Inequalities for Rational Functions with $s$-fold zeros at the Origin

Authors

  • B A Zargar
  • M H Gulzar
  • Rubia Akhter

DOI:

https://doi.org/10.5644/SJM.20.02.05

Keywords:

Rational Functions, s-fold zeros, Polynomial inequalities, Poles

Abstract

In this paper, we establish some inequalities for rational functions with prescribed poles having $s$ fold zeros at the origin. Our results generalize and refine the results of A. Aziz and W. M. Shah [2], A. Aziz and B. A. Zargar [3] and other known rational inequalities.

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References

[1] Abdul Aziz and W. M. Shah, Some refinements of Bernstein type inequalities for rational functions. Glas. Mat. (52) 32(1997), 29-37.

[2] A. Aziz and W. M. Shah, Some properties of rational functions with prescribed poles and restricted zeros, Mathematica Balkanica 18 (2004), 33-40.

[3] Abdul Aziz and B. A. Zargar, Some properties of rational functions with prescribed poles, Canad. Math. Bull. 42(4),1999, 417-426.

[4] S. N. Bernstein, Sur e’ordre de la meilleure approximation des functions continues par des polynomes de degre’ donne’. Mem. Acad. R. Belg. 4, 1-103 (1912).

[5] M H Gulzar, B A Zargar and Rubia Akhter, Some inequalities for the rational functions with prescribed poles and restricted zeros, Electronical Journal of Mathematical Analysis and Application, 10, 275-282, (2022).

[6] M H Gulzar, B A Zargar and Rubia Akhter, Some inequalities for the rational functions with prescribed poles and restricted zeros, J. Anal 30, 35-41 (2022).

[7] P. D. Lax, Proof of conjecture of P. Erdos on the derivative of a polynomials, Bull. Amer .Math. ¨ Soc, 50(1994), 509-511. 240 B. A. ZARGAR, M. H. GULZAR, AND RUBIA AKHTER

[8] Q.I.Rahman and G.Schmeisser, Analytic theory of polynomials, 2002, Oxford Science Publications.

[9] A. L. Schaffer, Inequalities of A. Markoff and S. Bernstein for polynomials and rational functions, Bull. Amer. Math. Soc. 47(1941), 565-579.

[10] Xin Li, R.N.Mohapatra and R.S.Rodriguez, Bernstein-type inequalities for rational functions with prescribed poles, J. London Math. Soc. (51) 20(1995), 523-531.

[11] B A Zargar, M H Gulzar and Rubia Akhter, Inequalities for the rational functions with prescribed poles and restricted zeros, Adv. Inequal. Appl. 2021, Article ID 1, 2021.

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Published

10.03.2025

How to Cite

Zargar, B. A., Gulzar, M. H., & Akhter, R. (2025). Inequalities for Rational Functions with $s$-fold zeros at the Origin. Sarajevo Journal of Mathematics, 20(2), 231–240. https://doi.org/10.5644/SJM.20.02.05

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