Applications of Multivalent Functions Associated with Generalized Fractional Integral Operator

Affiliation(s)

Department of Mathematics Education, Daegu National University of Education, Daegu, South Korea.

Department of Mathematics Education, Daegu National University of Education, Daegu, South Korea.

ABSTRACT

By using a method based upon the Briot-Bouquet differential subordination, we investigate some subordination properties of the generalized fractional integral operator which was defined by Owa, Saigo and Srivastava [1]. Some interesting further consequences are also considered.

KEYWORDS

Multivalent Functions; Subordination; Gaussian Hypergeometric Function; Fractional Integral Operator

Multivalent Functions; Subordination; Gaussian Hypergeometric Function; Fractional Integral Operator

Cite this paper

J. Choi, "Applications of Multivalent Functions Associated with Generalized Fractional Integral Operator,"*Advances in Pure Mathematics*, Vol. 3 No. 1, 2013, pp. 1-5. doi: 10.4236/apm.2013.31001.

J. Choi, "Applications of Multivalent Functions Associated with Generalized Fractional Integral Operator,"

References

[1] S. Owa, M. Saigo and H. M. Srivastava, “Some Characterization Theorems for Starlike and Convex Functions Involving a Certain Fractional Integral Operator,” Journal of Mathematical Analysis and Applications, Vol. 140, No. 2, 1989, pp. 419-426. doi:10.1016/0022-247X(89)90075-9

[2] S. G. Samko, A. A. Kilbas and O. I. Marichev, “Fractional Integral and Derivatives, Theory and Applications,” Gordon and Breach, New York, Philadelphia, London, Paris, Montreux, Toronto, Melbourne, 1993.

[3] H. M. Srivastava and R. G. Buschman, “Theory and Applications of Convolution Integral Equations,” Kluwer Academic Publishers, Dordrecht, Boston, London, 1992.

[4] M. Saigo, “A Remark on Integral Operators Involving the Gauss Hypergeometric Functions,” Mathematical Reports, Kyushu University, Vol. 11, No. 2, 1977-1978, pp. 135-143.

[5] J. H. Choi, “Note on Differential Subordination Associated with Fractional Integral Operator,” Far East Journal of Mathematical Sciences, Vol. 26, No. 2, 2007, pp. 499- 511.

[6] I. B. Jung, Y. C. Kim and H. M. srivastava, “The Hardy Space of Analytic Functions Associated with Certain One-Parameter Families of Integral Operators,” Journal of Mathematical Analysis and Applications, Vol. 176, No. 1, 1993, pp. 138-147. doi:10.1006/jmaa.1993.1204

[7] J.-L. Liu, “Notes on Jung-Kim-Srivastava Integral Operator,” Journal of Mathematical Analysis and Applications, Vol. 294, No. 1, 2004, pp. 96-103. doi:10.1016/j.jmaa.2004.01.040

[8] R. M. EL-Ashwash and M. K. Aouf, “Some Subclasses of Multivalent Functions Involving the Extended Fractional Differintegral Operator,” Journal of Mathematical Inequalities, Vol. 4, No. 1, 2010, pp. 77-93.

[9] J. Patel, A. K. Mishra and H. M. Srivastava, “Classes of Multinalent Analytic Functions Involving the Dziok-Srivastava Operator,” Computers and Mathematics with Applications, Vol. 54, No. 5, 2007, pp. 599-616. doi:10.1016/j.camwa.2006.08.041

[10] S. S. Miller and P. T. Mocanu, “Differential Subordinations and Univalent Functions,” Michigan Mathematical Journal, Vol. 28, No. 2, 1981, pp. 157-172. doi:10.1307/mmj/1029002507

[11] 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

[12] R. J. Libera, “Some Classes of Regular Univalent Functions,” Proceedings of the American Mathematical Society, Vol. 16, No. 4, 1965, pp. 755-758. doi:10.1090/S0002-9939-1965-0178131-2

[13] H. M. Srivastava and S. Owa, Eds., “Current Topics in Analytic Function Theory,” World Scientific Publishing Company, Singapore, New Jersey, London, Hong Kong, 1992. doi:10.1142/1628

[1] S. Owa, M. Saigo and H. M. Srivastava, “Some Characterization Theorems for Starlike and Convex Functions Involving a Certain Fractional Integral Operator,” Journal of Mathematical Analysis and Applications, Vol. 140, No. 2, 1989, pp. 419-426. doi:10.1016/0022-247X(89)90075-9

[2] S. G. Samko, A. A. Kilbas and O. I. Marichev, “Fractional Integral and Derivatives, Theory and Applications,” Gordon and Breach, New York, Philadelphia, London, Paris, Montreux, Toronto, Melbourne, 1993.

[3] H. M. Srivastava and R. G. Buschman, “Theory and Applications of Convolution Integral Equations,” Kluwer Academic Publishers, Dordrecht, Boston, London, 1992.

[4] M. Saigo, “A Remark on Integral Operators Involving the Gauss Hypergeometric Functions,” Mathematical Reports, Kyushu University, Vol. 11, No. 2, 1977-1978, pp. 135-143.

[5] J. H. Choi, “Note on Differential Subordination Associated with Fractional Integral Operator,” Far East Journal of Mathematical Sciences, Vol. 26, No. 2, 2007, pp. 499- 511.

[6] I. B. Jung, Y. C. Kim and H. M. srivastava, “The Hardy Space of Analytic Functions Associated with Certain One-Parameter Families of Integral Operators,” Journal of Mathematical Analysis and Applications, Vol. 176, No. 1, 1993, pp. 138-147. doi:10.1006/jmaa.1993.1204

[7] J.-L. Liu, “Notes on Jung-Kim-Srivastava Integral Operator,” Journal of Mathematical Analysis and Applications, Vol. 294, No. 1, 2004, pp. 96-103. doi:10.1016/j.jmaa.2004.01.040

[8] R. M. EL-Ashwash and M. K. Aouf, “Some Subclasses of Multivalent Functions Involving the Extended Fractional Differintegral Operator,” Journal of Mathematical Inequalities, Vol. 4, No. 1, 2010, pp. 77-93.

[9] J. Patel, A. K. Mishra and H. M. Srivastava, “Classes of Multinalent Analytic Functions Involving the Dziok-Srivastava Operator,” Computers and Mathematics with Applications, Vol. 54, No. 5, 2007, pp. 599-616. doi:10.1016/j.camwa.2006.08.041

[10] S. S. Miller and P. T. Mocanu, “Differential Subordinations and Univalent Functions,” Michigan Mathematical Journal, Vol. 28, No. 2, 1981, pp. 157-172. doi:10.1307/mmj/1029002507

[11] 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

[12] R. J. Libera, “Some Classes of Regular Univalent Functions,” Proceedings of the American Mathematical Society, Vol. 16, No. 4, 1965, pp. 755-758. doi:10.1090/S0002-9939-1965-0178131-2

[13] H. M. Srivastava and S. Owa, Eds., “Current Topics in Analytic Function Theory,” World Scientific Publishing Company, Singapore, New Jersey, London, Hong Kong, 1992. doi:10.1142/1628