Polar Derivative Versions of Polynomial Inequalities
Author(s)
Barchand Chanam*
Affiliation(s)
Department of Basic Sciences and Humanities, National Institute of Technology, Manipur, India.
Department of Basic Sciences and Humanities, National Institute of Technology, Manipur, India.
ABSTRACT
Let
be a polynomial of degree n and for a complex number , let 
denote the polar derivative of the polynomial 
with respect to . In this paper, first we extend as well as generalize the result proved by Dewan and Mir [Inter. Jour. Math. and Math. Sci., 16 (2005), 2641-2645] to polar derivative. Besides, another result due to Dewan et al. [J. Math. Anal. Appl. 269 (2002), 489-499] is also extended to polar derivative.
Let
be a polynomial of degree n and for a complex number , let denote the polar derivative of the polynomial with respect to . In this paper, first we extend as well as generalize the result proved by Dewan and Mir [Inter. Jour. Math. and Math. Sci., 16 (2005), 2641-2645] to polar derivative. Besides, another result due to Dewan et al. [J. Math. Anal. Appl. 269 (2002), 489-499] is also extended to polar derivative.
Cite this paper
Chanam, B. (2015) Polar Derivative Versions of Polynomial Inequalities. Advances in Pure Mathematics, 5, 745-755. doi: 10.4236/apm.2015.512068.
Chanam, B. (2015) Polar Derivative Versions of Polynomial Inequalities. Advances in Pure Mathematics, 5, 745-755. doi: 10.4236/apm.2015.512068.
References
[1] Bernstein, S. (1926) Lecons Sur Les Propriétés extrémales et la meilleure approximation desfonctions analytiques d’une variable réele, Paris. [2] Lax, P.D. (1944) Proof of a Conjecture of P. Erdös on the Derivative of a Polynomial. Bulletin of the American Mathematical Society, 50, 509-513.
http://dx.doi.org/10.1090/S0002-9904-1944-08177-9 [3] Malik, M.A. (1969) On the Derivative of a Polynomial. Journal of the London Mathematical Society, 1, 57-60.
http://dx.doi.org/10.1112/jlms/s2-1.1.57 [4] Bidkham, M. and Dewan, K.K. (1992) Inequalities for a Polynomial and Its Derivative. Journal of Mathematical Analysis and Applications, 166, 319-324.
http://dx.doi.org/10.1016/0022-247X(92)90298-R [5] Dewan, K.K. and Mir, A. (2005) On the Maximum Modulus of a Polynomial and Its Derivatives. International Journal of Mathematics and Mathematical Sciences, 16, 2641-2645.
http://dx.doi.org/10.1155/IJMMS.2005.2641 [6] Aziz (1983) Inequalities for the Derivatives of a Polynomial. Proceedings of the American Mathematical Society, 89, 259-266.
http://dx.doi.org/10.1090/S0002-9939-1983-0712634-5 [7] Turán, P. (1939) über die ableitung von polynomen. Compositio Mathematica, 7, 89-95. [8] Govil, N.K. (1973) On the Derivative of Polynomial. Proceedings of the American Mathematical Society, 41, 543-546.
http://dx.doi.org/10.1090/S0002-9939-1973-0325932-8 [9] Dewan, K.K., Kaur, J. and Mir, A. (2002) Inequalities for the Derivative of a Polynomial. Journal of Mathematical Analysis and Applications, 269, 489-499.
http://dx.doi.org/10.1016/S0022-247X(02)00030-6 [10] Aziz, A. and Rather, N.A. (1998) A Refinement of a Theorem of Paul Turán Concerning Polynomials. Mathematical Inequalities & Applications, 1, 231-238.
