- Research Article
- Open access
- Published:
Markov Inequalities for Polynomials with Restricted Coefficients
Journal of Inequalities and Applications volume 2009, Article number: 808720 (2009)
Abstract
Essentially sharp Markov-type inequalities are known for various classes of polynomials with constraints including constraints of the coefficients of the polynomials. For and
we introduce the class
as the collection of all polynomials of the form
,
,
,
. In this paper, we prove essentially sharp Markov-type inequalities for polynomials from the classes
on
. Our main result shows that the Markov factor
valid for all polynomials of degree at most
on
improves to
for polynomials in the classes
on
.
1. Introduction
In this paper, always denotes a nonnegative integer;
and
always denote absolute positive constants. In this paper
will always denote a positive constant depending only on
the value of which may vary from place to place. We use the usual notation
to denote the Banach space of functions defined on
with the norms
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F808720/MediaObjects/13660_2008_Article_2010_Equ1_HTML.gif)
We introduce the following classes of polynomials. Let
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F808720/MediaObjects/13660_2008_Article_2010_Equ2_HTML.gif)
denote the set of all algebraic polynomials of degree at most with real coefficients. Let
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F808720/MediaObjects/13660_2008_Article_2010_Equ3_HTML.gif)
denote the set of all algebraic polynomials of degree at most with complex coefficients. For
we introduce the class
as the collection of all polynomials of the form
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F808720/MediaObjects/13660_2008_Article_2010_Equ4_HTML.gif)
So obviously
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F808720/MediaObjects/13660_2008_Article_2010_Equ5_HTML.gif)
The following so-called Markov inequality is an important tool to prove inverse theorems in approximation theory. See, for example, Duffin and Schaeffer [1], Devore and Lorentz [2], and Borwein and Erdelyi [3].
Markov inequality. The inequality
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F808720/MediaObjects/13660_2008_Article_2010_Equ6_HTML.gif)
holds for every
It is well known that there have been some improvements of Markov-type inequality when the coefficients of polynomial are restricted; see, for example, [3–7]. In [5], Borwein and Erdélyi restricted the coefficients of polynomials and improved the Markov inequality as in following form.
Theorem 1.1.
There is an absolute constant such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F808720/MediaObjects/13660_2008_Article_2010_Equ7_HTML.gif)
for every .
We notice that the coefficients of polynomials in only take three integers:
and
. So, it is natural to raise the question: can we take the coefficients of polynomials as more general integers, and the conclusion of the theorem still holds? This question was not posed by Borwein and Erdélyi in [5, 6]. Also, we have not found the study for the question by now. This paper addresses the question. We shall give an affirmative answer. Indeed, we will prove the following results.
Theorem 1.2.
There are an absolute constant and a positive constant
depending only on
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F808720/MediaObjects/13660_2008_Article_2010_Equ8_HTML.gif)
Our proof follows [6] closely.
Remark 1.3.
Theorem 1.2 does not contradict [6, Theorem 2.4] since the coefficients of polynomials in are assumed to be integers, in which case there is a room for improvement.
2. The Proof of Theorem
In order to prove our main results, we need the following lemmas.
Lemma 2.1.
Let and
. Suppose
,
is analytical inside and on the ellipse
, which has focal points
and
, and major axis
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F808720/MediaObjects/13660_2008_Article_2010_Equ9_HTML.gif)
Let be the ellipse with focal points
and
, and major axis
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F808720/MediaObjects/13660_2008_Article_2010_Equ10_HTML.gif)
Then there is an absolute constant such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F808720/MediaObjects/13660_2008_Article_2010_Equ11_HTML.gif)
Proof.
The proof of Lemma 2.1 is mainly based on the famous Hadamard's Three Circles Theorem and the proof [6, Corollary 3.2]. In fact, if one uses it with replaced by
and
replaced by
, Lemma 2.1 follows immediately from [6, Corollary 3.2].
Lemma 2.2.
Let with
,
. Suppose
and
. Then there is a constant
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F808720/MediaObjects/13660_2008_Article_2010_Equ12_HTML.gif)
Proof.
By Chebyshev's inequality, there is an such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F808720/MediaObjects/13660_2008_Article_2010_Equ13_HTML.gif)
for every with
. Therefore,
Because of the assumption on
, we can write
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F808720/MediaObjects/13660_2008_Article_2010_Equ14_HTML.gif)
Recalling the facts that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F808720/MediaObjects/13660_2008_Article_2010_Equ15_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F808720/MediaObjects/13660_2008_Article_2010_IEq63_HTML.gif)
, and we obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F808720/MediaObjects/13660_2008_Article_2010_Equ16_HTML.gif)
Now by Lemma 2.1 we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F808720/MediaObjects/13660_2008_Article_2010_Equ17_HTML.gif)
Let , then there is an absolute constant
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F808720/MediaObjects/13660_2008_Article_2010_Equ18_HTML.gif)
By Cauchy's integral formula and the above inequality, we obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F808720/MediaObjects/13660_2008_Article_2010_Equ19_HTML.gif)
The proof of Lemma 2.2 is complete.
