Asymptotic Growth of Moduli of m-th Derivatives of Algebraic Polynomials in Weighted Bergman Spaces on Regions Without Zero Angles

Abstract

In this paper, we study asymptotic bounds on the m-th derivatives of general algebraic polynomials in weighted Bergman spaces. We consider regions in the complex plane defined by bounded, piecewise, asymptotically conformal curves with strictly positive interior angles. We first establish asymptotic bounds on the growth in the exterior of a given unbounded region. We then extend our analysis to the closures of the region and derive the corresponding growth bounds. Combining these bounds with those for the corresponding exterior, we obtain comprehensive bounds on the growth of the m-th derivatives of arbitrary algebraic polynomials in the whole complex plane.

1. Introduction

Let $\mathbb{C}$ denote the complex plane; $G\subset \mathbb{C}$, with $0\in G$, be a finite region bounded by the Jordan curve $\Gamma :=\partial G$; $\Omega :=\overline{\mathbb{C}}\setminus \overline{G}=ext\Gamma$, where $\overline{\mathbb{C}}:=\mathbb{C}\cup \left\{\infty \right\}$ represents the extended complex plane. For $t\in \mathbb{C}$ and $\delta >0,$ let $\Delta (t,\delta ):=\left\{w\in \mathbb{C}:\left|w-t\right|>\delta \right\};$ $\Delta :=\Delta (0,1)$. Let $\Phi :\Omega \to \Delta$ be the univalent conformal mapping normalized by $\Phi (\infty )=\infty$ and ${lim}_{z\to \infty }{\displaystyle \frac{\Phi \left(z\right)}{z}}>0$; $\Psi :={\Phi }^{-1}$. For $t\ge 1$, the sets ${\Gamma }_t$, $G_t$ and ${\Omega }_t$ are defined as follows:

$${\Gamma }_t:=\left\{z:\left|\Phi \left(z\right)\right|=t\right\},{\Gamma }_1\equiv \Gamma ;$$

$$G_t:=int{\Gamma }_t:=G\cup \left\{z\in \mathbb{C}:1\le \left|\Phi \left(z\right)\right|<t\right\},{\Omega }_t:=ext{\Gamma }_t:=\left\{z\in \mathbb{C}:\left|\Phi \left(z\right)\right|>t\right\}.$$

For $z\in \mathbb{C}$ and some set $S\subset \mathbb{C},$ let

$$d(z,S):=dist(z,S)=inf\left\{\left|\zeta -z\right|:\zeta \in S\right\}.$$

We introduce the notation ${\wp }_n$ to designate the family of all algebraic polynomials $P_n\left(z\right)$, expressible as $P_n\left(z\right)={\sum }_{j=0}^n{a}_j{z}^j$, with complex coefficients $a_j$; their degree does not exceed n (that is, $deg{P}_n\le n$), where n is a natural number, $n\in \mathbb{N}$.

Let ${\left\{z_j\right\}}_{j=1}^l$ be a fixed, finite collection of distinct points located on $\Gamma$, ordered sequentially, without loss of generality, in the positive orientation. For a given constant $R_0$ satisfying $1<R_0<\infty$, consider the generalized Jacobi weight function defined by

$$h\left(z\right):=\left\{\begin{array}{cc}{h}_0\left(z\right){\displaystyle \prod _{j=1}^l}{\left|z-z_j\right|}^{{\gamma }_j},& z\in {\overline{G}}_{R_0},\\ 0,& z\in \mathbb{C}\setminus {\overline{G}}_{R_0},\end{array}\right.$$

(1)

where ${\gamma }_j>-2$ for each $j=1,2,\dots ,l$. Here, $h_0$ is a measurable function satisfying the inequality $h_0\left(z\right)\ge c_0(G,h)>0$ for all $z\in G_{R_0}$, where $c_0(G,h)$ is a positive constant depending only on G and h. Let $0<p\le \infty$ and $\sigma$ be the two-dimensional Lebesgue measure. For the Jordan region $G,$we introduce:

$$\begin{array}{ccc}{∥P_n∥}_p\hfill & :& ={∥P_n∥}_{A_p(h,G)}:={\left(\underset{G}{\int \int }h\left(z\right){\left|P_n\left(z\right)\right|}^{p}d{\sigma }_z\right)}^{1/p},0<p<\infty ,\hfill \\ {∥P_n∥}_{\infty }\hfill & :& ={∥P_n∥}_{A_{\infty }(1,G)}:=\underset{z\in \overline{G}}{max}\left|P_n\left(z\right)\right|,p=\infty ;A_p(1,G)=:A_p\left(G\right),\hfill \end{array}$$

(2)

and

$$\begin{array}{ccc}{∥P_n∥}_{{\mathcal{L}}_p(h,\Gamma )}\hfill & :& ={\left(\underset{\Gamma }{\int }h\left(z\right){\left|P_n\left(z\right)\right|}^p\left|dz\right|\right)}^{1/p}<\infty ,0<p<\infty ,\hfill \\ {∥P_n∥}_{{\mathcal{L}}_{\infty }(1,\Gamma )}\hfill & :& =\underset{z\in \Gamma }{max}\left|P_n\left(z\right)\right|,p=\infty ;{\mathcal{L}}_p(1,\Gamma )=:{\mathcal{L}}_p(\Gamma ),\hfill \end{array}$$

(3)

when $\Gamma$ is rectifiable.

