On Certain Subclasses of Multivalent Functions Associated with a Family of Linear Operators

Jae Ho Choi
DOI: 10.4236/apm.2011.14040                

Abstract

Making use of a linear operator I^λ_p(a,c), which is defined here by means of the Hadamard product (or convolution), we introduce some new subclasses of multivalent functions and investigate various inclusion properties of these subclasses. Some radius problems are also discussed.

Keywords

Multivalent Functions, Hadamard Product (or Convolution), Linear Operators, Radius Problem

Conflicts of Interest

The authors declare no conflicts of interest.

References

[1]	S. D. Bernardi, “Convex and Starlike Univalent Functions,” Transactions of the American Mathematical Society, Vol. 135, 1969, pp. 429-446. doi:10.1090/S0002-9947-1969-0232920-2
[2]	B. C. Carlson and D. B. Shaffer, “Starlike and Prestarlike Hypergeometric Functions,” SIAM Journal on Mathematical Analysis, Vol. 15, No. 4, 1984, pp. 737-745. doi:10.1137/0515057
[3]	N. K. Cho, O. S. Kwon and H. M. Srivastava, “Inclusion Relationships and Argument Properties for Certain Subclasses of Multivalent Functions Associated with a Family of Linear Operator,” Journal of Mathematical Analysis and Applications, Vol. 292, No. 2, 2004, pp. 470-483. doi:10.1016/j.jmaa.2003.12.026
[4]	A. W. Goodman, “Uni-valent Functions, Vol. I, II,” Polygonal Publishing House, Washington, 1983.
[5]	J.-L. Liu and K. I. Noor, “Some Properties of Noor Integral Operator,” Journal of Natural Ge-ometry, Vol. 21, 2002, pp. 81-90.
[6]	S. S. Miller, “Differen-tial Inequalities and Caratheodory Functions,” Bulletin of the American Mathematical Society, Vol. 81, 1975, pp. 78-81. doi:10.1090/S0002-9904-1975-13643-3
[7]	K. I. Noor, “On Subclasses of Close-to-Convex Functions of Higher Order,” International Journal of Mathematics and Mathematical Sci-ences, Vol. 15, No. 2, 1992, pp. 279-289. doi:10.1155/S016117129200036X
[8]	K. I. Noor, “On New Classes of Integral Operators,” Journal of Natural Geometry, Vol. 16, 1999, pp. 71-80.
[9]	K. I. Noor and M. Arif, “Gen-eralized Integral Operators Related with p-Valent Analytic Functions,” Mathematical Inequalities Applications, Vol. 12, No. 1, 2009, pp. 91-98.
[10]	J. Patal and N. E. Cho, “Some Classes of Analytic Functions Involving Noor Integral Opera-tor,” Journal of Mathematical Analysis and Applications, Vol. 312, No. 2, 2005, pp. 564-575. doi:10.1016/j.jmaa.2005.03.047
[11]	B. Pinchuk, “Functions with Bounded Boundary Rotation,” Israel Journal of Mathe-matics, Vol. 10, No. 1, 1971, pp. 7-16. doi:10.1007/BF02771515
[12]	S. Ponnusamy, “Differential Subordination and Bazilevic Functions,” Proceedings Mathe-matical Sciences, Vol. 105, No. 2, 1995, pp. 169-186. doi:10.1007/BF02880363
[13]	H. Saitoh, “A Linear Operator and Its Applications of First Order Differential Subordina-tions,” Mathematica Japonica, Vol. 44, 1996, pp. 31-38.
[14]	J. Sokó? and L. Trojnar-Spelina, “Convolution Properties for Certain Classes of Multivalent Functions,” Journal of Mathematical Analysis and Applications, Vol. 337, No. 2, 2008, pp. 1190-1197. doi:10.1016/j.jmaa.2007.04.055
[15]	H. M. Srivastava and S. Owa (Eds.), “Current Topics in analytic Function Theory,” World Scientific, Singapore, 1992.