Generalization of Certain Subclasses of Multivalent Functions with Negative Coefficients Defined by Cho-Kwon-Srivastava Operator

ABSTRACT

Making use of the Cho-Kwon-Srivastava operator, we introduce and study a certain*SC*_{n} (*j, p, λ, α, δ*) of *p*-valently analytic functions with negative coefficients. In this paper, we obtain coefficient estimates, distortion theorem, radii of close-to-convexity, starlikeness and convexity and modified Hadamard products of functions belonging to the class *SC*_{n} (*j, p, λ, α, δ*). Finally, several applications investigate an integral operator, and certain fractional calculus operators also considered.

Making use of the Cho-Kwon-Srivastava operator, we introduce and study a certain

KEYWORDS

Multivalent Functions, Cho-Kwon-Srivastava Operator, Modified-Hadamard Product, Fractional Calculus

Multivalent Functions, Cho-Kwon-Srivastava Operator, Modified-Hadamard Product, Fractional Calculus

Cite this paper

nullE. Elrifai, H. Darwish and A. Ahmed, "Generalization of Certain Subclasses of Multivalent Functions with Negative Coefficients Defined by Cho-Kwon-Srivastava Operator,"*Applied Mathematics*, Vol. 2 No. 10, 2011, pp. 1225-1235. doi: 10.4236/am.2011.210171.

nullE. Elrifai, H. Darwish and A. Ahmed, "Generalization of Certain Subclasses of Multivalent Functions with Negative Coefficients Defined by Cho-Kwon-Srivastava Operator,"

References

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[2] A. W. Goodman, “Univalent Functions, Vols. I and II,” Polygonal Pub-lishing House, Washington, 1983.

[3] S. Owa, “The Quasi-Hadamard Products of Certain Analytic Functions,” In: H. M. Srivastava and S. Owa, Eds., Current Topics in Analytic Function Theory, World Scientific Publishing Company, Singapore, 1992, pp. 234- 251.

[4] Z.-G. Wang, R. Aghalary, M. Darus and R. W. Ibrahim, “Some Properties of Certain Multivalent Analytic Functions Involving the Cho-Kwon-Srivastava Operator,” Journal of Mathematical and Computer Modelling, Vol. 49, No. 9-10, 2009, pp. 1969-1984.

[5] N. E. 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 Operators,” Journal of Mathematical Analysis and Applications, Vol. 292, No. 2, 2004, pp. 470-483. doi:10.1016/j.jmaa.2003.12.026

[6] R. Yamakawa, “Certain Subclasses of p-Valently Starlike Functions with Negative Coefficients,” In: H. M. Srivas- tava and S. Owa, Eds., Current Topics in Analytic Function Theory, World Scientific Publishing Company, Singapore, 1992, pp. 393-402.

[7] 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

[8] A. E. Li-vingston, “On the Radius of Univalence of Certain Ana-lytic Functions,” Proceedings of the American Mathe-matical Society, Vol. 17, No. 2, 1966, pp. 352-357. doi:10.1090/S0002-9939-1966-0188423-X

[9] H. M. Srivastava and S. Owa (Eds.), “Current Topics in Analytic Function Theory,” World Scientific Publishing Company, Singapore, 1992.

[10] S. Owa, “On Distortion Theorems. I,” Kyungpook Mathematical Journal, Vol. 18, 1978, pp. 55-59.

[11] H. M. Srivastava and M. K. Aouf, “A Certain Fractional Derivative Operator and Its Applications to a New Class of Analytic and Multivalent Functions with Negative Coefficients. I and II,” Journal of Mathematical Analysis and Applications, Vol. 171, No. 1, 1992, pp. 1-13. doi:10.1006/jmaa.1995.1197

[12] A. Schild and H. Sil-verman, “Convolutions of Univalent Functions with Neg-ative Coefficients,” Annales Universitatis Mariae Cu-rie-Sklodowska Section A, Vol. 29, 1975, pp. 99-107.

