SOME SUBCLASSES OF ANALYTIC FUNCTIONS INVOLVING THE GENERALIZED SRIVASTAVA-ATTIYA OPERATOR

Abstract

In the present paper, we introduce and investigate some new subclasses of multivalent analytic functions involving the generalized SrivastavaAttiya operator. Such results as inclusion relationships, subordination and superordination properties, integral-preserving properties and convolution properties are proved.

Keywords

• Analytic functions,Multivalent functions,Differential subordination,Superordination,Hadamard product (or convolution),Generalized Srivastava-Attiya operator,2000 AMS Classification: Primary: 30 C 45. Secondary: 30 C 80

References

1. Choi, J. and Srivastava, H. M. Certain families of series associated with the Hurwitz-Lerch Zeta function, Appl. Math. Comput. 170, 399–409, 2005.
2. Elrifai, E. A., Darwish, H. E. and Ahmed, A. R. Some applications of Srivastava-Attiya operator to p-valent starlike functions, J. Inequal. Appl. 2010, Article ID 790730, 1–11, 2010.
3. Ferreira, C. and L´opez, J. L. Asymptotic expansions of the Hurwitz-Lerch Zeta function, J. Math. Anal. Appl. 298, 210–224, 2004.
4. Garg, M., Jain, K. and Srivastava, H. M. Some relationships between the generalized Apostol- Bernoulli polynomials and Hurwitz-Lerch Zeta functions, Integral Transforms Spec. Funct. 17, 803–815, 2006.
5. Kutbi, M. A. and Attiya, A. A. Differential subordination result with the Srivastava-Attiya integral operator, J. Inequal. Appl. 2010, Article ID 618523, 1–10, 2010.
6. Lin, S. -D. and Srivastava, H. M. Some families of the Hurwitz-Lerch Zeta functions and associated fractional derivative and other integral representations, Appl. Math. Comput. 154, 725–733, 2004.
7. Lin, S.D., Srivastava, H. M. and Wang, P. -Y. Some expansion formulas for a class of gen- eralized Hurwitz-Lerch Zeta functions, Integral Transforms Spec. Funct. 17, 817–827, 2006. [8] Liu, J. -L. Subordinations for certain multivalent analytic functions associated with the gen- eralized Srivastava-Attiya operator, Integral Transforms Spec. Funct. 19, 893–901, 2008.
8. Liu, J. -L. Sufficient condtions for strongly starlike functions involving the generalized Srivastava-Attiya operator, Integral Transforms Spec. Funct. 22, 79–90, 2011.

9. Q.-M. Luo, Q. -M. and Srivastava, H. M. Some generalizations of the Apostol-Bernoulli and Apostol-Euler polynomials, J. Math. Anal. Appl. 308, 290–302, 2005.
10. Miller, S. S. and Mocanu, P. T. On some classes of first order differential subordinations, Michigan Math. J. 32, 185–195, 1985.
11. Miller, S. S. and Mocanu, P. T. Differential Subordination: Theory and Applications, in: (Series in Pure and Applied Mathematics 225, Marcel Dekker, New York, 2000).
12. Miller, S. S. and Mocanu, P. T. Subordinats of differential superordinations, Complex Var. Theory Appl. 48, 815–826, 2003.
13. Murugusundaramoorthy, G. A subclass of analytic functions associated with the Hurwitz- Lerch Zeta function, Hacet. J. Math. Stat. 39, 265–272, 2010.
14. Murugusundaramoorthy, G. Subordination results for spiral-like functions associated with the Srivastava-Attiya operator, Integral Transforms Spec. Funct. (2011), doi: 10.1080/10652469.2011.562501.
15. Noor, K. I. and Bukhari, S. Z. H. Some subclasses of analytic and spiral-like functions of complex order involving the Srivastava-Attiya integral operator, Integral Transforms Spec. Funct. 21, 907–916, 2010.
16. Prajapat, J. K. and Goyal, S. P. Applications of Srivastava-Attiya operator to the classes of strongly starlike and strongly convex functions, J. Math. Inequal. 3, 129–137, 2009.
17. Rˇaducanu, D. and Srivastava, H. M. A new class of analytic functions defined by means of a convolution operator involving the Hurwitz-Lerch Zeta function, Integral Transforms Spec. Funct. 18, 933–943, 2007.
18. Shanmugam, T. N., Ravichandran, V. and Sivasubramanian, S. Differential sandwich the- orems for subclasses of analytic functions, Aust. J. Math. Anal. Appl. 3, Article 8, 1–11, 2006 (electronic).
19. Singh, R. and Singh, S. Convolution properties of a class of starlike functions, Proc. Amer. Math. Soc. 106, 145–152, 1989.
20. Srivastava, H. M. and Attiya, A. A. An integral operator associated with the Hurwitz-Lerch Zeta function and differential subordination, Integral Transforms Spec. Funct. 18, 207–216, 2007.
21. Srivastava, H. M. and Choi, J. Series associated with the Zeta and related functions (Kluwer Academic Publishers, Dordrecht, Boston, London, 2001).
22. Sun, Y., Kuang, W. -P. and Wang, Z. -G. Properties for uniformly starlike and related functions under the Srivastava-Attiya operator, Appl. Math. Comput. 218, 3615–3623, 2011. [24] Wang, Z. -G., Li, Q. -G. and Jiang, Y. -P. Certain subclasses of multivalent analytic functions involving the generalized Srivastava-Attiya operator, Integral Transforms Spec. Funct. 21, 221–234, 2010.
23. Wang, Z. -G., Liu, Z. -H. and Sun, Y. Some properties of the generalized Srivastava-Attiya operator, Integral Transforms Spec. Funct. 23, 223–236, 2012.
24. Wen, F. and Liu, Z. A Copula-based correlation measure and its application in Chinese stock market, Int. J. Inf. Tech. Decis. 8, 1–15, 2009.
25. Wen, F. and Yang, X. Skewness of return distribution and coeffcient of risk premium, J. Syst. Sci. Complex. 22, 360–371, 2009.
26. Xiang, R. -G., Wang, Z. -G. and Darus, M. A family of integral operators preserving subor- dination and superordination, Bull. Malays. Math. Sci. Soc. (2) 33, 121–131, 2010.
27. Yuan, S. -M. and Liu, Z. -M. Some properties of two subclasses of k-fold symmetric functions associated with Srivastava-Attiya operator, Appl. Math. Comput. 218, 1136–1141, 2011.