In the theory of polynomial approximations of functions of a complex variable, the Bernstein–Walsh inequality occupies a significant place. It gives an upper bound on the growth of polynomials outside a given region and is expressed as follows [1]:

$${∥P_n∥}_{C\left({\overline{G}}_R\right)}\le {\left|\Phi \left(z\right)\right|}^n{∥P_n∥}_{C\left(\overline{G}\right)},\forall P_n\in {\wp }_n.$$

(4)

In particular, inequality (4) establishes that the enlargement of the region G to the extended region ${\overline{G}}_{1+c{n}^{-1}}$, where $c:=c\left(G\right)>0$, denotes a positive constant. It results in, at most, a constant-factor increase in ${∥P_n∥}_{\infty }$, while preserving the asymptotic growth rate with respect to n. The corresponding “symmetric” form of inequality (4) in the space ${\mathcal{L}}_p(\Gamma )$, associated with the constant function $h\left(z\right)\equiv 1$, is derived as follows [2]:

$${∥P_n∥}_{{\mathcal{L}}_p\left({\Gamma }_R\right)}\le {\left|\Phi \left(z\right)\right|}^{n+{\displaystyle \frac{1}{p}}}{∥P_n∥}_{{\mathcal{L}}_p(\Gamma )},\forall P_n\in {\wp }_n,p>0.$$

Subsequently, this estimate was generalized in ([3], Lemma 2.4) to the weighted space ${\mathcal{L}}_p(h,\Gamma )$, where the weight function is defined by (1) and the exponents ${\gamma }_j$ satisfy ${\gamma }_j>-1$ for all $j=1,2,\dots ,l$:

$${∥P_n∥}_{{\mathcal{L}}_p(h,{\Gamma }_R)}\le R^{n+{\displaystyle \frac{1+{\gamma }^{\ast }}{p}}}{∥P_n∥}_{{\mathcal{L}}_p(h,\Gamma )},{\gamma }^{\ast }=max\left\{0;{\gamma }_j:1\le j\le l\right\}.$$

(5)

An analog of the inequalities (4) and (5) in the space $A_p(h,G)$ was provided in [4] in the following form:

$${∥P_n∥}_{A_p(h,G_R)}\le c_1{R}^{{\ast }^{n+{\displaystyle \frac{1}{p}}}}{∥P_n∥}_{A_p(h,G)},R>1,p>0,$$

where G is a quasidisk (for the definition, see Section 2); $R^{*}:=1+c_2(R-1)$, with $c_2>0$ and $c_1=c_1(G,p,c_2)>0$ are constants independent of n and $R.$ In (([5], Theorem 1.1) the result was further extended and generalized to arbitrary Jordan regions G under the normalization $h\left(z\right)\equiv 1$:

$$\parallel P_n{\parallel }_{A_p\left(G_R\right)}\le c_3{R}^{n+{\displaystyle \frac{2}{p}}}{\parallel P_n\parallel }_{A_p\left(G_{R_1}\right)},\mathrm{for}R>R_1=1+{\displaystyle \frac{1}{n}},p>0,$$

where the constant

$$c_3={\left({\displaystyle \frac{2}{e^p-1}}\right)}^{{\displaystyle \frac{1}{p}}}\left(1+O\left({\displaystyle \frac{1}{n}}\right)\right),\mathrm{as}n\to \infty ,$$

is asymptotically sharp. N. Stylianopoulos [6] considered a modification of inequality (4) by replacing the norm ${∥P_n∥}_{C\left(\overline{G}\right)}$ with ${∥P_n∥}_{A_2\left(G\right)}$, and derived an alternative formulation valid for rectifiable quasicircles $\Gamma$ and arbitrary polynomials $P_n\in {\wp }_n$, as follows:

$$\left|P_n\left(z\right)\right|\le c_4{\displaystyle \frac{\sqrt{n}}{d(z,\Gamma )}}{∥P_n∥}_{A_2\left(G\right)}{\left|\Phi \left(z\right)\right|}^{n+1},z\in \Omega ,$$

