On Chebyshev polynomials in the complex plane

TL;DR

This paper investigates the uniform norm estimates of Chebyshev polynomials in the complex plane, providing exact bounds for certain geometric configurations and exploring properties related to uniformly perfect sets.

Contribution

It offers new precise estimates for Chebyshev polynomials on complex sets, including quasiconformal curves and uniformly perfect subsets of the real line.

Findings

Exact norm estimates for Chebyshev polynomials on quasiconformal curves

Norm bounds for Chebyshev polynomials on uniformly perfect sets

Extension of estimates to complex and real subsets

Abstract

The estimates of the uniform norm of the Chebyshev polynomials associated with a compact set $K$ in the complex plane are established. These estimates are exact (up to a constant factor) in the case where $K$ consists of a finite number of quasiconformal curves or arcs. The case where $K$ is a uniformly perfect subset of the real line is also studied.

Equations

$$||T_n|{|}_K\ge \text{mbox}cap(K{)}^n\text{mbox}and\underset{n\to \infty }{\lim }||T_n|{|}_K^{1/n}=\text{mbox}cap(K).$$

$$t_n(K):=\frac{||T_n|{|}_K}{\text{mbox}cap(K{)}^n}.$$

$$K={\cup }_{j=1}^m{K}^j;K^j\cap K^k=\varnothing \text{mbox}forj\ne k;\text{mbox}diam(K^j)>0.$$

$$\text{mbox}diam(S):=\underset{z,\zeta \in S}{\sup }|z-\zeta |,S\subset \mathbf{C}.$$

$$t_n(K)=O(\log n)\text{mbox}forn\to \infty .$$

$$\text{mbox}diam(L(z_1,z_2))\le {\beta }_L|z_2-z_1|,$$

$$t_n(K)=O(1)\text{mbox}forn\to \infty .$$

$$||F_n|{|}_{K_{\ast }}>n^{\alpha },n\in \Lambda .$$

$$K\cap \{\zeta :{\gamma }_{K}r\le |z-\zeta |\le r\}\ne \varnothing ,0<r<\text{mbox}diam(K).$$

$$\text{mbox}cap(K\cap \{\zeta :|\zeta -z|\le r\})\ge {\lambda }_{K}r,0<r<\text{mbox}diam(K).$$

$$t_n(K)=O(n^c)\text{mbox}forn\to \infty .$$

$$|K\cap (x-r,x+r)|\ge {\eta }_{K}r,0<r<\text{mbox}diam(K).$$

$$d(z,S):=\text{mbox}dist(\{z\},S):=\underset{\zeta \in S}{\inf }|z-\zeta |,z\in \mathbf{C},S\subset \mathbf{C}.$$

$${\Phi }_{\Omega }(z,z_0):=\exp (g_{\Omega }(z,z_0)+i{\tilde{g}}_{\Omega }(z,z_0))$$

$$g_{\Omega }(z):=0,z\in K,$$

$$K_s:=\{z\in \mathbf{C}:g(z)\le s\},{\Omega }_s:=\overline{\mathbf{C}}\setminus K_s,s>0.$$

$$g_{{\Omega }_s}(z)=g_{\Omega }(z)-s,{\Phi }_{{\Omega }_s}(z)=e^{-s}{\Phi }_{\Omega }(z).$$

$$W_n(z):={\Phi }_{\Omega }(z{)}^n,z\in {\Omega }_{1/n^2}.$$

$$W_n(z):={\Phi }_{\Omega }(z{)}^n\prod _{l=1}^q{\Phi }_{{\Omega }_{1/n^2}}(z,z_{l,n}{)}^{-1},z\in {\Omega }_{1/n^2},$$

$$F_n(z):=\frac{1}{2\pi i}{\int }_{C_n}\frac{W_n(\zeta )}{\zeta -z}d\zeta ,z\in \mathbf{C},$$

$$\left|\prod _{l=1}^q{\Phi }_{{\Omega }_{1/n^2}}(\infty ,z_{l,n})\right|$$

$$F_n(z)=W_n(z)+\frac{1}{2\pi i}{\int }_{C_n^{\prime }}\frac{W_n(\zeta )}{\zeta -z}d\zeta ,$$

