Subordination properties of certain classes of multivalently analytic functions. (English) Zbl 1201.30019

Srivastava, H. M.; Lashin, A. Y.

Math. Comput. Modelling 52, No. 3-4, 596-602 (2010).

Summary: By using a method based upon the principle of the Briot-Bouquet differential subordination, we obtain a sharp result for \(p\)-valently starlikeness of certain normalized and multivalently analytic functions. Our investigation of such families of multivalently analytic functions is motivated essentially by a recent study by M. Nunokawa, S. Owa, T. Sekine, R. Yamakawa, H. Saitoh and J. Nishiwaki [Int. J. Math. Math. Sci. 2007, Article ID 72393, 5 p. (2007; Zbl 1139.30305)]. We also give an answer to the open problem which was posed in the aforementioned article by Nunokawa et al.

MSC:

30C45

Special classes of univalent and multivalent functions of one complex variable (starlike, convex, bounded rotation, etc.)

Keywords:

analytic functions; multivalent functions; starlike and convex functions; \(p\)-valently starlike and convex functions; Briot-Bouquet differential subordination; generalized hypergeometric function

Citations:

Zbl 1139.30305

References:

[1]	Goodman, A. W., On the Schwarz-Christoffel transformation and \(p\)-valent functions, Trans. Amer. Math. Soc., 68, 204-223 (1950) · Zbl 0037.05502
[2]	Nunokawa, M.; Owa, S.; Sekine, T.; Yamakawa, R.; Saitoh, H.; Nishiwaki, J., On certain multivalent functions, Internat. J. Math. Math. Sci., 2007, 1-5 (2007), Article ID 72393 · Zbl 1139.30305
[3]	Miller, S. S.; Mocanu, P. T., Univalent solutions of Briot-Bouquet differential subordinations, J. Differential Equations, 56, 297-309 (1985) · Zbl 0507.34009
[4]	Miller, S. S.; Mocanu, P. T., Differential Subordinations: Theory and Applications, Series on Monographs, Textbooks and Lecture Notes in Pure and Applied Mathematics, Vol. 225 (2000), Marcel Dekker: Marcel Dekker New York, Basel
[5]	Liu, M.-S.; Zhu, Y.-C.; Srivastava, H. M., Properties and characteristics of certain subclasses of starlike functions of order \(\beta \), Math. Comput. Modelling, 48, 402-419 (2008) · Zbl 1145.30306
[6]	Srivastava, H. M.; Eker, S. S., Some applications of a subordination theorem for a class of analytic functions, Appl. Math. Lett., 21, 394-399 (2008) · Zbl 1138.30014
[7]	Srivastava, H. M.; Eker, S. S.; Şeker, B., A certain convolution approach for subclasses of analytic functions with negative coefficients, Integral Transforms Spec. Funct., 20, 687-699 (2009) · Zbl 1219.26010
[8]	Srivastava, H. M.; Yang, D.-G.; Xu, N-E., Subordinations for multivalent analytic functions associated with the Dziok-Srivastava operator, Integral Transforms Spec. Funct., 20, 581-606 (2009) · Zbl 1170.30006
[9]	Wang, Z.-G.; Jiang, Y.-P.; Srivastava, H. M., Some subclasses of meromorphically multivalent functions associated with the generalized hypergeometric function, Comput. Math. Appl., 57, 571-586 (2009) · Zbl 1165.30344