Arestov's theorems on Bernstein's inequality

TL;DR

This paper presents a straightforward and partly novel proof of Arestov's extension of Bernstein's inequality in Lp spaces, utilizing an approach related to Mahler's inequality for polynomials.

Contribution

It introduces a simplified proof of Arestov's theorem, extending Boyd's approach from algebraic to trigonometric polynomials.

Findings

Proof of Arestov's extension of Bernstein's inequality in Lp for all p ≥ 0

Extension of Boyd's approach to trigonometric polynomials

Simplification of existing proof techniques

Abstract

We give a simple, elementary, and at least partially new proof of Arestov's famous extension of Bernstein's inequality in $L_p$ to all $p \geq 0$. Our crucial observation is that Boyd's approach to prove Mahler's inequality for algebraic polynomials $P_n \in {\mathcal P}_n^c$ can be extended to all trigonometric polynomials $T_n \in {\mathcal T}_n^c$.

Equations

$$T_n(z)=\sum _{j=-n}^n{a}_j{z}^j,a_j\in \mathbb{C},z\in \mathbb{C}\setminus \{0\}.$$

$$P_n(z)=\sum _{j=0}^n{a}_j{z}^j,a_j\in \mathbb{C},z\in \mathbb{C}.$$

$$\Vert Q{\Vert }_0=M_0(Q):=\exp \left(\frac{1}{2\pi }{\int }_0^{2\pi }\log |Q(e^{it})|dt\right)$$

$$\Vert Q{\Vert }_0=M_0(Q)=\underset{p\to 0+}{\lim }{M}_p(Q),$$

$$\Vert Q{\Vert }_p=M_p(Q):={\left(\frac{1}{2\pi }{\int }_0^{2\pi }{\left|Q(e^{it})\right|}^{p}dt\right)}^{1/p},p>0.$$

$$\Vert Q{\Vert }_{\infty }=M_{\infty }(Q):=\underset{t\in [0,2\pi ]}{\max }|Q(e^{it})|=\underset{p\to \infty }{\lim }{M}_p(Q).$$

$$M_0(Q)=|c|\prod _{k=1}^n\max \{1,|z_k|\}$$

$$Q(z)=c\prod _{k=1}^n(z-z_k),c,z_k\in \mathbb{C}.$$

$$\Vert T_n^{\prime }{\Vert }_{\infty }\le n\Vert T_n{\Vert }_{\infty },T_n\in {\text{Cal}T}_n^c,$$

$$\Vert T_n{\Vert }_p\le n\Vert T_n{\Vert }_p,T_n\in {\text{Cal}T}_n,$$

$$\Vert T_n^{\prime }{\Vert }_0\le n\Vert T_n{\Vert }_0,T_n\in {\text{Cal}T}_n^c.$$

$${\int }_{\partial D}\log |T_n^{\prime }(z)/n||dz|\le {\int }_{\partial D}\log |T_n(z)||dz|,T_n\in {\text{Cal}T}_n^c.$$

$${\int }_{\partial D}{\log }^{+}|T_n^{\prime }(z)/n||dz|\le {\int }_{\partial D}{\log }^{+}|T_n(z)||dz|,T_n\in {\text{Cal}T}_n^c.$$

$$\Vert T_n^{\prime }{\Vert }_p\le n\Vert T_n{\Vert }_p,T_n\in {\text{Cal}T}_n^c,$$

$$P_{2n}(z)=c\prod _{j=1}^{2n}(z-z_j),0\ne c\in \mathbb{C},z_j\in D.$$

$$\frac{a{S}_n^{\prime }(a)}{S_n(a)}=\sum _{j=1}^{2n}\frac{a}{a-z_j}-\frac{an}{a}=\sum _{j=1}^{2n}\frac{1}{1-z_j/a}-n.$$