$$|{\alpha }_n|$$

$$K_{1,1,k,n}(\zeta ,z),\zeta \in \Omega \setminus \{\infty \},z\in K,$$

$$\left|\frac{1}{\zeta -z}-K_{1,1,k,n}(\zeta ,z)\right|\le c_1\frac{|{\tilde{\zeta }}_{1/n}-\zeta {|}^k}{|\zeta -z||{\tilde{\zeta }}_{1/n}-z{|}^k}{\left(1+\left|\frac{\zeta -z}{{\tilde{\zeta }}_{1/n}-z}\right|\right)}^k,$$

$${\tilde{\zeta }}_{\delta }:={\Phi }_{\Omega }^{-1}((1+\delta ){\Phi }_{\Omega }(\zeta )),\delta >0.$$

$$\frac{|{\tilde{w}}_{1/n}-w|}{|{\tilde{w}}_{1/n}|-1}<\frac{1}{32}.$$

$$\frac{|{\tilde{\zeta }}_{1/n}-\zeta |}{|{\tilde{\zeta }}_{1/n}-z|}\le 16\frac{|{\tilde{w}}_{1/n}-w|}{|{\tilde{w}}_{1/n}|-1}<\frac{1}{2},$$

$$\left|\frac{1}{\zeta -z}-K_{1,1,k,n}(\zeta ,z)\right|\le c_2\frac{d(\zeta ,K{)}^k}{|\zeta -z{|}^{k+1}},c_2=c_2(K,k).$$

$$p_{n,k,\zeta ,K}(z)=\sum _{l=0}^n{a}_{l,k,K}(\zeta )z^l,$$

Full text

[TABLE]

If more information is known about the geometry of $K$, (1.2) can be improved, for example, in the following way. A Jordan curve $L\subset{\bf C}$ is called quasiconformal (see [1] or [16, p. 100]) if for every $z_{1},z_{2}\in L$,

[TABLE]

where $L(z_{1},z_{2})$ is the smaller subarc of $L$ between $z_{1}$ and $z_{2}$, a constant $\beta_{L}>1$ depends only on $L$. Any subarc of a quasiconformal curve is called a quasiconformal arc.

[TABLE]

The estimate (1.4) was proved by other methods in [34] and recently in [31] (for sufficiently smooth $\partial K^{j}$), in [32] (for piecewise sufficiently smooth $\partial K^{j}$), and in [4] (for quasismooth in the sense of Lavrentiev $\partial K^{j}$).

The question whether (1.4) does hold for a general continuum seems to be still open. In the Oberwolfach meeting (see [14] or [20, p. 365]) Pommerenke asked about an example for a continuum $K$ such that the sequence $\{t_{n}(K)\}$ is unbounded. It is mentioned in [20, p. 365] that “D. Wrase in Karlsruhe has shown that an example constructed by J. Clunie [9] for a different purpose has the required property”. But we could not find the proof of this result.

Moreover, in the case where $K$ is a continuum, one of the major sources for estimates of $t_{n}(K)$ are Faber polynomials $F_{n}(z)=F_{n}(z,K)$ associated with $K$ (see [25], [27]). Gaier [11, Theorem 2], using the same example by Clunie, [9] has shown that there exist a continuum $K_{*}$ bounded by a quasiconformal curve with $\mbox{ cap}(K_{*})=1$, a positive constant $\alpha$ and an infinite set $\Lambda\subset{\bf N}$ such that for the (monic) polynomial $F_{n}(z)=F_{n}(z,K_{*})$ we have

[TABLE]

Note that the first result of this kind (without the restriction on $K_{*}$ to be a quasidisk) was proved by Pommerenke [19].

Hence, Theorem 1 and Theorem 2 reveal the essential difference between the Chebyshev and the Faber polynomials. It is worth pointing out that the case of multiply connected $\Omega$ presents a more delicate problem (see for example [34]).

Let now $K\subset{\bf R}$, where ${\bf R}$ is the real line, consist of an infinite number of components. According to [13, Theorem 4.4] in this case $\{t_{n}(K)\}$ can increase faster than any sequence $\{t_{n}\}$ satisfying $t_{n}\geq 1$ and $\lim_{n\to\infty}(\log t_{n})/n=0$. Therefore, in order to have particular bounds for $t_{n}(K)$ some additional assumptions on $K$ are needed. We assume that $K$ is uniformly perfect, which according to Beardon and Pommerenke [6] means that there exists a constant $0<\gamma_{K}<1$ such that for $z\in K$,

[TABLE]

The classical Cantor set is an example of a uniformly perfect set. Pommerenke [21] has shown that uniformly perfect sets can be described using a density condition in terms of the logarithmic capacity. Namely, $K$ is uniformly perfect if and only if there exists a constant $0<\lambda_{K}<1$ such that for $z\in K$,

[TABLE]

Note that sets satisfying (1.5) play a significant role in the solution of the inverse problem of the constructive theory of functions of a complex variable. We refer to [28] where they are called $c$-dense sets. Other interesting properties of the uniformly perfect sets can be found in [12, pp. 343–345].

