Simple proofs of Bernstein-type inequalities (original) (raw)

Bernstein-Type Integral Inequalities for a Certain Class of Polynomials-II

Mediterranean Journal of Mathematics, 2020

In this paper, we establish some Bernstein-type integral inequalities for a certain class of polynomials involving the polar derivative. Our results generalize some known polynomial inequalities and include as special cases several interesting generalizations and refinements of some L γ inequalities for polynomials as well.

On an inequality of S. Bernstein

AIP Conference Proceedings, 2022

For a polynomial w(ζ) of degree m having all its zeros on |ζ | = η, η ≤ 1, Govil proved that max |ζ |=1 |w (ζ)| ≤ m η m + η m−1 max |ζ |=1 |w(ζ)|. Under the same hypotheses, Dewan and Mir improved the above inequality and proved max |ζ |=1 |w (ζ)| ≤ m η m m|a m |η 2 + |a m−1 | m|a m | 1 + η 2 + 2|a m−1 | max |ζ |=1 |w(ζ)|. We extend both the above inequalities to their respective polar derivative versions.

Bernstein and Markov type inequalities for generalized nonnegative polynomials

Canadian Journal of Mathematics, 1991

Generalized polynomials are defined as products of polynomials raised to positive real powers. The generalized degree can be introduced in a natural way. Several inequalities holding for ordinary polynomials are expected to be true for generalized polynomials, by utilizing the generalized degree in place of the ordinary one. Based on Remez-type inequalities on the size of generalized polynomials, we establish Bernstein and Markov type inequalities for generalized non-negative polynomials, obtaining the best possible result up to a multiplicative absolute constant.

Bernstein-Type Integral Inequalities for a Certain Class of Polynomials

Mediterranean Journal of Mathematics, 2019

Bernstein inequalities for polynomials with constrained roots

We prove Bernstein type inequalities for algebraic polynomials on the finite interval I := [−1, 1] and for trigonometric polynomials on R when the roots of the polynomials are outside of a certain domain of the complex plane. The case of real vs. complex coefficients are handled separately. In case of trigonometric polynomials with real coefficients and root restriction, the L psituation will also be considered. In most cases, the sharpness of the estimates will be shown.

Bernstein type inequalities for rational functions

Indian Journal of Pure and Applied Mathematics, 2015

In this paper, we consider a more general class of rational functions r(s(z)) of degree mn, where s(z) is a polynomial of degree m and prove some sharp results concerning to Bernstein type inequalities for rational functions.

On Bernstein's inequality

Journal of Mathematical Analysis and Applications, 1987

Our result includes as a special case ErdGs conjecture first proved by Lax. Also it sharpens and includes as a special case the inequality due to Malik that if p,(z) # 0 for IzI <K, K> 1, then lipLl/ <(n/(1 + K)) lip,ll, fi' 1987 Academic press. IIK

Simple proofs of Bernstein-type inequalities (original) (raw)

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