Turan type inequalities for rational functions with prescribed poles

Document Type : Research Paper

Authors

1 Naseem Bagh Srinagar

2 Department of Mathematics, University of Kashmir , Srinagar, Jammu and Kashmir, India

3 University of Kashmir

Abstract

In this paper, we establish some inequalities for rational functions with prescribed poles having t-fold zeros at the origin. The estimates obtained generalise as well as refine some known results for rational functions and in turn, produce extensions of some polynomial inequalities earlier proved by Turan, Jain etc.

Keywords

[1] A. Aziz and Q.M. Dawood, Inequalities for a polynomial and its derivative, J. Approx. Theory 54 (1998) 306–313.
[2] A. Aziz and W.M. Shah, Some refinements of Bernstein type inequalities for rational functions, Glas. Mat. Ser. iii 32 (1997) 29–37.
[3] A. Aziz and W.M. Shah, Some properties of rational functions with prescribed poles and restricted zeros, Math. Balkanica (N.S) 18 (2004) 33–40.
[4] A. Aziz and B.A. Zargar, Some properties of rational functions with prescribed poles, Canad. Math. Bull. 42 (1999) 417–426.
[5] S. N. Bernstein, Sur l’ordre de la meilleure approximation des functions continues par des polynˆomes de degr´e donn´e, Mem. Acad. R. Belg. 4 (1912) 1–103.
[6] V. K. Jain, Inequalities for a polynomial and its derivative, Proc. Indian Acad. Sci. (Math. Sci.) 110 (2000) 137–146.
[7] Xin Li, A comparison inequality for rational functions, Proc. Amer. Math. Soc. 139 (2011) 1659–1665.
[8] Xin Li, R. N. Mohapatra and R. S.Rodriguez, Bernstein type inequalities for rational functions with prescribed poles, J. Lond. Math. Soc. 51 (1995) 523–531.
[9] G.V. Milovanovi´c and A. Mir, Comparison inequalities between rational functions with prescribed poles, Rev. R. Acad. Cienc. Exactas. Fis. Nat. Ser. A. Mat. 115 (2021) 1–13.
[10] P. Turan, Uber die Ableitung von polynomen ¨ , Compositio Math. 7 (1939) 89–95 .
Volume 13, Issue 1
March 2022
Pages 1003-1009
  • Receive Date: 15 April 2021
  • Revise Date: 15 August 2021
  • Accept Date: 21 August 2021