(6)

where a constant $c_4=c_4(\Gamma )>0$ depends only on $\Gamma .$

This approach enables the study of the growth of $\left|P_n\left(z\right)\right|$ over closed bounded regions, unbounded regions, and consequently throughout the whole complex plane, with respect to the norm ${∥P_n∥}_p$. A related inequality for points $z\in G$ can be obtained by applying the mean value theorem: for each $p>0$, there exists a constant $c_5=c_5\left(G\right)>0$, independent of n and z, such that for any Jordan region G and polynomial $P_n\in {\wp }_n$,

$$\left|P_n\left(z\right)\right|\le c_5{\left({\displaystyle \frac{1}{d(z,\Gamma )}}\right)}^{2/p}{∥P_n∥}_{A_p\left(G\right)},z\in G.$$

(7)

In particular, for the Bergman polynomials, $K_n\left(z\right)=K_n(z,h,G)$ satisfies $deg{K}_n=n$, which are orthonormal with respect to a weight function h on G. Similar estimates have been established (see [7,8,9,10]). Specifically, in the weighted space $A_2(h,G)$, the following bound holds:

$$\left|K_n\left(z\right)\right|\le c_6{\displaystyle \frac{n^{-\beta }}{d^{\alpha }(z,\Gamma )}},z\in G,$$

where $c_6$, $\alpha =\alpha (G,h)>0$, and $\beta =\beta (G,h)>0$ are constants depending only on G and h. By combining inequalities (6) and (7), we thus obtain a global estimate for the growth of $\left|P_n\left(z\right)\right|$ in $\mathbb{C}\setminus \Gamma$.

In this paper, we further advance the line of inquiry initiated in [11,12], which concerns the derivation of pointwise estimates for the derivatives $\left|P_n^{\left(m\right)}\left(z\right)\right|$, with $m=1,2,\dots$, within certain classes of unbounded regions characterized by nonzero interior angles. More precisely, we aim to establish inequalities of the following form:

$$\left|P_n^{\left(m\right)}\left(z\right)\right|\le {\eta }_n{∥P_n∥}_p,z\in \Omega ,$$

(8)

where the function ${\eta }_n={\eta }_n(G,h,p,m,z)$ exhibits a growth to infinity as $n\to \infty$, with its asymptotic behavior intimately governed by the geometric features of the region G and the analytic properties of the associated weight function h. Analogous results of the type (8) for some norms and for different unbounded regions were obtained by N.A. Lebedev, P.M. Tamrazov, V.K. Dzjadyk (see, for example, [13], pp. 418–428), [14], p. 383, F. G. Abdullayev et al. [15] (for $m=0,$ $p>1$ and regions with piecewise Dini-smooth boundary), (for $m=0,$ $p>0$ and regions bounded by asymptotically conformal curve), [16] (for $m=0,$ $p>0$ and regions with piecewise smooth boundary with interior angles), [17] (for $m=0,$ $p>0$ and regions bounded by piecewise asymtotically conformal curves) and the references cited therein. Similar problems were studied with respect to the space ${\mathcal{L}}_p(h,G)$ in [18] (for $m=0,$ $p>0;$ $\Gamma -$rectifiable asymptotically conformal curve). For the $m\ge 1$, the estimates of the type (8) were investigated in [12] (for $m\ge 1,$ $p>1$ and quasidisks with an additional functional condition), [19] (for $m=1,$ $p>1$ and regions with piecewise smooth boundary with interior zero and nonzero angles) and the references cited therein. Note that these works relied on a recurrence formula to mean that the inequality for each derivative was derived by estimating the previous derivative.

In this study, the estimate of the inequality of the type (8) is obtained without using the recurrence formula.

To further investigate the behavior of $\left|P_n^{\left(m\right)}\left(z\right)\right|$ throughout the complex plane, it is necessary to consider a Bernstein–Markov–Nikolskii-type inequality for the specified regions:

$$\parallel P_n^{\left(m\right)}{\parallel }_{\infty }\le {\lambda }_n{\parallel P_n\parallel }_p,m=1,2,\dots ,$$

(9)

where ${\lambda }_n:={\lambda }_n(G,h,p,m)>0$ is a constant that tends to infinity as $n\to \infty$, and depends on the characteristics of the region G and the weight function h. The study of inequalities of the form (9) began with the seminal works [20,21,22]. Similar investigations were later extended in a number of publications. Recently, such inequalities have been examined for $m\ge 0$ and various functional spaces, as seen in [13] (pp. 257–263), [23,24,25,26] (pp. 122–133), [27,28,29,30,31], and others. Further research includes [32] (for $h\equiv 1$, $m=0$, $p>0$, and regions bounded by piecewise asymptotically conformal curves), [33] (for $m\ge 1$, $p>0$, and $\kappa$-quasidisks), [34] (for $m=0$, $p>0$, and regions with piecewise smooth boundaries containing interior cusps), [16] (for $m=0$, $p>0$, and regions with piecewise smooth boundaries with interior angles), and [35] (for $m\ge 1$, $p>1$, and regions with piecewise Dini-smooth boundaries having interior zero angles).