[TABLE]

Following Carleson [7] we say that a compact set $K\subset{\bf R}$ is homogeneous if there is a constant $\eta_{K}>0$ such that for all $x\in K$,

[TABLE]

Here, $|S|$ is the linear measure (length) of a (Borel) set $S\subset{\bf C}$ (see [22, p. 129]). The Cantor sets of positive length are examples of homogeneous sets (see [18, p. 125]). Recently Christiansen, Simon, and Zinchenko [8] have shown that for the homogeneous subsets of the real line the term $O(n^{c})$ in (1.6) can be replaced by $O(1)$. It is worth pointing out that there is a principal difference between the above mentioned classes of compact sets, i.e., $K$ is the Parreau-Widom set in the case of the homogeneous $K\subset{\bf R}$ and it is not, in general, the Parreau-Widom set in the case of the uniformly perfect $K$. See [8] for more details.

In what follows, we use the convention that $c,c_{1},\ldots$ denote positive constants (different in different sections) that are either absolute or they depend only on $K$; otherwise, the dependence on other parameters is explicitly stated. For the nonnegative functions $a$ and $b$ we write $a\preceq b$ if $a\leq cb$, and $a\asymp b$ if $a\preceq b$ and $b\preceq a$ simultaneously.

We also use the additional notation

[TABLE]

2. The basic potential-theoretic functions

Let $K$ be as in (1.1). Following Widom [34], we extend the concept of Faber polynomials to the case of compact sets with the finite number of connected components. Since in [34] all $\partial K^{j}$ are sufficiently smooth curves, we need to add some purely technical details. Denote by $g_{\Omega}(z,z_{0}),\>z,z_{0}\in\Omega,$ the Green function for $\Omega$ with pole at $z_{0}$. It has a multiple-valued harmonic conjugate $\tilde{g}_{\Omega}(z,z_{0})$. Thus, the analytic function

[TABLE]

is also multiple-valued. We write $g_{\Omega}(z)$, $\tilde{g}_{\Omega}(z)$, and $\Phi_{\Omega}(z)$ in the case $z_{0}=\infty$.

Let

[TABLE]

[TABLE]

Then for $z\in\Omega_{s}$,

[TABLE]

For $n\in{\bf N}$, if $\Phi_{\Omega}(z)^{n}$ is single-valued in $\Omega_{1/n^{2}}$, we set

[TABLE]

If $\Phi_{\Omega}(z)^{n}$ is multiple-valued in $\Omega_{1/n^{2}}$, then according to [34, pp. 159, 211] there exist $q\leq m-1$ points $z_{1,n},\ldots,z_{q,n}\in\Omega_{1/n^{2}}$ such that the function

[TABLE]

is single-valued in $\Omega_{1/n^{2}}$. Moreover, all $z_{l,n}$ lie in the convex hull of $K_{1/n^{2}}$.

In both cases we consider the entire function

[TABLE]

where $C_{n}\subset\Omega_{1/n^{2}}\setminus\{\infty\}$ is a Jordan curve, oriented in the positive direction, containing $K_{1/n^{2}}$ and $z$ in its interior.

Since all points $z_{l,n}$ are in the convex hull of $K_{1}$, by the symmetry property of the Green function, we obtain

[TABLE]

For $z\in\Omega_{1/n^{2}}\setminus\{\infty\}$, let $C_{n}^{\prime}\subset\Omega_{1/n^{2}}\setminus\{\infty\}$ be any Jordan curve, oriented in the positive direction, containing $K_{1/n^{2}}$ in its interior and $z$ in its exterior. Since by the Cauchy formula

[TABLE]

we see that $F_{n}(z)=\alpha_{n}z^{n}+\ldots$ is a polynomial with the property

[TABLE]

Now let $K$ consist of one component, i.e., $m=1$ and let $\Phi_{\Omega}:\Omega\to{\bf D}^{*}:=\{w:|w|>1\}$ be the Riemann conformal mapping with $\Phi_{\Omega}(\infty)=\infty,\Phi_{\Omega}^{\prime}(\infty)>0$. We follow a technique of [10, Chapter IX], [5, p. 387] and for $k,n\in{\bf N},k\geq 2$, consider the Dzjadyk polynomial kernel

[TABLE]

which is a polynomial with respect to $z$ of degree $(k+3)k(n-1)-1$ with continuous coefficients depending on $\zeta$.

