# Markov Inequalities for Polynomials with Restricted Coefficients

- Feilong Cao
^{1}Email author and - Shaobo Lin
^{2}

**2009**:808720

**DOI: **10.1155/2009/808720

© F. Cao and S. Lin. 2009

**Received: **13 November 2008

**Accepted: **15 April 2009

**Published: **4 May 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

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

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.

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.

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.

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.

Proof.

The proof of Lemma 2.2 is complete.

Proof of Theorem 1.2.

we distinguish four cases.

Case 1.

Case 2.

Case 3.

Case 4.

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.

## Declarations

### 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).

## Authors’ Affiliations

## 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/S0002-9947-1941-0005942-4MathSciNetView ArticleMATHGoogle Scholar - DeVore RA, Lorentz GG:
*Constructive Approximation, Grundlehren der Mathematischen Wissenschaften*.*Volume 303*. Springer, Berlin, Germany; 1993:x+449.MATHGoogle Scholar - Borwein P, Erdélyi T:
*Polynomials and Polynomial Inequalities, Graduate Texts in Mathematics*.*Volume 161*. Springer, New York, NY, USA; 1995:x+480.View ArticleMATHGoogle Scholar - Borwein PB:
**Markov's inequality for polynomials with real zeros.***Proceedings of the American Mathematical Society*1985,**93**(1):43–47.MathSciNetMATHGoogle Scholar - 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:1009761214134View ArticleMathSciNetMATHGoogle Scholar - 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)89074-6MathSciNetView ArticleMATHGoogle Scholar - Borwein P, Erdélyi T, Kós G:
**Littlewood-type problems on .***Proceedings of the London Mathematical Society*1999,**79**(1):22–46. 10.1112/S0024611599011831MathSciNetView ArticleMATHGoogle Scholar

## Copyright

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