In this context, by combining the inequalities (8) and (9), we establish an upper bound for the growth of the m-th derivatives $\left|P_n^{\left(m\right)}\left(z\right)\right|$, where $m=1,2,\dots$, across the whole complex plane. Specifically, the growth of these derivatives is governed by the following inequality:

$$\left|P_n^{\left(m\right)}\left(z\right)\right|\le c_7{∥P_n∥}_p\left\{\begin{array}{cc}{\lambda }_n,& z\in {\overline{G}}_R,\\ {\eta }_n,& z\in {\Omega }_R,\end{array}\right.$$

(10)

where $R=1+c{n}^{-1}$, with $0<c<1$, and $c_7=c_7(G,p)>0$ are constants that do not depend on n, h, or $P_n$. Furthermore, we have ${\lambda }_n\to \infty$ and ${\eta }_n\to \infty$ as $n\to \infty$. It is important to note that these constants depend on the geometric properties of the region G and the weight function h.

2. The Regions

Throughout this paper, the symbols $c,c_0,c_1,c_2,\dots$ represent positive constants, and the symbols ${\epsilon }_0,{\epsilon }_1,{\epsilon }_2,\dots$ represent sufficiently small positive constants, which may differ across relations. These constants may depend on G in general and, on parameters that are not essential for the argument; otherwise, any such dependence will be explicitly stated. For any $k\ge 0$ and $m>k$, with $m,k\in \mathbb{N}\cup \left\{0\right\},$ the notation $i=\overline{k,m}$ means $i=k,k+1,\dots ,m$.

For any $\delta >0$ and arbitrary $t,w\in \mathbb{C}$, let $B(w,\delta ):=$ $\left\{t:\left|t-w\right|<\delta \right\},$ and let $\phi :G\to$ $B:=B(0,1)$ be a conformal and univalent map, normalized by $\phi \left(0\right)=0$ and ${\phi }^{\prime }\left(0\right)>0.$ Also, let $\psi :={\phi }^{-1}.$

Following ([36], p. 97) (see also [37]), a Jordan curve (or arc) $\Gamma$ is called K-quasiconformal ($K>1$) if there exists a K-quasiconformal mapping f of the region $D\supset \Gamma$ such that $f(\Gamma )$ is a circle (or a line segment). There are other equivalent definitions of quasiconformal curves (quasicircles) in the literature (see, for example, [38], p. 81; [39], p. 107). Here, we present the geometric definition.

Let S be a Jordan curve, and let $z=z\left(s\right),$ with $s\in [0,$ $\left|S\right|],$ denote the natural representation of S where $\left|S\right|:=mesS$. Let $z_1,z_2\in S$ be arbitrary points, and let $S(z_1,z_2)\subset S$ denote the subarc of S with endpoints $z_1$ and $z_2$, having the shorter diameter. The curve S is a quasicircle if and only if the “three-point property”

$$\underset{z_1,z_2\in S;z\in S(z_1,z_2)}{max}{\displaystyle \frac{\left|z_1-z\right|+\left|z-z_2\right|}{\left|z_1-z_2\right|}}<\infty$$

(11)

is satisfied ([38], p. 81; [39], p. 107).

Now, using (11), we define a new class of curves to be considered in this paper. A Jordan curve S is called asymptotically conformal [39,40], if the property

$$\underset{z_1,z_2\in S;z\in S(z_1,z_2)}{max}{\displaystyle \frac{\left|z_1-z\right|+\left|z-z_2\right|}{\left|z_1-z_2\right|}}\to 1, \left|z_1-z_2\right|\to 0,$$

(12)

is satisfied. Denote by $AC$ the class of asymptotically conformal curves. We say that $G\in AC$ if $\Gamma :=\partial G$ is an asymptotically conformal curve.

Various properties of asymptotically conformal curves have been studied in ([39], pp. 246–250), [41,42,43,44], and the references provided therein. As can be seen from (12), the value on the right side is a constant on ellipses with foci at points $z_1$ and $z_2$. According to the (11) “three point property”, every asymptotically conformal curve is quasiconformal. Every smooth curve is an asymptotically conformal curve, but corners are not allowed. Asymptotic conformal curves can be even more pathologically complicated. For example, they can contain “horn” arcs with infinite turns. Another complicated example of asymptotically conformal curves can be found in ([39], pp. 246–250). Quasicircles, as well as asymptotically conformal curves, can be non-rectifiable (see, for example, [45], p. 146; [46], p. 42).