According to [5, p. 389, Theorem 2.4] we have

[TABLE]

where $c_{1}=c_{1}(K,k)$ and

[TABLE]

Let $w:=\Phi_{\Omega}(\zeta),\tilde{w}_{1/n}:=w(1+1/n)$. A straightforward calculation shows that for $n>32^{2}$ and $|w|\geq 1+32/n$, we have

[TABLE]

Therefore, [5, p. 23, Lemma 2.3] implies

[TABLE]

i.e.,

[TABLE]

We summarize our reasoning as follows. Given $k\in{\bf N}$, there exist sufficiently large constants $n_{0}=n_{0}(k)$ and $c_{3}=c_{3}(k)$ such that for any integer $n>n_{0}$ and $\zeta$ with $|\Phi_{\Omega}(\zeta)|-1\geq c_{3}/n$, there exists a polynomial

[TABLE]

where $a_{l,k,K}$ are continuous functions of $\zeta$, satisfying

[TABLE]

Indeed, to get (2.4) we can take

[TABLE]

Furthermore, by virtue of (2.3), for $\zeta\in\Omega$ with $c_{4}\leq|\Phi_{\Omega}(\zeta)|-1\leq c_{5}$ we have

[TABLE]

Let $K$ now be as in (1.1) with $m>1$. Denote by $r_{K}>0$ any fixed number such that $K_{r_{K}}$ consists of exactly $m$ components, i.e.,

[TABLE]

Let $\Omega^{j}:=\overline{{\bf C}}\setminus K^{j}$. The maximum principle for the appropriate linear combination of harmonic functions $g_{\Omega}$ and $\log|\Phi_{\Omega^{j}}|$ in $K_{r_{K}}^{j}\setminus K^{j}$ shows that

[TABLE]

For sufficiently large $v\in{\bf N}$, $\zeta\in K_{r_{K}}^{j}\setminus K^{j}$ with $|\Phi_{\Omega^{j}}(\zeta)|-1\geq c_{3}/v$, and $z\in K^{l},l=1,\ldots m$, by virtue of (2.4) and (2.5), applied for the continuum $K^{l}$, we have

[TABLE]

Here $c_{7}:=c_{2}+c_{6}$.

For $\zeta$ as in (2.7) and $l=1,\ldots,m,$ consider functions

[TABLE]

[TABLE]

so that

[TABLE]

Since $h_{l}$ can be extended analytically to $K_{r_{K}}$, by the Walsh approximation theorem [33, pp. 75-76] there is $u_{0}=u_{0}(K)\in{\bf N}$, such that for any integer $u>u_{0}$, there exists a polynomial $q_{u,l}\in{\bf P}_{u}$ satisfying

[TABLE]

For sufficiently large $n$ and $\zeta\in K_{r_{K}}^{j}\setminus K^{j}$ with $|\Phi_{\Omega^{j}}(\zeta)|-1\geq c^{*}/n\geq c_{3}/v$, where $v$ and the constant $c^{*}$ are to be chosen later, consider the polynomial

[TABLE]

Let

[TABLE]

Since for $z\in K^{p},p=1,\ldots,m$,

[TABLE]

by (2.7), (2.8), and the Bernstein-Walsh lemma (see [33, p. 77] or [24, p. 153]), we obtain the following estimates:

if $l=j$, then

[TABLE]

if $l\neq j$, then

[TABLE]

[TABLE]

Therefore, for the polynomial

[TABLE]

according to (2.9) and (2.10) for $\zeta$ as in (2.7) and $z\in K^{p}$, we obtain:

if $p=j$, then

[TABLE]

if $p\neq j$, then

[TABLE]

Let

[TABLE]