$$\text{Re}\left(\frac{1}{1-z_j/a}\right)>\frac{1}{2},j=1,2,\dots ,2n.$$

$$\text{Re}\left(\frac{a{S}_n^{\prime }(a)}{S_n(a)}\right)>\frac{2n}{2}-n=0.$$

$$|T_n(z)|\le |V_n(z)|,z\in \partial D,$$

$$|T_n^{\prime }(z)|\le |V_n^{\prime }(z)|,z\in \partial D.$$

$$S_n(z):=(V_n-\alpha T_n)(z)=z^{-n}(R_{2n}-\alpha Q_{2n})(z).$$

$$|Q_{2n}(z)|\le |R_{2n}(z)|,z\in \partial D,$$

$$|\alpha Q_{2n}(z)|<|R_{2n}(z)|,z\in \partial D.$$

$$S_n^{\prime }(z)=V_n^{\prime }(z)-\alpha T_n^{\prime }(z)\ne 0,z\in \partial D,$$

$$|T_n^{\prime }(z)|\le |V_n^{\prime }(z)|,z\in \partial D.$$

$$T_n(z)=\sum _{j=-n}^n{a}_j{z}^j,a_j\in \mathbb{C},z\in \mathbb{C}\setminus \{0\}.$$

$$\Vert T_n^{\prime }{\Vert }_0=n\Vert T_n{\Vert }_0=n|a_n|.$$

$$T_n(z)=a_n{z}^{-n}\prod _{j=1}^m(z-z_j)\prod _{j=m+1}^{2n}(z-z_j).$$

$$V_n(z):=a_n{z}^{-n}\prod _{j=1}^m(1-\overline{z_j}z)\prod _{j=m+1}^{2n}(z-z_j).$$

$$\Vert T_n^{\prime }{\Vert }_0\le \Vert V_n^{\prime }{\Vert }_0=n\Vert V_n{\Vert }_0=n|a_n|\prod _{j=1}^m|\overline{z_j}|=n|a_n|\prod _{j=1}^m|z_j|=n\Vert T_n{\Vert }_0.$$

Full text

Arestov’s theorems on Bernstein’s inequality

Tamás Erdélyi

Department of Mathematics, Texas A&M University, College Station, Texas 77843, College Station, Texas 77843

terdelyimath.tamu.edu

(April 26, 2019 )

We give a simple, elementary, and at least partially new proof of Arestov’s famous extension of Bernstein’s inequality in $L_{p}$ to all $p\geq 0$. Our crucial observation is that Boyd’s approach to prove Mahler’s inequality for algebraic polynomials $P_{n}\in{\Cal{P}}_{n}^{c}$ can be extended to all trigonometric polynomials $T_{n}\in{\Cal{T}}_{n}^{c}$.

Bernstein’s inequality in $L_{p}$ for all $p\geq 0$, Arestov’s theorem

††support: 2010 Mathematics Subject Classifications. 41A17

1. Introduction and Notation

Let ${\Cal{T}}_{n}^{c}$ be the collection of all trigonometric polynomials $T_{n}$ of the form

[TABLE]

Let ${\Cal{P}}_{n}^{c}$ be the collection of all algebraic polynomials $P_{n}$ of the form

[TABLE]

Let $D$ denote the open unit disk, and let $\partial D$ denote the unit circle of the complex plane. We define the Mahler measure (geometric mean of $Q$ on ${\partial D}$)

[TABLE]

for bounded measurable functions $Q$ on $\partial D$. It is well known, see [HL-52], for instance, that

[TABLE]

where

[TABLE]

It is also well known that for a function $Q$ continuous on $\partial D$ we have

[TABLE]

It is a simple consequence of the Jensen formula that

[TABLE]

for every polynomial of the form

[TABLE]

Bernstein’s inequality

[TABLE]

plays a crucial role in proving inverse theorems of approximation as well as many other results in approximation theory. See [BE-95], for instance. As far as the history of Bernstein’s inequality is concerned we refer to Nevai’s lovely papers [N-14] and [N-19] and the references in them. We do not repeat the full story here. In 1981 Arestov [A-81] proved that

