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Markov Inequalities for Polynomials with Restricted Coefficients
Journal of Inequalities and Applications volume 2009, Article number: 808720 (2009)
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 .
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
We introduce the following classes of polynomials. Let
denote the set of all algebraic polynomials of degree at most with real coefficients. Let
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
The following so-called Markov inequality is an important tool to prove inverse theorems in approximation theory. See, for example, Duffin and Schaeffer , Devore and Lorentz , and Borwein and Erdelyi .
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 , Borwein and Erdélyi restricted the coefficients of polynomials and improved the Markov inequality as in following form.
There is an absolute constant such that
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.
There are an absolute constant and a positive constant depending only on such that
Our proof follows  closely.
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.
Let and . Suppose , is analytical inside and on the ellipse , which has focal points and , and major axis
Let be the ellipse with focal points and , and major axis
Then there is an absolute constant such that
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].
Let with , . Suppose and . Then there is a constant such that
By Chebyshev's inequality, there is an such that
for every with . Therefore, Because of the assumption on , we can write
Recalling the facts that
, and we obtain
Now by Lemma 2.1 we have
Let , then there is an absolute constant such that
By Cauchy's integral formula and the above inequality, we obtain
The proof of Lemma 2.2 is complete.
Proof of Theorem 1.2.
Noting and the fact
proved by , we only need to prove the upper bound. To obtain
we distinguish four cases.
. Let be an arbitrary number in , then
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
So, the last inequality and imply
Now using Taylor's Theorem, Lemma 2.2 with , the above inequality, and the fact , we obtain
and . Let . We have , where and are the major axis and minor axis of , respectively, and . Let , we see
The solution of equation is
It is obvious that
So, and the assumption of Lemma 2.2 imply
And from (2.17) and Cauchy's integral formula, it follows that for every ,
and there holds
. Applying Lemma 2.1 with and , we obtain that there is constant such that
Indeed, noting that
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.
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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).
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Cao, F., Lin, S. Markov Inequalities for Polynomials with Restricted Coefficients. J Inequal Appl 2009, 808720 (2009). https://doi.org/10.1155/2009/808720
- Positive Constant
- Focal Point
- Major Axis
- Minor Axis
- Absolute Constant