Sandwich Theorems for Multivalent Analytic Functions Associated with Differential Operator


Download PDF

Authors: A. K. WANAS AND A. L. ALINA

DOI: 10.46793/KgJMat2101.007W

Abstract:

The purpose of this paper is to derive subordination and superordination results involving differential operator for multivalent analytic functions in the open unit disk. These results are applied to obtain sandwich results. Our results extend corresponding previously known results.

Keywords:

Multivalent functions, differential subordination, differential superoordination, dominant, subordinant, differential operator.

References:

[1]   R. M. Ali, V. Ravichandran, M. H. Khan and K. G. Subramanian, Differential sandwich theorems for certain analytic functions, Far East J. Math. Sci. 15(1) (2004), 87–94.

[2]   F. M. Al-Oboudi, On univalent functions defined by a generalized Salagean operator, Int. J. Math. Math. Sci. 27 (2004), 1429–1436.

[3]   K. Al-Shaqsi and M. Darus, On univalent functions with respect to K-symmetric points defined by a generalization Ruscheweyh derivative operators, J. Anal. Appl. 7 (2009), 53–61.

[4]   T. Bulboacă, Classes of first order differential superordinations, Demonstr. Math. 35(2) (2002), 287–292.

[5]   T. Bulboacă, A class of superordination-preserving integral operators, Indag. Math. (N.S.) 13(3) (2002), 301–311.

[6]   M. Darus and K. Al-Shaqsi, Differential sandwich theorems with generalized derivative operator, Int. J. Comput. Math. Sci. 22 (2008), 75–78.

[7]   M. Darus and K. Al-Shaqsi, Differential sandwich theorems with generalized derivative operator, Int. J. Math. Comput. Sci. 38 (2008), 11–14.

[8]   E. A. Eljamal and M. Darus, Majorization for certain classes of analytic functions defined by a new operator, CUBO 14(1) (2012), 119–125.

[9]   S. P. Goyal, P. Goswami and H. Silverman, Subordination and superordination results for a class of analytic multivalent functions, Int. J. Math. Math. Sci. (2008), Article ID 561638, 1–12.

[10]   S. S. Miller and P. T. Mocanu, Differential Subordinations: Theory and Applications, Series on Monographs and Textbooks in Pure and Applied Mathematics 225, Marcel Dekker Inc., New York, Basel, 2000.

[11]   S. S. Miller and P. T. Mocanu, Subordinants of differential superordinations, Complex Variables 48(10) (2003), 815–826.

[12]   G. Murugusundaramoorthy and N. Magesh, Differential subordinations and superordinations for analytic functions defined by Dziok-Srivastava linear operator, Journal of Inequalities in Pure and Applied Mathematics 7(4) (2006), Article ID 152, 1–20.

[13]    V. O. Nechita, Differential subordinations and superordinations for analytic functions defined by the generalized Salagean derivative operator, Acta Univ. Apulensis 16 (2008), 14–156.

[14]   D. Răducanu and V. O. Nechita, A differential sandwich theorem for analytic functions defined by the generalized Salagean operator, Aust. J. Math. Anal. Appl. 9(1) (2012), 1–7.

[15]   St. Ruscheweyh, New certain for univalent functions, Proc. Amer. Math. Soc. 49 (1975), 109–115.

[16]   G. Salagean, Subclasses of univalent functions, in: C. A. Cazacu, N. Boboc, M. Jurchescu and I. Suciu (Eds.), Complex Analysis - Fifth Romanian-Finnish Seminar, Lecture Notes in Mathematics 1013, Springer-Verlag, Berlin, 1983, 362–732.

[17]   T. N. Shanmugam, V. Ravichandran and S. Sivasubramanian, Differential sandwich theorems for some subclasses of analytic functions, Aust. J. Math. Anal. Appl. 3(1) (2006), 1–11.

[18]   T. N. Shanmugam, C. Ramachandran, M. Darus and S. Sivasubramanian, Differential sandwich theorems for some subclasses of analytic functions involving a linear operator, Acta Math. Univ. Comenianae 74(2) (2007), 287–294.

[19]   T. N. Shanmugam, S. Sivasubramanian and H. Silverman, On sandwich theorems for some classes of analytic functions, Int. J. Math. Math. Sci. (2006), Article ID 29684, 1–13.