Mapping Properties of Generalized Robertson Functions under Certain Integral Operators

ABSTRACT

In the present article, certain classes of generalized p-valent Robertson functions are considered. Mapping properties of these classes are investigated under certain p-valent integral operators introduced by Frasin recently.

In the present article, certain classes of generalized p-valent Robertson functions are considered. Mapping properties of these classes are investigated under certain p-valent integral operators introduced by Frasin recently.

Cite this paper

M. Arif, W. Ul-Haq and M. Ismail, "Mapping Properties of Generalized Robertson Functions under Certain Integral Operators,"*Applied Mathematics*, Vol. 3 No. 1, 2012, pp. 52-55. doi: 10.4236/am.2012.31009.

M. Arif, W. Ul-Haq and M. Ismail, "Mapping Properties of Generalized Robertson Functions under Certain Integral Operators,"

References

[1] L. Spacek, “Prispěvek k Teorii Funkei Prostych,” ?asopis pro pěstováni matematiky a fysiky, Vol. 62, No. 2, 1933, pp. 12-19.

[2] M. S. Robertson, “Univalent Functions f(z) for wich zf'(z) Is Spiral-Like,” Michigan Mathematical Journal, Vol. 16, No. 2, 1969, pp. 97-101.

[3] K. S. Padmanabhan and R. Parvatham, “Properties of a Class of Functions with Bounded Boundary Rotation,” Annales Polonici Mathematici, Vol. 31, No. 1, 1975, pp. 311-323.

[4] B. Pinchuk, “Functions with Bounded Boundary Rotation,” Israel Journal of Mathematics, Vol. 10, No. 1, 1971, pp. 7-16. doi:10.1007/BF02771515

[5] O. Tammi, “On the Maximization of the Coefficients of Schlicht and Related Functions,” Annales Academiae Scientiarum Fennicae. Series A I. Mathematica, Vol. 114, No. 1, 1952, 51 pages.

[6] V. Paatero, “Uber Gebiete von Beschrankter Randdrehung,” Annales Academiae Scientiarum Fennnicae, Vol. 37-39, No. 9, 1933.

[7] M. Arif, M. Ayaz and S. I. Ali Shah, “Radii Problems for Certain Classes of Analytic Functions with Fixed Second Coefficients,” World Applied Sciences Journal, Vol. 13, No. 10, 2011, pp. 2240-2243.

[8] K. I. Noor, “On Some Subclasses of Fuctions with Bounded Boundary and Bounded Radius Rotation,” Pan American Mathematical Journal, Vol. 6, No. 1, 1996, pp. 75-81.

[9] K. I. Noor, M. Arif and W. Haq, “Some Properties of Certain Integral Opertors,” Acta Universitatis Apulensis, Vol. 21, 2010, pp. 89-95.

[10] K. I. Noor, M. Arif and A. Muhammad, “Mapping Properties of Some Classes of Analytic Functions under an Integral Operator,” Journal of Mathematical Inequalities, Vol. 4, No. 4, 2010, pp.593-600.

[11] K. I. Noor, W. Haq, M. Arif and S. Mustafa, “On Bounded Boundary and Bounded Radius Rotations,” Journal of Inequalities and Applications, 2009, Article ID: 813687.

[12] K. I. Noor, S. N. Malik, M. Arif and M. Raza, “On Bounded Boundary and Bounded Radius Rotation Related with Janowski Function,” World Applied Sciences Journal, Vol. 12, No. 6, 2011, pp. 895-902.

[13] B. A. Frasin, “New General Integral Operators of p-Valent Functions,” Journal of Inequatilies Pure and Applied Mathematics, Vol. 10, No. 4, 2009

[14] D. Breaz and N. Breaz, “Two Integral Operators,” Studia Universitatis Babes-Bolyai, Mathematica, Clunj-Napoca, Vol. 47, No. 3, 2002, pp. 13-21.

[15] D. Breaz, S. Owa and N. Breaz, “A New Integral Univalent Operator,” Acta Universitatis Apulensis, Vol. 16, 2008, pp. 11-16.

[16] N. Breaz, V. Pescar and D. Breaz, “Univalence Criteria for a New Integral Operator,” Mathematical and Computer Modelling, Vol. 52, No. 1-2, 2010, pp. 241-246. doi:10.1016/j.mcm.2010.02.013

[17] B. A. Frasin, “Convexity of Integral Operators of p-Valent Functions,” Mathematical and Computer Modelling, Vol. 51, No. 5-6, 2010, pp. 601-605.

[18] B. A. Frasin, “Some Sufficient Conditions for Certain Integral Operators,” Journal of Mathematics and Inequalities, Vol. 2. No. 4, 2008, pp. 527-535.

[19] G. Saltik, E. Deniz and E. Kadioglu, “Two New General p-Valent Integral Operators,” Mathematical and Computer Modelling, Vol. 52, No. 9-10, 2010, pp. 1605-1609. doi:10.1016/j.mcm.2010.06.025

[20] R. M. Ali and V. Ravichandran, “Integral Operators on Ma-Minda Type Starlike and Convex Functions,” Mathematical and Computer Modelling, Vo. 53, No. 5-6, 2011, pp. 581-586. doi:10.1016/j.mcm.2010.09.007

[21] S. S. Miller, P. T. Mocanu and M. O. Reade, “Starlike Integral Operators,” Pacific Journal of Mathematics, Vol. 79, No. 1, 1978, pp.157-168.

