On Bernstein-Type Inequalities for Rational Functions with Prescribed Poles

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Authors: A. MIR

DOI: 10.46793/KgJMat2104.615M


In this paper, we shall use a parameter β and obtain some Bernstein-type inequalities for rational functions with prescribed poles which generalize the results of Qasim and Liman and Li, Mohapatra and Rodriguez and others.


Rational function, polynomial, poles, zeros.


[1]   A. Aziz and Q. M. Dawood, Inequalities for a polynomial and its derivative, J. Approx. Theory 54 (1988), 306–313.

[2]   S. Bernstein, Sur l’ordre de la meilleure approximation des fonctions continues par des polynômes de degré donné, Mem. Cl. Sci. Acad. Roy. Belg. 4 (1912), 1–103.

[3]   P. Borwein and T. Erdélyi, Polynomials and Polynomial Inequalities, Springer-Verlag, New York, 1995.

[4]   M. I. Ganzburg, Sharp constants in V. A. Markov-Bernstein type inequalities of different metrics, J. Approx. Theory 215 (2017), 92–105.

[5]   M. I. Ganzburg and S. Y. Tikhonov, On sharp constants in Bernstein-Nikolskii inequalities, Constr. Approx. 45 (2017), 449–466.

[6]   I. Qasim and A. Liman, Bernstein type inequalities for rational functions, Indian J. Pure Appl. Math. 46 (2015), 337–348.

[7]   S. Kalmykov and B. Nagy, Higher Markov and Bernstein inequalities and fast decreasing polynomials with prescribed zeros, J. Approx. Theory 226 (2018), 34–59.

[8]   S. Kalmykov, B. Nagy and V. Totik, Bernstein- and Markov-type inequalities for rational functions, Acta Math. 219 (2017), 21–63.

[9]   P. D. Lax, Proof of a conjecture of P. Erdös on the derivative of a polynomial, Bull. Amer. Math. Soc. 50 (1944), 509–513.

[10]   M. A. Malik, On the derivative of a polynomial, J. Lond. Math. Soc. 1 (1969), 57–60.

[11]   G. V. Milovanović, D. S. Mitrinović and Th. M. Rassias, Topics in Polynomials: Extremal Problems, Inequalities, Zeros, World Scientific, Singapore, 1994.

[12]   P. Turán, Über die Ableitung von Polynomen, Compos. Math. 7 (1939), 89–95.

[13]   Xin Li, A comparison inequality for rational functions, Proc. Amer. Math. Soc. 139 (2011), 1659–1665.

[14]   X. 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.