A Jordan arc ℓ is called an asymptotically conformal arc if it is part of some asymptotically conformal curve.

We will here introduce a new class of regions bounded by a piecewise asymptotically conformal curve that has non-zero “angles” at the points of connection of the boundary arcs for consideration.

Definition 1 

([17]). We say that a Jordan region $G\in PAC({\nu }_1,...,{\nu }_l),0<{\nu }_j<2,j=\overline{1,l},$ if $\Gamma =\partial G$ is the union of finitely many asymptotically conformal arcs ${\left\{{\Gamma }_j\right\}}_{j=1}^l,$ connected at the points ${\left\{{\zeta }_j\right\}}_{j=1}^l\in \Gamma$ such that the curve Γ is locally asymptotically conformal at the point ${\zeta }_0\in \Gamma \setminus {\left\{{\zeta }_j\right\}}_{j=1}^l$ and, for each junction point ${\zeta }_j\in \Gamma ,$ $j=\overline{1,l},$ where two arcs ${\Gamma }_{j-1}$ and ${\Gamma }_j$, there exists $r_j:=r_j(\Gamma ,{\zeta }_j)>0$ and ${\nu }_j:={\nu }_j(\Gamma ,{\zeta }_j),0<{\nu }_j<2,$ such that, for some $0\le {\theta }_0<2$, a closed maximal wide circular sector

$$S({\zeta }_j;r_j,{\nu }_j):=\left\{\xi :\xi ={\zeta }_j+r_j{e}^{i\theta \pi },{\theta }_0<\theta <{\theta }_0+{\nu }_j\right\}$$

of radius $r_j$ and opening ${\nu }_j\pi$ lies in $\overline{G}=\overline{int\Gamma }$ with vetrex at the point ${\zeta }_j.$

Definition 2. 

We say that a Jordan region $G\in PAC\left(\nu \right),$ if $G\in PAC({\nu }_1,...,{\nu }_l),$ $0<{\nu }_j<2,j=\overline{1,l},$ where $\nu =min\{{\nu }_j:0<{\nu }_j<2,j=\overline{1,l}\}.$

It is clear from Definition 1 (Definition 2), that every region $G\in PAC({\nu }_1,\dots ,{\nu }_l),$ where $0<{\nu }_j<2,$ for $j=\overline{1,l},$ (or $G\in PAC\left(\nu \right)$) may have “singularity” at the junction points ${\left\{z_i\right\}}_{i=1}^l\in \Gamma$. If the region does not have such a “singularity” (in which case we set ${\nu }_i=1,$ for $i=\overline{1,l}$), then $PAC(1,\dots ,1)\equiv AC$. By the notation $G\in PAC({\nu }_1,\dots ,{\nu }_l),$ where $0<{\nu }_j\le 1,$ for $j=\overline{1,l},$ we mean that there is at least one $j_0$ with $0<j_0<l$, such that $0<{\nu }_{j_0}<1$. If no such $j_0$ exists, then $G\in AC.$ Furthermore, from this definition, it immediately follows that $PAC({\nu }_1,{\nu }_2,...,{\nu }_j,...,{\nu }_l)\subset PAC({\nu }_1,{\nu }_2,...,{\tilde{\nu }}_j,...,{\nu }_l),$ if ${\tilde{\nu }}_j\ge {\nu }_j,$ for any $j=\overline{1,l}.$

According to the (11), every piecewise asymptotically conformal curve (without any cusps) is quasiconformal ([38], p. 100).

Throughout this work, we will assume that the points ${\left\{z_i\right\}}_{i=1}^l\in \Gamma$, as defined in (1), and ${\left\{{\zeta }_i\right\}}_{i=1}^l\in \Gamma$, as defined in Definition 1, coincide. Without loss of generality, we also assume that the points ${\left\{z_i\right\}}_{i=1}^l$ are ordered in the positive direction on the curve $\Gamma .$

3. Main Results

In this section, we will summarize our results. Assume that the curve $\Gamma$ has a “singularity” at each of its boundary points ${\left\{{\zeta }_j\right\}}_{j=1}^l$, i.e., $0<{\nu }_j<1,$ for all $j=\overline{1,l,}$ and that the weight function $h$ has a “singularity” at each of the same boundary points, i.e., ${\gamma }_j\ne 0$ for some $j=\overline{1,l}.$ First, we will study the estimation of the type (8) for the region $G\in PAC\left(\nu \right)$. Next, taking into account estimates of the type (9) for a closed region $\overline{G}\in PAC\left(\nu \right)$, we will find estimates of the type (10) for the whole complex plane.

