On the univalence of an integral operator

Abstract

In this paper the method of Loewner chains is used to derive a fairly general and flexible univalence criterion for an integral operator. Two examples involving Bessel and hypergeometric functions are given. Our results include a number of known or new univalence criteria.

Keywords

• Analytic functions,integral operator,Loewner chain,univalence criterion

References

1. [1] L. V. Ahlfors,Sufficient conditions for quasiconformal extension, Ann. Math. Studies., 79 (1974), 23-29.
2. [2] A. Baricz, B. A. Frasin ,Univalence of integral operator involving Bessel functions, Appl. Math. Lett., 23(4)(2010), 371-376.
3. [3] J. Becker, Löwnersche differential gleichung und quasikonform fortsetzbare schlichte functionen, J. Reine Angew. Math.255 (1972), 23-43.
4. [4] J. Becker, Conformal mappings with quasiconformal extensions, Aspects of Contemporary Complex Analysis, Ed. by D. A. Brannan and J. G. Clunie, Acad. Press, 1980, 37-77.
5. [5] D. Breaz, N. Breaz, H. M. Srivastava, An extension of the univalent condition for a family of integral operators, Appl. Math. Lett., 22(2009), 41-44.
6. [6] E. Deniz, H. Orhan, H. M. Srivastava, Some sufficient conditions for univalence of certain families of integral operators involving generalized Bessel functions, Taiwanese J. Math., 15(2)(2011), 883-971.
7. [7] E. Deniz, D. Raducanu, H. Orhan, On the univalence of an integral operator defined by Hadamard product, Appl. Math. Lett., 25(2012), 179-184.
8. [8] B. A. Frasin, Certain sufficient conditions for univalence of two integral operators, European J. Pure Appl. Math., 3(6)(2010), 1141-1149.

9. [9] S. Moldoveanu, N. N. Pascu, Integral operators which preserve the univalence, Mathematica (Cluj), 32(55)(1990), 159-166.
10. [10] H. Ovesea, An extension of Lewandowski’s univalence criterion, Demonstratio Math., XXIX, 4(1996), 699-706.
11. [11] H. Ovesea, A generalization of Ruscheweyh’s univalence criterion, J. Math. Anal. Appl., 258 (2001), 102-109.
12. [12] N. N. Pascu, An improvement of Becker’s univalence criterion, Commemorative Session Simion Stoilov, Univ. of Brasov, (Preprint) (1987), 43-48.
13. [13] V. Pescar, A new generalization of Ahlfors and Becker’s criterion of univalence, Bull. Math. Malaysian Math. Soc. (Second Series), 19 (1996), 53-54.
14. [14] V. Pescar, D. Breaz, The Univalence of Integrals Operators, Prof. M. Drinov Acad. Publish. House, Sofia, 2008.
15. [15] Ch. Pommerenke, Über die Subordination analytischer Funktionen, J. Reine Angew Math., 218 (1965), 159-173.
16. [16] Ch. Pommerenke, Univalent Functions, Vandenhoeck Ruprecht in Göttingen, 1975.
17. [17] D. Raducanu, H. Orhan, E. Deniz, On some sufficient conditions for univalence, An. St. Univ. Ovidius, Constanµa, Ser. Math., 18(2)(2010), 217-222.
18. [18] H. M. Srivastava, E. Deniz, H. Orhan, Some general univalence criteria for a family of integral operators, Appl. Math. Comp., 215(2010), 3696-3701.
19. [19] D. Tan, Quasiconformal extension and univalency criteria, Mich. Math. J., 39(1992), 163- 172.