Subordination Results of p-Valent Functions Defined by Linear Operator

[1]

**R. M. EL-Ashwah**, Department of Mathematics, Faculty of Science, Damietta University, New Damietta, Egypt.

[2]

**M. E. Drbuk**, Department of Mathematics, Faculty of Science, Damietta University, New Damietta, Egypt.

In this paper, we investigate some interesting differential subordination properties for certain subclasses of p-valent functions which are defined here by means of linear operator defined by El-Ashwah and Drbuk [11]. Further, few interesting convolution properties are obtained.

Analytic Functions, Convex Functions, Linear Operators, Differential Subordination

[1]

M. Abramowitz and I. A. Stegun, Hand book of mathematical functions, Dover Publications, Inc., New York, (1971).

[2]

F. M. Al-Oboudi, On univalent functions defined by a generalized Salagean operator, Indian J. Math. Math. Sci. 25-28 (2004), 1429--1436.

[3]

M . K. Aouf, R. M. El-Ashwah and S. M. El-Deeb, Some inequalities for certain -valent functions involving an extended multiplier transformations, Proc. Pakistan Acad. Sci., 46 (2009), no. 4, 217-221.

[4]

M. K. Aouf and A. O. Mostafa, On a subclass of n-p-valentprestarlike functions, Comput. Math. Appl., (2008), no. 55, 851-861.

[5]

M. K. Aouf, A. O. Mostafa and R. M. El-Ashwah, Sandwich theorems for P -valent functions defined by a certain integral operator, Math. Comput. Modelling, 53 (2011), 1647-1653.

[6]

M . K. Aouf and T. M. Seoudy, On differential sandwich theorems of analytic functions defined by generalized Salagean integral operator, Appl. Math. Letter 24 (2011), 1364-1368.

[7]

A. Catas, On certain classes of p-valent functions defined by multiplier transformations, in Proc. Book of the International Symposium on Geometric Functions Theory and Applications, Istanbul, Turkey, (August 2007), 241-250.

[8]

N. E. Cho and T. H. Kim, Multiplier transformations and strongly close-to-convex functions, Bull. Korean Math. Soc., 40 (2003), no. 3, 399-410.

[9]

N. E. Cho and H. M. Srivastava, Argument estimates of certain analytic functions defined by a class of multiplier transformations, Math. Comput. Modeling, 37 (2003), no. 1-2, 39-49.

[10]

R. M. El-Ashwah and M. K. Aouf, Some properties of new integral operator, Acta Univ. Apulensis, 24 (2010), 51-61.

[11]

R. M. El-Ashwah and M. E. Drbuk, Subordination Properties of p-Valent Functions defined by Linear Operators, British J. Math. Comput. Sci., 4 (2014), no. 21, 3000-3013 .

[12]

T. M. Flett, The dual of an identity of Hardy and Littlewood and some related inequalities, J. Math. Anal. Appl., 38 (1972), 746-765.

[13]

D. J. Hallenbeck and St. Rusheweyh, Subordination by convex functions, Proc. Amer. Math. Soc., 52 (1995), 191-195.

[14]

M. Kamali and H. Orhan, On a subclass of certianstarlike functions with negative coefficients, Bull. Korean Math. Soc., 41 (2004), no. 1, 53-71.

[15]

V. Kiryakova, Generalized Fractional Calculus and Applications, Pitman Research Notes in Mathematics Series, 301, John Willey & Sons, Inc. New York, 1994.

[16]

S. S. Kumar, H. C. Taneja and V. Ravichandran, Classes multivalent functions defined by Dziok-Srivastava linear operaor and multiplier transformations, Kyungpook Math. J., (2006), no. 46, 97-109.

[17]

S. S. Miller and P. T. Mocanu, Differential Subordinations. Theory and Applications, in monographs and textbooks in pure and applied mathematics, 225, Marcel Dekker, New York, (2000).

[18]

S. S. Miller and P. T. Mocanu, Differential superordinations and univalent functions, Michigan Math. J., 28 (1981), no. 2, 157--171.

[19]

J. Patel, Inclusion relations and convolution properties of certain subclasses of analytic functions defined by generalized Salagean operator, Bull. Belg. Math. Soc. Simon Stevin, 15 (2008), 33-47.

[20]

J. Patel and P. Sahoo, Certain subclasses of multivalent analytic functions, Indian J. Pure Appl. Math., 34 (2003), 487-500.

[21]

J. K. Prajapat, Subordination and superordination preserving properties for generalized multiplier transformation operator, Math. Comput. Modeling, 55 (2012), 1456-1465.

[22]

G. S. Salagean, Subclasses of univalent functions, Lecture Notes in Math. (Springer-Verlag) 1013(1983), 362-372.

[23]

H. M. Srivastava, M. K. Aouf and R. M. El-Ashwah, Some inclusion relationships associated with a certain class of integral operators, Asian European J. Math., 3 (2010), no. 4,667-684.

[24]

H. M. Srivastava and S. Owa (Eds.), Curent topics in analytic function theory, World Scientific, Singapore, (1992).

[25]

H. M. Srivastava, K. Suchithra, B. Adolf Stephen and S. Sivasubramanian, Inclusion and neighborhood properties of certian subclasses of multivalent functions of complex order, J. Ineq. Pure Appl. Math., 7(2006), no. 5, Art. 191 ,1-8.

[26]

J. Stankiewicz and z. Stankiewicz, Some applications of the Hadamard convolution in the theory of functions, Ann. Univ. Mariae Curie-Sklodowska Sect. A, 40 (1986), 251-265.