Before we formulate the main results, let us introduce some notations that we will need. For $0<{\delta }_j<{\delta }_0:={\displaystyle \frac{1}{4}}min\left\{\left|z_i-z_j\right|:i,j=1,2,...,l,i\ne j\right\}$, let $\Omega (z_j,{\delta }_j):=\Omega \cap \left\{z:\left|z-z_j\right|\le {\delta }_j\right\};$ $\delta :={\displaystyle \underset{1\le j\le l}{min}}{\delta }_j;$ Let $w_j:=\Phi \left(z_j\right),$ ${\phi }_j:=arg{w}_j.$ Without loss of generality, we will assume that ${\phi }_l<2\pi .$ For $\eta :=min\left\{{\eta }_j,j=\overline{1,l}\right\},$ where ${\eta }_j={\displaystyle \underset{t\in \partial \Phi (\Omega (z_j,{\delta }_j))}{min}}\left|t-w_j\right|>0,$ let us set:

$$\begin{array}{ccc}{\Delta }_j\left({\eta }_j\right)\hfill & :& =\left\{t:\left|t-w_j\right|\le {\eta }_j\right\}\subset \Phi (\Omega (z_j,{\delta }_j)),\hfill \\ \Delta \left(\eta \right)\hfill & :& =\bigcup _{j=1}^l{\Delta }_j\left(\eta \right),{\widehat{\Delta }}_j=\Delta \setminus \Delta \left({\eta }_j\right);\widehat{\Delta }\left(\eta \right):=\Delta \setminus \Delta \left(\eta \right);{\Delta }_1^{\prime }:={\Delta }_1^{\prime }\left(1\right),\hfill \\ {\Delta }_1^{\prime }\left(\rho \right)\hfill & :& =\left\{t=\rho e^{i\theta }:\rho \ge 1,{\displaystyle \frac{{\phi }_0+{\phi }_1}{2}}\le \theta <{\displaystyle \frac{{\phi }_1+{\phi }_2}{2}}\right\},\hfill \\ {\Delta }_j^{\prime }\hfill & :& ={\Delta }_j^{\prime }\left(1\right),{\Delta }_j^{\prime }\left(\rho \right):=\left\{t=\rho e^{i\theta }:\rho \ge 1,{\displaystyle \frac{{\phi }_{j-1}+{\phi }_j}{2}}\le \theta <{\displaystyle \frac{{\phi }_j+{\phi }_0}{2}}\right\};j=\overline{2,l}\hfill \end{array}$$

where ${\phi }_0=2\pi -{\phi }_l;{\Omega }_{\rho }^j:=\Psi \left({\Delta }_j^{\prime }\left(\rho \right)\right),{\Gamma }_{\rho }^j:={\Gamma }_{\rho }\cap {\Omega }_{\rho }^j;{\Omega }_{\rho }=\bigcup _{j=1}^l{\Omega }_{\rho }^j,\rho \ge 1.$

Throughout this paper, we denote by $R:=1+{\displaystyle \frac{{\epsilon }_0}{n}};$

$${\gamma }^{\ast }:=max\left\{0;{\gamma }_j^{\ast }\right\},{\gamma }_j^{\ast }:=max\left\{0;{\gamma }_j\right\},j=\overline{1,l};$$

(13)

$${\nu }_j^{\ast }:=\left\{\begin{array}{cc}2-{\nu }_j,& \mathrm{if}0<{\nu }_j<1,\\ 1+\epsilon ,& \mathrm{if}{\nu }_j=1,\end{array}\right.;{\tilde{\nu }}^{\ast }:=min\left\{{\nu }_j^{\ast }\right\},{\nu }^{\ast }:=\left\{\begin{array}{cc}2-\nu ,& \mathrm{if}0<\nu <1,\\ 1+\epsilon ,& \mathrm{if}\nu =1,\end{array}\right.;$$

(14)

for all $j=\overline{1,l}.$

Theorem 1. 

Let $p\ge 2;$ $G\in PAC({\nu }_1,...,{\nu }_l),$ $0<{\nu }_j\le 1,j=\overline{1,l},$ and $h\left(z\right)$ be defined by (1). Then, for any $P_n\in {\wp }_n,n\in \mathbb{N},$ $n\ge m,$ $m=1,2,...,$

$$\left|P_n^{\left(m\right)}\left(z\right)\right|\le c_1{\displaystyle \frac{{\left|\Phi \left(z\right)\right|}^{n-m+1}}{d(z,\Gamma )}}{J}_{n,p}^1\left(m\right){∥P_n∥}_p,z\in {\Omega }_R^j,$$

