Weighted analogues of Bernstein-type inequalities on several intervals
© Akturk and Lukashov; licensee Springer. 2013
Received: 7 February 2013
Accepted: 11 October 2013
Published: 7 November 2013
We give weighted analogues of Bernstein-type inequalities for trigonometric polynomials and rational functions on several intervals.
MSC:41A17, 42A05, 41A20.
Keywordsweighted polynomial inequalities inequalities for derivatives of rational functions
Inequalities for polynomials have been a classical object of studies for more than one century. Modern expositions can be found in books and surveys [1–3] and . Recently weighted analogues of classical polynomial inequalities were considered (see, for instance, [5, 6] and ). Other ways of generalizations are in replacing the domain of polynomials by more complicated (disconnected) sets and (or) in considering polynomials in more general Chebyshev systems. The main goal of the paper is to give simple proofs of weighted analogues of Bernstein-type inequalities on several intervals. They are inspired by weighted Bernstein-type inequalities from Section 5.2, E.4 in . It turns out that for disconnected sets similar ideas allow to write down weighted versions with an explicit constant.
for the set of trigonometric polynomials with real (complex) coefficients; as a weight w, we consider an arbitrary continuous positive function on a suitable set, is the uniform norm on this set.
The first theorem is a weighted analogue of the Bernstein-type inequality on several intervals.
for every polynomial , .
Next result is a weighted version of the Bernstein-type inequality for trigonometric polynomials on several intervals.
for every polynomial , . Inequality (6) is sharp in the sense that it is not possible to replace n in (6) by for arbitrary .
be the circular arc on the unit circle of central angle 2α and with a midpoint at 1. Next result is a weighted version of the inequality from .
for every polynomial , .
where is the Green function of the domain and n is the exterior normal at ζ (see, for example, ).
Our last result is an extension of Rusak’s inequality [, p.57] to the case of several intervals.
where , .
Remark 1 A Markov-type inequality, which is obtained by a similar method, was announced in the conference .
In the following we use several auxiliary results.
Lemma 1 
where is given by (13).
For further reference, it is convenient to give a particular case of a version of Lemma 1 from .
- 1.The trigonometric polynomial deviates least from zero on , with respect to the sup-norm among all trigonometric polynomials of degree with leading coefficients cosψ and sinψ, i.e.,(21)
has the maximal possible number of extremum points on ℰ.
- 2.For every , the equilibrium measures of the arcs are positive rational numbers. More precisely,(22)
- 3.There is a real trigonometric polynomial of order such that for a constant ,(23)
where is given by (5).
the numbers are equal to the number of zeros of on , ;
- (b)the polynomial may also be written in terms of as(24)
Lemma 4 
Proof We want to present here a different proof of the lemma which uses the representations of extremal polynomials in (24).
Now equality (26) follows from the representation (24). Uniqueness of ’s follows from the uniqueness of extremal trigonometric polynomials in Lemma 3.
The case of is proved then similarly to the proof of [, Corollary 5.1.5]. The theorem is sharp even for the case . Namely, we cannot replace the multiplier n by with any in the right-hand side of (6).
Equality (44) is equivalent to . □
The first author would like to thank the Scientific and Technological Research Council of Turkey (TUBİTAK) for the financial support. The authors are deeply grateful to reviewers for their careful reading of the manuscript and remarks which helped to improve the presentation.
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