[22] Y. J. Kim and E. P. Merkes, “On an Integral of Powers of a Spirallike Function,” Kyungpook Mathematical Journal, Vol. 12, No. 2, 1972, pp. 249-252.

[23] N. N. Pascu and V. Pescar, “On the Integral Operators of Kim-Merkes and Pfaltzgraff,” Mathematica, Universitatis Babes-Bolyai Cluj-Napoca, Vol. 32, No. 2, 1990, pp. 185-192.

[1] L. Spacek, “Prispěvek k Teorii Funkei Prostych,” ?asopis pro pěstováni matematiky a fysiky, Vol. 62, No. 2, 1933, pp. 12-19.

[2] M. S. Robertson, “Univalent Functions f(z) for wich zf'(z) Is Spiral-Like,” Michigan Mathematical Journal, Vol. 16, No. 2, 1969, pp. 97-101.

[3] K. S. Padmanabhan and R. Parvatham, “Properties of a Class of Functions with Bounded Boundary Rotation,” Annales Polonici Mathematici, Vol. 31, No. 1, 1975, pp. 311-323.

[4] B. Pinchuk, “Functions with Bounded Boundary Rotation,” Israel Journal of Mathematics, Vol. 10, No. 1, 1971, pp. 7-16. doi:10.1007/BF02771515

[5] O. Tammi, “On the Maximization of the Coefficients of Schlicht and Related Functions,” Annales Academiae Scientiarum Fennicae. Series A I. Mathematica, Vol. 114, No. 1, 1952, 51 pages.

[6] V. Paatero, “Uber Gebiete von Beschrankter Randdrehung,” Annales Academiae Scientiarum Fennnicae, Vol. 37-39, No. 9, 1933.

[7] M. Arif, M. Ayaz and S. I. Ali Shah, “Radii Problems for Certain Classes of Analytic Functions with Fixed Second Coefficients,” World Applied Sciences Journal, Vol. 13, No. 10, 2011, pp. 2240-2243.

[8] K. I. Noor, “On Some Subclasses of Fuctions with Bounded Boundary and Bounded Radius Rotation,” Pan American Mathematical Journal, Vol. 6, No. 1, 1996, pp. 75-81.

[9] K. I. Noor, M. Arif and W. Haq, “Some Properties of Certain Integral Opertors,” Acta Universitatis Apulensis, Vol. 21, 2010, pp. 89-95.

[10] K. I. Noor, M. Arif and A. Muhammad, “Mapping Properties of Some Classes of Analytic Functions under an Integral Operator,” Journal of Mathematical Inequalities, Vol. 4, No. 4, 2010, pp.593-600.

[11] K. I. Noor, W. Haq, M. Arif and S. Mustafa, “On Bounded Boundary and Bounded Radius Rotations,” Journal of Inequalities and Applications, 2009, Article ID: 813687.

[12] K. I. Noor, S. N. Malik, M. Arif and M. Raza, “On Bounded Boundary and Bounded Radius Rotation Related with Janowski Function,” World Applied Sciences Journal, Vol. 12, No. 6, 2011, pp. 895-902.

[13] B. A. Frasin, “New General Integral Operators of p-Valent Functions,” Journal of Inequatilies Pure and Applied Mathematics, Vol. 10, No. 4, 2009

[14] D. Breaz and N. Breaz, “Two Integral Operators,” Studia Universitatis Babes-Bolyai, Mathematica, Clunj-Napoca, Vol. 47, No. 3, 2002, pp. 13-21.

[15] D. Breaz, S. Owa and N. Breaz, “A New Integral Univalent Operator,” Acta Universitatis Apulensis, Vol. 16, 2008, pp. 11-16.

[16] N. Breaz, V. Pescar and D. Breaz, “Univalence Criteria for a New Integral Operator,” Mathematical and Computer Modelling, Vol. 52, No. 1-2, 2010, pp. 241-246. doi:10.1016/j.mcm.2010.02.013

[17] B. A. Frasin, “Convexity of Integral Operators of p-Valent Functions,” Mathematical and Computer Modelling, Vol. 51, No. 5-6, 2010, pp. 601-605.

[18] B. A. Frasin, “Some Sufficient Conditions for Certain Integral Operators,” Journal of Mathematics and Inequalities, Vol. 2. No. 4, 2008, pp. 527-535.

[19] G. Saltik, E. Deniz and E. Kadioglu, “Two New General p-Valent Integral Operators,” Mathematical and Computer Modelling, Vol. 52, No. 9-10, 2010, pp. 1605-1609. doi:10.1016/j.mcm.2010.06.025

[20] R. M. Ali and V. Ravichandran, “Integral Operators on Ma-Minda Type Starlike and Convex Functions,” Mathematical and Computer Modelling, Vo. 53, No. 5-6, 2011, pp. 581-586. doi:10.1016/j.mcm.2010.09.007

[21] S. S. Miller, P. T. Mocanu and M. O. Reade, “Starlike Integral Operators,” Pacific Journal of Mathematics, Vol. 79, No. 1, 1978, pp.157-168.

[22] Y. J. Kim and E. P. Merkes, “On an Integral of Powers of a Spirallike Function,” Kyungpook Mathematical Journal, Vol. 12, No. 2, 1972, pp. 249-252.

[23] N. N. Pascu and V. Pescar, “On the Integral Operators of Kim-Merkes and Pfaltzgraff,” Mathematica, Universitatis Babes-Bolyai Cluj-Napoca, Vol. 32, No. 2, 1990, pp. 185-192.