Proof of Theorem 1.2.
Noting and the fact
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F808720/MediaObjects/13660_2008_Article_2010_Equ20_HTML.gif)
proved by [6], we only need to prove the upper bound. To obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F808720/MediaObjects/13660_2008_Article_2010_Equ21_HTML.gif)
we distinguish four cases.
Case 1.
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F808720/MediaObjects/13660_2008_Article_2010_IEq68_HTML.gif)
. Let be an arbitrary number in
, then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F808720/MediaObjects/13660_2008_Article_2010_Equ22_HTML.gif)
Case 2.
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F808720/MediaObjects/13660_2008_Article_2010_IEq71_HTML.gif)
and , where
and
denotes the number of zeros of
at 1. Let
be a positive integer. If
satisfies the assumptions, then
, and
. Therefore, Markov inequality implies
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F808720/MediaObjects/13660_2008_Article_2010_Equ23_HTML.gif)
Hence
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F808720/MediaObjects/13660_2008_Article_2010_Equ24_HTML.gif)
So, the last inequality and imply
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F808720/MediaObjects/13660_2008_Article_2010_Equ25_HTML.gif)
Now using Taylor's Theorem, Lemma 2.2 with , the above inequality, and the fact
, we obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F808720/MediaObjects/13660_2008_Article_2010_Equ26_HTML.gif)
Case 3.
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F808720/MediaObjects/13660_2008_Article_2010_IEq83_HTML.gif)
and . Let
. We have
, where
and
are the major axis and minor axis of
, respectively, and
. Let
, we see
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F808720/MediaObjects/13660_2008_Article_2010_Equ27_HTML.gif)
Denote
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F808720/MediaObjects/13660_2008_Article_2010_Equ28_HTML.gif)
The solution of equation is
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F808720/MediaObjects/13660_2008_Article_2010_Equ29_HTML.gif)
It is obvious that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F808720/MediaObjects/13660_2008_Article_2010_Equ30_HTML.gif)
So, and the assumption of Lemma 2.2 imply
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F808720/MediaObjects/13660_2008_Article_2010_Equ31_HTML.gif)
And from (2.17) and Cauchy's integral formula, it follows that for every ,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F808720/MediaObjects/13660_2008_Article_2010_Equ32_HTML.gif)
and there holds
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F808720/MediaObjects/13660_2008_Article_2010_Equ33_HTML.gif)
Case 4.
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F808720/MediaObjects/13660_2008_Article_2010_IEq95_HTML.gif)
. Applying Lemma 2.1 with and
, we obtain that there is constant
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F808720/MediaObjects/13660_2008_Article_2010_Equ34_HTML.gif)
Indeed, noting that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F808720/MediaObjects/13660_2008_Article_2010_Equ35_HTML.gif)
we get the result want to be proved by a simple modification of the proof of Lemma 2.2. We omit the details. The proof of Theorem 1.2 is complete.
References
Duffin RJ, Schaeffer AC: A refinement of an inequality of the brothers Markoff. Transactions of the American Mathematical Society 1941,50(3):517–528. 10.1090/S0002-9947-1941-0005942-4
DeVore RA, Lorentz GG: Constructive Approximation, Grundlehren der Mathematischen Wissenschaften. Volume 303. Springer, Berlin, Germany; 1993:x+449.
Borwein P, Erdélyi T: Polynomials and Polynomial Inequalities, Graduate Texts in Mathematics. Volume 161. Springer, New York, NY, USA; 1995:x+480.
Borwein PB: Markov's inequality for polynomials with real zeros. Proceedings of the American Mathematical Society 1985,93(1):43–47.
Borwein P, Erdélyi T: Markov- and Bernstein-type inequalities for polynomials with restricted coefficients. The Ramanujan Journal 1997,1(3):309–323. 10.1023/A:1009761214134
Borwein P, Erdélyi T: Markov-Bernstein type inequalities under Littlewood-type coefficient constraints. Indagationes Mathematicae 2000,11(2):159–172. 10.1016/S0019-3577(00)89074-6
Borwein P, Erdélyi T, Kós G: Littlewood-type problems on . Proceedings of the London Mathematical Society 1999,79(1):22–46. 10.1112/S0024611599011831
Acknowledgments
The research was supported by the National Natural Science Foundition of China (no. 90818020) and the Natural Science Foundation of Zhejiang Province of China (no. Y7080235).
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 2.0 International License (https://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
About this article
Cite this article
Cao, F., Lin, S. Markov Inequalities for Polynomials with Restricted Coefficients. J Inequal Appl 2009, 808720 (2009). https://doi.org/10.1155/2009/808720
Received:
Revised:
Accepted:
Published:
DOI: https://doi.org/10.1155/2009/808720