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Weighted analogues of Bernstein-type inequalities on several intervals
Journal of Inequalities and Applications volume 2013, Article number: 487 (2013)
We give weighted analogues of Bernstein-type inequalities for trigonometric polynomials and rational functions on several intervals.
MSC:41A17, 42A05, 41A20.
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.
Throughout the paper, we use the notations
for the set of algebraic polynomials and
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.
Theorem 1 Let E be a set consisting of a finite number of disjoint intervals, , , then there exists depending on w and E such that
for every polynomial , .
Next result is a weighted version of the Bernstein-type inequality for trigonometric polynomials on several intervals.
Theorem 2 Let w be any function which is continuous and positive on
Then there exists depending on w and ℰ such that
for every polynomial , . Inequality (6) is sharp in the sense that it is not possible to replace n in (6) by for arbitrary .
Now let , and let
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 .
Theorem 3 With the above notations and for any continuous positive function w, there exists depending on w and α such that
for every polynomial , .
Next we recall the definition of the harmonic measure of a set at a point relative to the domain ,
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.
Theorem 4 Let be a complex-valued algebraic fraction
where , is a real polynomial of degree ν which is positive on , satisfying the condition
and is a differentiable function on E. Then the estimate
is valid. Here
. If x is not a multiple root of , then the equality sign is valid only for algebraic fractions
at the points x satisfying
in the case when
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 
Consider any algebraic fraction
where , and is a real polynomial of degree ν which is positive on , . Then
where is given by (13).
For further reference, it is convenient to give a particular case of a version of Lemma 1 from .
Lemma 2 The following inequality holds for any trigonometric polynomial and , ℰ is a real compact subset of :
The following assertions are equivalent.
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.
For every , the equilibrium measures of the arcs are positive rational numbers. More precisely,(22)
There is a real trigonometric polynomial of order such that for a constant ,(23)
where is given by (5).
If any of those assertions is valid, then
the numbers are equal to the number of zeros of on , ;
the polynomial may also be written in terms of as(24)
Lemma 4 
The density of the equilibrium measure from (20), , is given by
where , and , , , are uniquely determined by
Proof We want to present here a different proof of the lemma which uses the representations of extremal polynomials in (24).
(1) Suppose firstly , , . Then by Lemma 3 the function
is a real trigonometric polynomial of order N. If we take a derivative, we get
where , , are zeros of and there is a real trigonometric polynomial of order such that
Moreover, has a maximal number of deviation points, and inner zeros of its derivative coincide with zeros of , and has one zero at each gap , . Hence
so we have
Now equality (26) follows from the representation (24). Uniqueness of ’s follows from the uniqueness of extremal trigonometric polynomials in Lemma 3.
Proof of Theorem 2 First consider . By the Weierstrass approximation theorem, for any , there is such that
where are given by (26) in Lemma 4. Hence
and, using Lemmas 2 and 4, we have
where . Now, for every and , provided is sufficiently small, such that , we get
and because of , we obtain, for sufficiently small ,
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).
Take , . Then we have , and
Take , then
for sufficiently large n such that
Proof of Theorem 4 Firstly we consider the case when the numerator has real coefficients. Put ; using Lemma 1, we obtain
The validity of the estimate for complex-valued algebraic fractions is proved by the same trick as in [, Corollary 5.1.5]. Equality sign in the last inequality in (43) is valid only for the function , if (16) holds [13, 20]. Equality sign in the second inequality in (43) then holds only for the same function at those points where
Equality (44) is equivalent to . □
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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.
The authors declare that they have no competing interests.
All authors jointly worked on the results and they read and approved the final manuscript.
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Akturk, M.A., Lukashov, A. Weighted analogues of Bernstein-type inequalities on several intervals. J Inequal Appl 2013, 487 (2013). https://doi.org/10.1186/1029-242X-2013-487
- weighted polynomial inequalities
- inequalities for derivatives of rational functions