güney .
(6) For , and , where and , we obtain the integral operator introduced and studied by Pescar .
In order to derive our main results, we have to recall the following univalence criteria.
Lemma 5 (see ). Let with , with . If satisfies for all then the integral operator is in the class .
Lemma 6 (Generalized Schwarz Lemma, see ). (Generalized Schwarz Lemma) Let the function be analytic in the disk , with for fixed . If has one zero with multiplicity order bigger that for , then Equality can hold only if where is constant.
2. Univalence Conditions for
Theorem 7. Let for all , and with If for all , and then the integral operator defined by (11) is analytic and univalent in .
Proof. From the definition of the operator it can be observed that and for , we have We define the function by the form Therefore Differentiating logarithmically and multiplying by on both sides of (29) Thus we have So Since , and for all , then from the Schwarz Lemma and (10), we obtain which, in the light of the hypothesis (24), yields Applying Lemma (1) for the function we obtain that is univalent.
Taking , and in Theorem 7, we have the following.
Taking (for all),, and in Theorem 7, we have the following.
Corollary 9. Let for all and with If and for all , then the integral operator defined by (13) is analytic and univalent in .
Theorem 10. Let for all , and with If for all , and then the integral operator defined by (11) is analytic and univalent in .
Proof. From the proof of Theorem 7, we have Thus we have From this result and using the proof of Theorem 7 we obtain Since, then we have Applying Lemma (4) for the function we obtain that is univalent.
Taking (for all),, and in Theorem 10, we have the following.
Corollary 11. Let for all ; , and with If for all then the integral operator defined by (13) is analytic and univalent in .
Letting and in Corollary 11, we have the following.
Corollary 12. Let , and with If then the integral operator defined by (17) is analytic and univalent in .
The work presented here was partially supported by GUP-2012-023 and UKM-DLP-2011-050.
- G. Gasper and M. Rahman, Basic Hypergeometric Series, vol. 35 of Encyclopedia of Mathematics and its Applications, Cambridge University Press, Cambridge, UK, 1990.
- A. Mohammed and M. Darus, “A generalized operator involving the -hypergeometric function,” Mathematici Vesnik, Available online 10.06.2012, 12 pages.
- J. Dziok and H. M. Srivastava, “Classes of analytic functions associated with the generalized hypergeometric function,” Applied Mathematics and Computation, vol. 103, no. 1, pp. 1–13, 1999.
- S. Ruscheweyh, “New criteria for univalent functions,” Proceedings of the American Mathematical Society, vol. 49, pp. 109–115, 1975.
- B. C. Carlson and D. B. Shaffer, “Starlike and prestarlike hypergeometric functions,” SIAM Journal on Mathematical Analysis, vol. 15, no. 4, pp. 737–745, 1984.
- S. D. Bernardi, “Convex and starlike univalent functions,” Transactions of the American Mathematical Society, vol. 135, pp. 429–446, 1969.
- R. J. Libera, “Some classes of regular univalent functions,” Proceedings of the American Mathematical Society, vol. 16, pp. 755–758, 1965.
- A. E. Livingston, “On the radius of univalence of certain analytic functions,” Proceedings of the American Mathematical Society, vol. 17, pp. 352–357, 1966.
- H. Exton, q-Hypergeometric Functions and Applications, Ellis Horwood, Chichester, UK, 1983.
- H. A. Ghany, “-derivative of basic hypergeometric series with respect to parameters,” International Journal of Mathematical Analysis, vol. 3, no. 33-36, pp. 1617–1632, 2009.
- 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.
- C. Selvaraj and K. R. Karthikeyan, “Sufficient conditions for univalence of a general integral operator,” Acta Universitatis Apulensis, no. 17, pp. 87–94, 2009.
- 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.
- D. Breaz and N. Breaz, “Two integral operators,” Studia Universitatis Babes-Bolyai, Mathematica, Cluj-Napoca, vol. 47, no. 3, pp. 13–19, 2002.
- D. Breaz and H. Ö. Güney, “The integral operator on the classes and ,” Journal of Mathematical Inequalities, vol. 2, no. 1, pp. 97–100, 2008.
- V. Pescar, “A new generalization of Ahlfors's and Becker's criterion of univalence,” Malaysian Mathematical Society Bulletin, vol. 19, no. 2, pp. 53–54, 1996.
- N. N. Pascu, “On a univalence criterion. II,” in Itinerant Seminar on Functional Equations, Approximation and Convexity (Cluj-Napoca, 1985), vol. 85, pp. 153–154, Babeş-Bolyai University, Cluj-Napoca, Romania, 1985.
- N. N. Pascu, “An improvement of Becker's univalence criterion,” in Proceedings of the Commemorative Session: Simion Stoïlow, pp. 43–48, University of Braşov, Braşov, Romania, 1987.
- Z. Nehari, Conformal Mapping, Dover, New York, NY, USA, 1975.
- D. Breaz and H. Ö. Güney, “On the univalence criterion of a general integral operator,” Journal of Inequalities and Applications, vol. 2008, Article ID 702715, 8 pages, 2008.
- G. I. Oros, G. Oros, and D. Breaz, “Sufficient conditions for univalence of an integral operator,” Journal of Inequalities and Applications, vol. 2008, Article ID 127645, 7 pages, 2008.