(15)

holds, where $c_1=c_1(G,\gamma ,m,p)>0$ is a constant independent of n and $z;$

$$J_{n,p}^1\left(m\right):=\left\{\begin{array}{ccc}{n}^{\left({\displaystyle \frac{{\gamma }_j+2}{p}}+m-1\right){\nu }_j^{\ast }},& 2\le p<1+{\displaystyle \frac{{\nu }_j^{\ast }}{{\nu }_j^{\ast }+1}}\left({\gamma }_j+1\right),& {\gamma }_j>{\displaystyle \frac{1}{{\nu }_j^{\ast }}},\\ n^{m{\nu }_j^{\ast }}{\left(nlnn\right)}^{1-{\displaystyle \frac{1}{p}}},& p=1+{\displaystyle \frac{{\nu }_j^{\ast }}{{\nu }_j^{\ast }+1}}\left({\gamma }_j+1\right),& {\gamma }_j>{\displaystyle \frac{1}{{\nu }_j^{\ast }}},\\ n^{m{\nu }_j^{\ast }+1-{\displaystyle \frac{1}{p}}},& p>1+{\displaystyle \frac{{\nu }_j^{\ast }}{{\nu }_j^{\ast }+1}}\left({\gamma }_j+1\right),& {\gamma }_j>{\displaystyle \frac{1}{{\nu }_j^{\ast }}},\\ n^{m{\nu }_j^{\ast }+1-{\displaystyle \frac{1}{p}}},& p\ge 2,& -2<{\gamma }_j\le {\displaystyle \frac{1}{{\nu }_j^{\ast }}}.\end{array}\right.$$

Theorem 2. 

Let $1\le p<2;$ $G\in PAC({\nu }_1,...,{\nu }_l),$ $0<{\nu }_j\le 1, j=\overline{1,l},$ and $h\left(z\right)$ be defined by (1). Then, for any $P_n\in {\wp }_n,n\in \mathbb{N},$ $n\ge m$, $m=1,2,...,$

$$\left|P_n^{\left(m\right)}\left(z\right)\right|\le c_2{\displaystyle \frac{{\left|\Phi \left(z\right)\right|}^{n-m+1}}{d(z,\Gamma )}}{J}_{n,p}^2\left(m\right){∥P_n∥}_p,z\in {\Omega }_R^j,$$

(16)

holds, where $c_2=c_2(G,\gamma ,m,p)>0$ is a constant independent of n and $z;$

$$J_{n,p}^2\left(m\right):=\left\{\begin{array}{ccc}{n}^{\left({\displaystyle \frac{{\gamma }^{\ast }+2}{p}}+m-1\right){\nu }^{\ast }},& 1\le p<1+{\displaystyle \frac{{\nu }_j^{\ast }}{{\nu }_j^{\ast }+1}}\left({\gamma }_j^{\ast }+1\right),& -2<{\gamma }_j\le {\displaystyle \frac{1}{{\nu }_j^{\ast }}},\\ n^{m{\nu }^{\ast }}{\left(nlnn\right)}^{1-{\displaystyle \frac{1}{p}}},& p=1+{\displaystyle \frac{{\nu }_j^{\ast }}{{\nu }_j^{\ast }+1}}\left({\gamma }_j^{\ast }+1\right),& -2<{\gamma }_j\le {\displaystyle \frac{1}{{\nu }_j^{\ast }}},\\ n^{m{\nu }^{\ast }+1-{\displaystyle \frac{1}{p}}},& 1+{\displaystyle \frac{{\nu }_j^{\ast }}{{\nu }_j^{\ast }+1}}\left({\gamma }_j^{\ast }+1\right)<p<2,& -2<{\gamma }_j\le {\displaystyle \frac{1}{{\nu }_j^{\ast }}},\\ n^{\left({\displaystyle \frac{{\gamma }_j+2}{p}}+m-1\right){\nu }_j^{\ast }},& 1\le p<2,& {\gamma }_j>{\displaystyle \frac{1}{{\nu }_j^{\ast }}}.\end{array}\right.$$

Theorems 1 and 2 give “local” estimates in the unbounded “circular half-sectors” ${\Omega }_R^j$, constructed for a point $z_j\in {\Gamma }^j.$ Combining estimates over all such circular half-sectors ${\Omega }_R^j,$ $j=\overline{1,l},$ we obtain global estimates for the whole ${\Omega }_R$.

Corollary 1. 