[TABLE]

for all $p\geq 0$, extending the result known only for $p\geq 1$ for a long time. Simpler proof of Bernstein’s inequality in $L_{p}$ for all $p\geq 0$ have been given by Golitschek and Lorentz in [GL-89] which is presented in the book [DL-93] by DeVore and Lorentz. A very elegant and even more simplified proof was published recently in [QZ-19] by Queffélec and Zarouf. A central part of their proof is to prove (1.1) for $p=0$ first. Mahler [M-61] showed (1.1) for $p=0$ but only for polynomials $P_{n}\in{\Cal{P}}_{n}^{c}$, and he gave a rather involved proof. Mahler’s inequality was also posed as a problem by Vaaler in the Problems section of the American Mathematical Monthly and solved by Boyd [VB-91] using an elementary theorem of Bernstein. We note that Glazyrina [G-05] proved a sharp Markov-type inequality for algebraic polynomials in $L_{0}$ on finite subintervals of the real line.

In this note we give a simple, elementary, and at least partially new proof of Arestov’s famous extension of Bernstein’s inequality in $L_{p}$ to all $p\geq 0$. Our crucial observation is that Boyd’s approach to prove Mahler’s inequality for algebraic polynomials $P_{n}\in{\Cal{P}}_{n}^{c}$ can be extended to all trigonometric polynomials $T_{n}\in{\Cal{T}}_{n}^{c}$.

Theorem 1.1

We have

[TABLE]

Equivalently

[TABLE]

Theorem 1.2

With the notation $\log^{+}|a|:=\max\{\log|a|,0\}$ we have

[TABLE]

Theorem 1.3

We have

[TABLE]

for every $p>0$.

2. Lemmas

To prove the Theorem 1.1 we need two lemmas.

Lemma 2.1

Associated with $S_{n}\in{\Cal{T}}_{n}^{c}$ let $P_{2n}\in{\Cal{P}}_{2n}^{c}$ be defined by $P_{2n}(z)=z^{n}S_{n}(z)$. If $P_{2n}$ has each of its $2n$ zeros in $D$, then $S_{n}^{\prime}$ has each of its zeros in $D$ as well. The same is true if $D$ is replaced by the closed unit disk $\overline{D}$.

Demonstration Proof

We prove the lemma for $D$, the case of the closed unit disk $\overline{D}$ follows from this by a straightforward limiting argument. Suppose $a\notin D$, that is, $a\in{\mathbb{C}}$ and $|a|\geq 1$. Suppose also that $S_{n}\in{\Cal{T}}_{n}^{c}$ and $P_{2n}(z):=z^{n}S_{n}(z)$ has each of its $2n$ zeros in $D$, that is

[TABLE]

We have

[TABLE]

Observe that $|z_{j}/a|<1$ for each $j=1,2,\ldots,2n$, and hence

[TABLE]

Combining (2.1) and (2.2) we obtain

[TABLE]

We conclude that $S_{n}^{\prime}(a)\neq 0$. ∎

Lemma 2.2

Associated with $V_{n}\in{\Cal{T}}_{n}^{c}$ let $R_{2n}\in{\Cal{P}}_{2n}^{c}$ be defined by $R_{2n}(z)=z^{n}V_{n}(z)$ and suppose that $R_{2n}$ has each of its $2n$ zeros in the closed unit disk $\overline{D}$. If $T_{n}\in{\Cal{T}}_{n}$ and

[TABLE]

then

[TABLE]

Demonstration Proof

Without loss of generality we may assume that $R_{2n}$ has each of its $2n$ zeros in $D$, the case when $R_{2n}$ has each of its $2n$ zeros in the closed unit disk $\overline{D}$ follows from this by a straightforward limiting argument. Let $Q_{2n}\in{\Cal{P}}_{2n}^{c}$ be defined by $Q_{2n}(z):=z^{n}T_{n}(z)\,.$ Let $\alpha\in{\mathbb{C}}$, $|\alpha|<1$, and

[TABLE]

