A study on Becker’s univalence criteria. (English) Zbl 1220.30015

Summary: We study univalence properties for certain subclasses of univalent functions \(\mathfrak K\), \(\mathfrak K_2\), \(\mathfrak K_3\), and \(S(p)\). These subclasses are associated with a generalized integral operator. An extended Becker type univalence criterion is studied for these subclasses.


30C45 Special classes of univalent and multivalent functions of one complex variable (starlike, convex, bounded rotation, etc.)
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[1] S. Owa, “On the distortion theorems. I,” Kyungpook Mathematical Journal, vol. 18, no. 1, pp. 53-59, 1978. · Zbl 0401.30009
[2] S. Owa and H. M. Srivastava, “Univalent and starlike generalized hypergeometric functions,” Canadian Journal of Mathematics, vol. 39, no. 5, pp. 1057-1077, 1987. · Zbl 0611.33007 · doi:10.4153/CJM-1987-054-3
[3] 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.
[4] 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. · Zbl 1072.30009 · doi:10.1155/S0161171204108090
[5] G. \cS. S\ual\uagean, “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. · Zbl 0531.30009 · doi:10.1007/BFb0066543
[6] 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. · Zbl 0987.30508
[7] 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. · Zbl 1050.30007 · doi:10.1016/S0895-7177(03)80004-3
[8] 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. · Zbl 1032.30007 · doi:10.4134/BKMS.2003.40.3.399
[9] 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. · Zbl 1132.30010 · doi:10.1016/j.jmaa.2007.05.087
[10] 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. · Zbl 1279.30008
[11] 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.
[12] 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. · Zbl 1104.30016
[13] 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. · Zbl 1131.30323
[14] 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. · Zbl 1149.30015 · doi:10.1155/2008/957042
[15] 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. · Zbl 1163.30304 · doi:10.1016/j.aml.2007.11.008
[16] D. Breaz and N. Breaz, “Two integral operators,” Universitatis Babe\cs-Bolyai, vol. 47, no. 3, pp. 13-19, 2002. · Zbl 1027.30018
[17] 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. · Zbl 0324.30034
[18] J. Becker, “Löwnersche differentialgleichung und Schlichtheitskriterien,” Mathematische Annalen, vol. 202, pp. 321-335, 1973. · Zbl 0236.30024 · doi:10.1007/BF01433462
[19] 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. · Zbl 0880.30020
[20] V. Singh, “On a class of univalent functions,” International Journal of Mathematics and Mathematical Sciences, vol. 23, no. 12, pp. 855-857, 2000. · Zbl 0962.30004 · doi:10.1155/S0161171200001824
[21] 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. · Zbl 1207.41012 · doi:10.1016/j.camwa.2010.10.012
[22] A. Mohammed and M. Darus, “A new integral operator for meromorphic functions,” Acta Universitatis Apulensis, no. 24, pp. 231-238, 2010. · Zbl 1224.30062
[23] A. Mohammed, M. Darus, and D. Breaz, “Some properties for certain integral operators,” Acta Universitatis Apulensis, no. 23, pp. 79-89, 2010. · Zbl 1265.30077
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