Note that $v\geq n/c_{9}$. To be sure that (2.11) and (2.12) hold we need to have $c_{3}/v\leq c^{*}/n$ which dictates the choice $c^{*}:=c_{3}c_{9}.$

Thus, using the Löwner inequality (see [5, p. 359, Corollary 2.5]), $d(\zeta,K^{j})\geq c_{10}/n^{2},c_{10}=c_{10}(K,k),$ we obtain a polynomial

[TABLE]

satisfying, by virtue of (2.11) and (2.12), for $\zeta\in K^{j}_{r_{K}}\setminus K^{j}$ with $|\Phi_{\Omega^{j}}(\zeta)|-1\geq c^{*}/n$, where $c^{*}=c^{*}(K,k)$ and $n>n_{1}=n_{1}(K,k)$, the inequality

[TABLE]

where $c_{11}=c_{11}(K,k)$.

3. Chebyshev polynomials for a system of continua

We start with the proof of the following estimate.

[TABLE]

where $c^{*}$ and $n_{1}$ are the constants from (2.13), $c_{1}=c_{1}(K,k)$, and

[TABLE]

Proof. Let $F_{n}$ be defined by (2.1). By our assumption $n$ is so large that the curves $S_{n,j}:=L_{c^{*}/n,j}\subset\Omega_{1/n^{2}}$ are mutually disjoint. Let $S_{n}:=\cup_{j=1}^{m}S_{n,j}$.

By [5, p. 23, Lemma 2.3], for $\zeta\in\Omega^{j}$, $w:=\Phi_{\Omega^{j}}(\zeta)$, and $\Psi_{\Omega^{j}}:=\Phi_{\Omega^{j}}^{-1}$, we have

[TABLE]

Therefore,

[TABLE]

By the Cauchy formula

[TABLE]

We can certainly assume that $k>1$. Consider polynomial $F^{*}_{n}(z)=\alpha_{n}z^{n}+\ldots\in{\bf P}_{n}$ defined as follows

[TABLE]

where $s_{n-1,k,\zeta}\in{\bf P}_{n-1}$ satisfies (2.13).

Since by (2.6), for $\zeta\in S_{n}$,

[TABLE]

where $c_{2}=c_{2}(K,k)$, according to (2.13) and (3.3), for $z\in K^{j}$, we obtain

[TABLE]

where $c_{l}=c_{l}(K,k),l=3,4.$

Making use of (2.2) and the obvious inequality $t_{n}(K)\mbox{ cap}(K)^{n}\leq||F_{n}^{*}||_{K}/|\alpha_{n}|$ we finally obtain (3.1).

$\Box$

Proof of Theorem 1. Changing the variable in the integrals from (3.1) and using (3.2), for sufficiently large $n$, we obtain

[TABLE]

Furthermore, since by [27, Chapter IX, §4, Lemma 3],

[TABLE]

the inequalities (3.1) (with $k=1$) and (3.4) imply (1.2).

$\Box$

Theorem 2 is a particular case of a more general result which we describe below. Let $K$ consist of one component, i.e., $m=1$, and let $\Omega$ be a John domain which can be defined as follows (see [22, p. 98]). For a crosscut $\gamma\subset\Omega\setminus\{\infty\}$ of $\Omega$ let $H(\gamma)$ be a bounded component of $\Omega\setminus\gamma$. We say that $\gamma$ is a circular crosscut if $\gamma\subset\Omega\cap C(z,r)$ for some $z\in\partial\Omega=\partial K,r>0$, and $z\in\overline{H(\gamma)}$. Here $C(z,r):=\{\zeta:|\zeta-z|=r\}$. Then $\Omega$ is a John domain if there exists a constant $\lambda_{\Omega}>1$ such that for any circular crosscut $\gamma$ of $\Omega$,

[TABLE]

By virtue of (1.3) the complement of a quasiconformal arc as well as the unbounded Jordan domain with a quasiconformal boundary both are John domains.

According to (3.5) the function $\Psi_{\Omega}$ has a continuous extension to $\overline{{\bf D}^{*}}$ which we denote by the same letter $\Psi_{\Omega}$. Next, we assume that $\partial K$ is piecewise quasiconformal, i.e., there exist

[TABLE]

such that each $J_{l}:=\Psi_{\Omega}(J^{\prime}_{l}),l=1,\ldots,p$, where $J_{l}^{\prime}:=\{e^{i\theta}:\theta_{l}\leq\theta\leq\theta_{l+1}\}$ is a quasiconformal arc.