[13] O. Altintas, H. Irmak and H. M. Srivastava, “Fractional Calculus and Certain Starlike Functions with Negative Coefficients,” Computers and Mathematics with Applications, Vol. 30, No. 2, 1995, pp. 9-15. doi:10.1016/0898-1221(95)00073-8

[14] M.-P. Chen, H. Irmak and H. M. Srivastava, “Some Families of Multiva-lently Analytic Functions with Negative Coefficients,” Journal of Mathematical Analysis and Applications, Vol. 214, No. 2, 1997, pp. 674-690. doi:10.1006/jmaa.1997.5615

[15] H. M. Srivastava and S. Owa (Eds.), “Univalent Functions, Fractional Calculus, and Their Applications,” Ellis Horwood Limited, Chiche-ster, 1989.

[1] P. L. Duren, “Univalent functions,” In: Grundlehen der Mathematischen Wissenschaften, Vol. 259, Springer- Verlag, New York, 1983.

[2] A. W. Goodman, “Univalent Functions, Vols. I and II,” Polygonal Pub-lishing House, Washington, 1983.

[3] S. Owa, “The Quasi-Hadamard Products of Certain Analytic Functions,” In: H. M. Srivastava and S. Owa, Eds., Current Topics in Analytic Function Theory, World Scientific Publishing Company, Singapore, 1992, pp. 234- 251.

[4] Z.-G. Wang, R. Aghalary, M. Darus and R. W. Ibrahim, “Some Properties of Certain Multivalent Analytic Functions Involving the Cho-Kwon-Srivastava Operator,” Journal of Mathematical and Computer Modelling, Vol. 49, No. 9-10, 2009, pp. 1969-1984.

[5] N. E. 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 Operators,” Journal of Mathematical Analysis and Applications, Vol. 292, No. 2, 2004, pp. 470-483. doi:10.1016/j.jmaa.2003.12.026

[6] R. Yamakawa, “Certain Subclasses of p-Valently Starlike Functions with Negative Coefficients,” In: H. M. Srivas- tava and S. Owa, Eds., Current Topics in Analytic Function Theory, World Scientific Publishing Company, Singapore, 1992, pp. 393-402.

[7] 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

[8] A. E. Li-vingston, “On the Radius of Univalence of Certain Ana-lytic Functions,” Proceedings of the American Mathe-matical Society, Vol. 17, No. 2, 1966, pp. 352-357. doi:10.1090/S0002-9939-1966-0188423-X

[9] H. M. Srivastava and S. Owa (Eds.), “Current Topics in Analytic Function Theory,” World Scientific Publishing Company, Singapore, 1992.

[10] S. Owa, “On Distortion Theorems. I,” Kyungpook Mathematical Journal, Vol. 18, 1978, pp. 55-59.

[11] H. M. Srivastava and M. K. Aouf, “A Certain Fractional Derivative Operator and Its Applications to a New Class of Analytic and Multivalent Functions with Negative Coefficients. I and II,” Journal of Mathematical Analysis and Applications, Vol. 171, No. 1, 1992, pp. 1-13. doi:10.1006/jmaa.1995.1197

[12] A. Schild and H. Sil-verman, “Convolutions of Univalent Functions with Neg-ative Coefficients,” Annales Universitatis Mariae Cu-rie-Sklodowska Section A, Vol. 29, 1975, pp. 99-107.

[13] O. Altintas, H. Irmak and H. M. Srivastava, “Fractional Calculus and Certain Starlike Functions with Negative Coefficients,” Computers and Mathematics with Applications, Vol. 30, No. 2, 1995, pp. 9-15. doi:10.1016/0898-1221(95)00073-8

[14] M.-P. Chen, H. Irmak and H. M. Srivastava, “Some Families of Multiva-lently Analytic Functions with Negative Coefficients,” Journal of Mathematical Analysis and Applications, Vol. 214, No. 2, 1997, pp. 674-690. doi:10.1006/jmaa.1997.5615

[15] H. M. Srivastava and S. Owa (Eds.), “Univalent Functions, Fractional Calculus, and Their Applications,” Ellis Horwood Limited, Chiche-ster, 1989.