It follows from (2.3) that

[TABLE]

and hence $|\alpha|<1$ and the fact that $R_{2n}\in{\Cal{P}}_{2n}^{c}$ does not vanish on $\partial D$ imply that

[TABLE]

Therefore Rouche’s Theorem implies that the polynomial $P_{2n}:=R_{2n}-\alpha Q_{2n}\in{\Cal{P}}_{2n}^{c}$ and $R_{2n}$ has the same number of zeros in $D$, that is, $R_{2n}-\alpha Q_{2n}$ has each of its $2n$ zeros in $D$. By Lemma 2.1 and (2.4) we can deduce that $S_{n}^{\prime}=V_{n}^{\prime}-\alpha T_{n}^{\prime}$ has each of its zeros in $D$. In particular,

[TABLE]

for all $\alpha\in{\mathbb{C}}$, $|\alpha|<1$. We conclude that

[TABLE]

∎

3. Proof of Theorems 1.1, 1.2, and 1.3

Demonstration Proof of Theorem 1.1

Associated with $T_{n}\in{\Cal{T}}_{n}^{c}$ let $P_{2n}\in{\Cal{P}}_{2n}^{c}$ be defined by $P_{2n}(z)=z^{n}T_{n}(z)$. Without loss of generality we may assume that $P_{2n}$ has exactly $2n$ complex zeros, the case when $P_{2n}$ has less than $2n$ complex zeros follows from this by a straightforward limiting argument.

Case 1. Suppose $P_{2n}$ has all its $2n$ zeros in the closed unit disk $\overline{D}$. It follows from Lemma 2.1 that $T_{n}^{\prime}$ has all its zeros in the closed unit disk $\overline{D}$. Let

[TABLE]

Using Jensen’s formula and the multiplicative property of the Mahler measure we can easily deduce that

[TABLE]

Case 2. Suppose that some of the zeros of $P_{2n}$ are outside the closed unit disk $\overline{D}$. Let $z_{1},z_{2},\ldots,z_{m}$ be the zeros of $T_{n}$ outside the closed unit disk $\overline{D}$ and let $z_{m+1},z_{m+2},\ldots,z_{2n}$ be the zeros of $T_{n}$ in the closed unit disk $\overline{D}$. We have

[TABLE]

We define

[TABLE]

Observe that (2.3) holds, $V_{n}\in{\Cal{T}}_{n}^{c}$, and $R_{2n}\in{\Cal{P}}_{2n}^{c}$ defined by $R_{2n}(z):=z^{n}V_{n}(z)$ has each of its $2n$ zeros in the closed unit disk $\overline{D}$. Using Lemma 2.2, the (in)equality of the theorem in Case 1, and Jensen’s formula, we obtain

[TABLE]

∎

Demonstration Proof of Theorem 1.2

We follow the argument given in [QZ-19] to base our proof on Theorem 1.1. It is well-known, and by applying Jensen’s formula it is ieasy to see, that

[TABLE]

and hence

[TABLE]

Let $T_{n}\in{\Cal{T}}_{n}^{c}$, $w\in\partial D$, and $E_{n}(z):=z^{n}$. Applying Theorem 1.1 with $T_{n}$ replaced by $T_{n}+wE_{n}$ we obtain

[TABLE]

Integrating both sides on $\partial D$ with respect to $|dw|$, then using Fubini’s theorem and (3.1), we get the theorem ∎

Demonstration Proof of Theorem 1.3

We follow the argument given in [QZ-19] to base our proof on Theorem 1.2. Observe that

[TABLE]

Indeed, the integration by parts formula gives

[TABLE]

For the sake of brevity we will use the notation $d\mu(a)=p^{2}a^{p-1}\,da$. Using (3.2), Fubini’s theorem, Theorem 1.2, and Fubini’s theorem again, we obtain

[TABLE]

∎

4. Acknowledgement

The author thanks Herve Queffélec and Paul Nevai for checking the details of the proof in this paper and for their suggestions to make the paper more readable.