Let

[TABLE]

[TABLE]

By [3, Lemma 2],

[TABLE]

Moreover, according to [3, (4.14)],

[TABLE]

Here for any arc or unbounded curve $\Gamma\subset{\bf C}$ and $\zeta_{1},\zeta_{2}\in\Gamma$, we denote by $\Gamma(\zeta_{1},\zeta_{2})$ the bounded subarc of $\Gamma$ between these points.

Thus, by virtue of (3.6) and (3.7), the curve $L_{l}^{*}:=\partial\Omega_{l}=\Gamma_{l}\cup J_{l}\cup\Gamma_{l+1}$ satisfies

[TABLE]

i.e., by the Ahlfors criterion (see [16, p. 100]), $L_{l}^{*}$ is quasiconformal. Since by the same Ahlfors criterion $\partial\Omega_{l}^{\prime}=\Gamma_{l}^{\prime}\cup J_{l}^{\prime}\cup\Gamma^{\prime}_{l+1}$ is also quasiconformal, the restriction of $\Phi_{\Omega}$ to $\Omega_{l}$ can be extended to a $Q_{l}$-quasiconformal homeomorphism $\Phi_{l}:{\bf C}\to{\bf C}$ with some $Q_{l}\geq 1$ (see [16, p. 98]).

The following result describes the distortion properties of $\Phi_{l}$ and the inverse mapping $\Phi^{-1}_{l}$ which both are $Q$-quasiconformal with $Q:=\max_{l=1,\ldots,p}Q_{l}$.

[TABLE]

where $c_{j}=c_{j}(c_{5},Q),j=6,7$.

We claim that for $z\in\partial K\setminus J_{l}$,

[TABLE]

Indeed, let $z^{\prime}_{l}\in\partial\Omega_{l}$ be such that $|z-z^{\prime}_{l}|=d(z,\Omega_{l})$. The nontrivial case arises when $z^{\prime}_{l}\not\in J_{l}$, i.e., $z^{\prime}_{l}\in\Gamma_{k}$ for $k=l$ or $k=l+1$. Then by (3.6) we obtain

[TABLE]

which yields (3.8).

For $z\in\partial K$, denote by $z_{l}^{*}$ any point of $J_{l}$ with the property $|z-z_{l}^{*}|=d(z,J_{l})$. We claim that

[TABLE]

Indeed, by (3.8),

[TABLE]

and (3.9) follows.

For $\zeta\in\Omega\setminus\{\infty\}$ denote by $\zeta_{K}:=\Psi_{\Omega}(\Phi_{\Omega}(\zeta)/|\Phi_{\Omega}(\zeta)|)$ the “projection” of $\zeta$ on $K$. As an immediate application of Lemma 2, for $\zeta\in\Omega_{l}$ and $z\in J_{l}$, we have

[TABLE]

Now let $K$ be as in (1.1). We assume that each $\Omega^{j}$ is a John domain and each $\partial K^{j}$ is piecewise quasiconformal, i.e., each $\partial K^{j}$ consists of $p_{j}$ quasiconformal arcs $J_{l,j},l=1,\ldots,p_{j}$ as described above. Let $\Phi_{l,j}$ be the appropriate quasiconformal homeomorphism of ${\bf C}$ which is conformal in a subdomain $\Omega_{l}^{j}$ of $\Omega^{j}$ with $J_{l,j}\subset\partial\Omega^{j}_{l}$. For $z\in\partial K^{j}$, denote by $z_{l,j}^{*}$ the nearest to $z$ point of $J_{l,j}$ and let $w_{l,j}:=\Phi_{l,j}(z_{l,j}^{*}),J_{l,j}^{\prime}:=\Phi_{l,j}(J_{l,j}).$

According to (3.2), Lemma 2 with $F=\Psi_{\Omega^{j}}$ restricted to $\Phi_{\Omega^{j}}(\Omega_{l}^{j})$ and the triplet of points $\tau,\tau/|\tau|,w_{l,j}$, as well as (3.9), for $z\in\partial K^{j}$, $s=c^{*}/n$, and sufficiently large $n$, we obtain

[TABLE]

if we fix $k$ satisfying $k+1>Q$.

Comparing the last estimate with Lemma 1 we obtain the following statement.

This theorem yields Theorem 2.