http://dx.doi.org/10.7153/mia-01-21 [11] Rather, N.A. (1998) Extremal Properties and Location of the Zeros of Polynomials. Ph.D. Thesis, University of Kashmir, Srinagar. [12] Govil, N.K., Rahman, Q.I. and Schmeisser, G. (1979) On the Derivative of a Polynomial. Illinois Journal of Mathematics, 23, 319-329. [13] Jain, V.K. (1994) Converse of an Extremal Problem in Polynomials II. Jourmal of the Indian Mathematical Society, 60, 41-47. [14] Qazi, M.A. (1992) On the Maximum Modulus of Polynomials. Proceedings of the American Mathematical Society, 115, 337-343.
http://dx.doi.org/10.1090/S0002-9939-1992-1113648-1 [15] Dewan, K.K. and Kaur, J. (1994) On the Maximum Modulus of Polynomials. Journal of Mathematical Analysis and Applications, 181, 493-497.
http://dx.doi.org/10.1006/jmaa.1994.1038
[1] Bernstein, S. (1926) Lecons Sur Les Propriétés extrémales et la meilleure approximation desfonctions analytiques d’une variable réele, Paris. [2] Lax, P.D. (1944) Proof of a Conjecture of P. Erdös on the Derivative of a Polynomial. Bulletin of the American Mathematical Society, 50, 509-513.
http://dx.doi.org/10.1090/S0002-9904-1944-08177-9 [3] Malik, M.A. (1969) On the Derivative of a Polynomial. Journal of the London Mathematical Society, 1, 57-60.
http://dx.doi.org/10.1112/jlms/s2-1.1.57 [4] Bidkham, M. and Dewan, K.K. (1992) Inequalities for a Polynomial and Its Derivative. Journal of Mathematical Analysis and Applications, 166, 319-324.
http://dx.doi.org/10.1016/0022-247X(92)90298-R [5] Dewan, K.K. and Mir, A. (2005) On the Maximum Modulus of a Polynomial and Its Derivatives. International Journal of Mathematics and Mathematical Sciences, 16, 2641-2645.
http://dx.doi.org/10.1155/IJMMS.2005.2641 [6] Aziz (1983) Inequalities for the Derivatives of a Polynomial. Proceedings of the American Mathematical Society, 89, 259-266.
http://dx.doi.org/10.1090/S0002-9939-1983-0712634-5 [7] Turán, P. (1939) über die ableitung von polynomen. Compositio Mathematica, 7, 89-95. [8] Govil, N.K. (1973) On the Derivative of Polynomial. Proceedings of the American Mathematical Society, 41, 543-546.
http://dx.doi.org/10.1090/S0002-9939-1973-0325932-8 [9] Dewan, K.K., Kaur, J. and Mir, A. (2002) Inequalities for the Derivative of a Polynomial. Journal of Mathematical Analysis and Applications, 269, 489-499.
http://dx.doi.org/10.1016/S0022-247X(02)00030-6 [10] Aziz, A. and Rather, N.A. (1998) A Refinement of a Theorem of Paul Turán Concerning Polynomials. Mathematical Inequalities & Applications, 1, 231-238.
http://dx.doi.org/10.7153/mia-01-21 [11] Rather, N.A. (1998) Extremal Properties and Location of the Zeros of Polynomials. Ph.D. Thesis, University of Kashmir, Srinagar. [12] Govil, N.K., Rahman, Q.I. and Schmeisser, G. (1979) On the Derivative of a Polynomial. Illinois Journal of Mathematics, 23, 319-329. [13] Jain, V.K. (1994) Converse of an Extremal Problem in Polynomials II. Jourmal of the Indian Mathematical Society, 60, 41-47. [14] Qazi, M.A. (1992) On the Maximum Modulus of Polynomials. Proceedings of the American Mathematical Society, 115, 337-343.
http://dx.doi.org/10.1090/S0002-9939-1992-1113648-1 [15] Dewan, K.K. and Kaur, J. (1994) On the Maximum Modulus of Polynomials. Journal of Mathematical Analysis and Applications, 181, 493-497.
http://dx.doi.org/10.1006/jmaa.1994.1038