ON CLASSES OF MULTIVALENT FUNCTIONS INVOLVING LINEAR OPERATOR AND MULTIPLIER TRANSFORMATIONS

Year 2008, Volume: 37 Issue: 2, 115 - 127, 01.02.2008

H. A. Al-kharsani N. M. Al-areefi

Abstract

Using the results of first order differential subordinations and superordinations, we define and discuss new classes of p-valent functions involving the Dziok-Srivastava operator and multiplier transformation. 

Keywords

Analytic function, p-valent function, Differential subordinates, Differential superordinations, Dziok-Srivastava linear operator, Multiplier transformation, 2000 AMS Classification: 30 C 80

ON CLASSES OF MULTIVALENT FUNCTIONS INVOLVING LINEAR OPERATOR AND MULTIPLIER TRANSFORMATIONS

Abstract

References

• Al-Kharsani, H. A. and Al-Khal, R. A. Generalized integral operator and univalent functions, Int. J. Math. Anal. 1 (15), 709–718, 2007.
• Bernardi, S. D. Convex and starlike univalent functions, Trans. Amer. Math. Soc. 135, 429–446, 1969.
• Bulboac˘a, T. A class of superordination-preserving integral operators, Indag. Math. (N.S.) 13(3), 301–311, 2002.
• Bulboac˘a, T. Classes of first-order differential superordinations, Demonstratio Math. 35 (2), 287–292, 2002.
• Carlson, B. C. and Shaffer, D. B. Starlike and prestarlike hypergeometric functions, SIAM J. Math. Anal. 15 (4), 737–745, 1984.
• Cho, N. E. and Srivastava, H. M. Argument estimates of certain analytic functions defined by a class of multiplier transformations, Math. Comput. Modeling 37 (1-2), 39–43, 2003.
• Cho, N. E. and Kim, T. H. Multiplier transformations and strongly close-to-convex func- tions, Bull. Korean Math. Soc. 40 (3), 399–410, 2003.
• Dziok, J. and Srivastava, H. M. Certain subclasses of analytic functions associated with the generalized hypergeometric functions, Integral Transforms Spec. Funct. 14 (1), 7–18, 2003. [9] Hohlov, Ju. E. Operators and operations on the class of univalent functions, Izv. Vyssh. Uchebn. Zaved. Math. 10 (197), 83–89, 1978.
• Kwon, O. S. and Cho, N. E. Inclusion properties for certain subclasses of analytic functions associated with the Dziok-Srivastava operator, J. Ineq. and Appl. 27, 10 pp, 2007.
• Libera, R. J. Some classes of regular univalent functions, Proc. Amer. Math. Soc. 16, 755– 758, 1965.
• Livingston, A. E. On the radius of univalence of certain analytic functions, Proc. Amer. Math. Soc. 17, 352–357, 1966.
• Miller, S. S. and Mocanu, P. T. Differential Subordinations: Theory and Applications (Mono- graphs and Textbooks in Pure and Applied Mathematics, No. 225, Marcel Dekker, New York, 2000). [14] Miller, S. S. and Mocanu, P. T. Subordinants of differential superordinations, Complex Var. Theory Appl. 48 (10), 815–826, 2003.
• Noor, K. I. and Noor, M. A. On integral operators, J. Math. Anal. and Appl. 23 (8), 341–352, 1999. [16] Noor, K. I. On some analytic functions defined by a multiplier transformation, Int. J. Math. & Math. Scie., 9 pp, 2007.
• Owa, S. On the distoriton theorems I, Kyungpook Math. J. 18 (1), 53–59, 1978.
• Owa, S. and Srivastava, H. M. Univalent and starlike generalized hypergeometric functions, Canad. J. Math. 39 (5), 1057–1077, 1987.
• Ruscheweyh, St. New criteria for univalent functions, Proc. Amer. Math. Soc. 49, 109–115, 1975. [20] S¸˘al˘agean, G. S. Subclasses of univalent functions in Complex analysis (Springer, Berlin 1, 1981), 362–372.
• Srivastava, H. M. Some families of fractional derivative and other linear operators associated with analytic univalent and multivalent functions in Analysis and its applications(Allied Publ., New Delhi, 2000), 209–243.
• Uralegaddi, B. A. and Somanatha, C. Certain classes of univalent functions in Current topics in analytic function theory (World Publishing, River Edge, NJ, 1992).