4. Chebyshev polynomials for uniformly perfect sets

We introduce some definitions and notations from geometric function theory. Let $K\subset{\bf R}$ be a uniformly perfect set satisfying

[TABLE]

The open (with respect to ${\bf R}$) set $I\setminus K$ consists of either a finite number $N\geq 1$ or an infinite number $N=\infty$ of disjoint open intervals, i.e.,

[TABLE]

where $(\alpha_{j},\beta_{j})\cap(\alpha_{k},\beta_{k})=\emptyset$ for $j\neq k$.

It follows immediately from (1.5) that $\Omega$ is regular (for the Dirichlet problem), see [23], [24], i.e., $g_{\Omega}$ extends continuously to $K$ and $g_{\Omega}(x):=0,\,x\in K$. Moreover, the Green function satisfies

[TABLE]

where constants $c_{1}$ and $\alpha$ could depend only on $\lambda_{K}$ from (1.5), see [15, Lemma 4.1] or [12, p. 119].

We need the Levin conformal mapping which can be defined as follows (for details, see [17], [2]). Consider the univalent in the upper half-plane ${\bf H}:=\{z:\Im z>0\}$ function

[TABLE]

where $\mu=\mu_{K}$ is the equilibrium measure for $K$. It maps ${\bf H}$ onto a vertical half-strip with $N$ slits parallel to the imaginary axis, i.e., the domain

[TABLE]

where $0<u_{j}=u_{j}(K)<\pi$ and $v_{j}=v_{j}(K)>0.$

The continuous extension of $\phi$ to $\overline{{\bf H}}$ satisfies the following boundary correspondence

[TABLE]

[TABLE]

[TABLE]

Note that in the last relation each point of $[u_{j},u_{j}+iv_{j})$ has two preimages on $[\alpha_{j},\beta_{j}]$.

The crucial fact is that $\phi$ satisfies

[TABLE]

For a horizontal crosscut $\gamma$ of $\Sigma_{K}$, i.e., an interval $\gamma=(a+ib,c+ib)\subset\Sigma_{k}$ with endpoints on $\partial\Sigma_{K}$, denote by $h(\gamma)$ its ”height”, that is, $h(\gamma):=b$.

[TABLE]

Proof. For convenience, let $u_{-1}:=0$ and $u_{0}:=\pi$. Let $\gamma=(u_{j}+ih(\gamma),u_{k}+ih(\gamma))$ and $R:=\{w=u+iv:u_{j}<u<u_{k},0<v<h(\gamma)\}.$ Denote by $\Gamma^{\prime}$ the family of crosscuts of $\Sigma_{K}\cap R$ which join $(u_{j},u_{k})$ to $\gamma$ and let $\Gamma^{*}$ be the family of crosscuts of the rectangle $R$ which join its horizontal boundary intervals. We refer to [1], [16], [12] for the basic properties of the module of a family of curves and arcs (such as conformal invariance, comparison principle, composition laws, etc.) We use these properties without further citation.

For the modules of $\Gamma^{\prime}$ and $\Gamma^{*}$ we have

[TABLE]

At the same time, we claim that for the module of $\Gamma:=\phi^{-1}(\Gamma^{\prime})$ the estimate

[TABLE]

holds.

Indeed, without loss of generality, we assume that $j,k\geq 1$ and $\beta_{j}-\alpha_{j}\leq\beta_{k}-\alpha_{k}$. The other particular cases may be handled in much the same way. Denote by $\Gamma_{1}$ the family of all crosscuts of

[TABLE]

which join $F_{1}:=K\cap[\beta_{j},2\beta_{j}-\alpha_{j}]$ with $[3\alpha_{j}-2\beta_{j},2\alpha_{j}-\beta_{j}]$. Since $\Gamma_{1}$ is “fewer and longer” than $\Gamma$, the comparison principle yields

[TABLE]

Note that by (1.5),

[TABLE]

Consider the conformal mapping of $G_{1}$ onto

[TABLE]

given by the function

[TABLE]

with the boundary correspondence

[TABLE]

[TABLE]

Since for $\beta_{j}\leq x_{1}<x_{2}\leq 2\beta_{j}-\alpha_{j}$,

[TABLE]

by the Fekete-Szegő Theorem (see [23, p. 153]) and (4.9) for the set $F_{2}:=f(F_{1})$ we have

[TABLE]

Furthermore, let $\Gamma_{2}:=f(\Gamma_{1})$ and denote by $\Gamma_{3}$ the family of all crosscuts of the annulus $\{\tau:r_{0}<|\tau|<1\}$ which join $F_{3}:=F_{2}\cup\overline{F_{2}}$, where $\overline{F_{2}}:=\{\tau:\overline{\tau}\in F_{2}\}$, with the circular boundary component $\{\tau:|\tau|=r_{0}\}$. By the symmetry principle $m(\Gamma_{3})=2m(\Gamma_{2})$. Now we apply Pfluger’s theorem (see [22, p. 212]) to obtain

[TABLE]

Therefore, the conformal invariance of the module yields

[TABLE]

which together with (4.8) implies (4.7).

At last, by virtue of the conformal invariance of the module, as well as (4.6) and (4.7), we have (4.5) with $c_{2}:=c_{3}^{-1}$.

$\Box$

Let now $1\leq N<\infty$. According to [34], for $n\in{\bf N}$, either $\Phi_{\Omega}(z)^{n}$ is single-valued or it is multiple-valued. In the first case, we set $W_{n}(z):=\Phi_{\Omega}(z)^{n}$ and in the second case there exist $q\leq N$ points $x_{1,n},\ldots,x_{q,n}\in I\setminus K$, such that

[TABLE]

is single-valued in $\Omega$. According to [34, pp. 159, 211] each complementary interval $(\beta_{j},\alpha_{j+1})$ cannot have more than one point from $\{x_{l,n}\}$.

Let polynomials $F_{n}(z)=F_{n}(z,K)$ be defined as in Section 2, i.e.,

[TABLE]

where $C_{n}\subset\Omega\setminus\{\infty\}$ is a Jordan curve, oriented in the positive direction, containing $K$ and $z$ in its interior.

By the Cauchy formula, for $z\in\Omega\setminus\{\infty\}$ and sufficiently small $t>0$, we have

[TABLE]

where $\tilde{K}_{t}:=\{\zeta\in\Omega:d(\zeta,K)=t\}$ consists of $N+1$ disjoint curves each surrounding exactly one component of $K$.

Passing to the limit, we obtain for $z\in\Omega$ with $|z|<2$,

[TABLE]

Here,

[TABLE]

According to (4.2) and (4.10), for $z$ with the property $g_{\Omega}(z)=1/n$, we have the inequality

[TABLE]

which by the maximum principle for $F_{n}$ in $K_{1/n}$ is also true for $z\in K$.

Note that $F_{n}(z)=\alpha_{n}z^{n}+\ldots$, where as in (2.2)

[TABLE]

Therefore, by (4.4),

[TABLE]

where

[TABLE]

and $v_{j}=v_{j}(K)$ are defined by (4.3).

Proof of Theorem 3. Applying linear transformation if necessary we always can assume that $K$ satisfies (4.1). By virtue of Theorem 2, the only nontrivial case arises when $K$ consists of infinitely many components. Consider

[TABLE]

It is worth pointing out that $K^{*}_{n}$ is uniformly perfect with $\lambda(K^{*}_{n})=\lambda(K)$. Moreover, by Lemma 3, $K_{n}^{*}$ consists of $N+1=N(K,n)+1\leq c_{5}n$ disjoint closed intervals and

[TABLE]

Let $F_{n}(z)=F_{n}(z,K_{n}^{*})$ be the Faber-Widom polynomial as above (constructed for $K_{n}^{*}$ instead of $K$). Denote by $v_{n,j}:=v_{j}(K_{n}^{*}),j=1,\ldots,N$, the quantities $v_{j}$ defined by (4.3) for $K_{n}^{*}$ instead of $K$. Note that

[TABLE]

For sufficiently large $n$, consider the sets

[TABLE]

[TABLE]

Since the number of elements in $\Lambda_{0}$ is at most $c_{5}n$ and by Lemma 3 the number of elements in $\Lambda_{k}$ is at most $c_{7}n2^{-k},$ we obtain

[TABLE]

Therefore, by (4.11) and (4.12)

[TABLE]

which implies (1.6).

$\Box$

Acknowledgements. Part of this work was done during the Fall of 2016 semester, while the author visited the Katholische Universität Eichstätt-Ingolstadt and the Julius Maximilian University of Würzburg. The author is also grateful to F. Nazarov for his helpful comments.