Let $p\ge 2;$ $G\in PAC\left(\nu \right),$ $0<\nu \le 1,$ and $h\left(z\right)$ be defined by (1). Then, for any $P_n\in {\wp }_n,n\in \mathbb{N},$ $n\ge m,$ $m=1,2,...,$

$$\left|P_n^{\left(m\right)}\left(z\right)\right|\le c_3{\displaystyle \frac{{\left|\Phi \left(z\right)\right|}^{n-m+1}}{d(z,\Gamma )}}{J}_{n,p}^3\left(m\right){∥P_n∥}_p,z\in {\Omega }_R,$$

(17)

holds, where $c_3=c_3(G,\gamma ,m,p)>0$ is a constant independent of n and $z;$

$$J_{n,p}^3\left(m\right):=\left\{\begin{array}{ccc}{n}^{\left({\displaystyle \frac{\gamma +2}{p}}+m-1\right){\nu }^{\ast }},& 2\le p<1+{\displaystyle \frac{{\nu }^{\ast }}{{\nu }^{\ast }+1}}\left(\gamma +1\right),& \gamma >{\displaystyle \frac{1}{{\nu }^{\ast }}},\\ n^{m{\nu }^{\ast }}{\left(nlnn\right)}^{1-{\displaystyle \frac{1}{p}}},& p=1+{\displaystyle \frac{{\nu }^{\ast }}{{\nu }^{\ast }+1}}\left(\gamma +1\right),& \gamma >{\displaystyle \frac{1}{{\nu }^{\ast }}},\\ n^{m{\nu }^{\ast }+1-{\displaystyle \frac{1}{p}}},& p>1+{\displaystyle \frac{{\nu }^{\ast }}{{\nu }^{\ast }+1}}\left(\gamma +1\right),& \gamma >{\displaystyle \frac{1}{{\nu }^{\ast }}},\\ n^{m{\nu }^{\ast }+1-{\displaystyle \frac{1}{p}}},& p\ge 2,& -2<\gamma \le {\displaystyle \frac{1}{{\nu }^{\ast }}}.\end{array}\right.$$

Corollary 2. 

Let $1\le p<2;$ $G\in PAC\left(\nu \right),$ $0<\nu \le 1,$ and $h\left(z\right)$ be defined by (1). Then, for any $P_n\in {\wp }_n,n\in \mathbb{N},$ $n\ge m$, $m=1,2,...,$

$$\left|P_n^{\left(m\right)}\left(z\right)\right|\le c_4{\displaystyle \frac{{\left|\Phi \left(z\right)\right|}^{n-m+1}}{d(z,\Gamma )}}{J}_{n,p}^4\left(m\right){∥P_n∥}_p,z\in {\Omega }_R,$$

(18)

holds, where $c_4=c_4(G,\gamma ,m,p)>0$ is a constant independent of n and $z;$

$$J_{n,p}^4\left(m\right):=\left\{\begin{array}{ccc}{n}^{\left({\displaystyle \frac{{\gamma }^{\ast }+2}{p}}+m-1\right){\nu }^{\ast }},& 1<p<1+{\displaystyle \frac{{\nu }^{\ast }}{{\nu }^{\ast }+1}}({\gamma }^{\ast }+1),& -2<\gamma \le {\displaystyle \frac{1}{{\nu }^{\ast }}},\\ n^{m{\nu }^{\ast }}{\left(nlnn\right)}^{1-{\displaystyle \frac{1}{p}}},& p=1+{\displaystyle \frac{{\nu }^{\ast }}{{\nu }^{\ast }+1}}({\gamma }^{\ast }+1),& -2<\gamma \le {\displaystyle \frac{1}{{\nu }^{\ast }}},\\ n^{m{\nu }^{\ast }+1-{\displaystyle \frac{1}{p}}},& 1+{\displaystyle \frac{{\nu }^{\ast }}{{\nu }^{\ast }+1}}({\gamma }^{\ast }+1)<p<2,& -2<\gamma \le {\displaystyle \frac{1}{{\nu }^{\ast }}},\\ n^{\left({\displaystyle \frac{\gamma +2}{p}}+m-1\right){\nu }^{\ast }},& 1<p<2,& \gamma >{\displaystyle \frac{1}{{\nu }^{\ast }}}.\end{array}\right.$$

Now, we can state estimates for $\left|P_n^{\left(m\right)}\left(z\right)\right|,$ $m\ge 1,$ in the bounded regions of the class $G\in PAC({\nu }_1,\dots ,{\nu }_l),$ $0<{\nu }_j\le 1,j=\overline{1,l}$.

Theorem 3. 

Let $p>0,G\in PAC({\nu }_1,...,{\nu }_l),$ $0<{\nu }_j\le 1, j=\overline{1,l},$ and $h\left(z\right)$ is defined as in (1). Then, for any $P_n\in {\wp }_n,n\in \mathbb{N}$, $n>m,$ $m\ge 0,$ and arbitrarily small $\epsilon >0$ there exists $c_5=c_5(G,p,{\gamma }_j,\epsilon )>0,$ such that

$$\left|P_n^{\left(m\right)}\left(z_j\right)\right|\le c_5{n}^{\left({\displaystyle \frac{2+{\gamma }_j}{p}}+m\right)(2-{\nu }_j^{\ast })}{∥P_n∥}_p,j=\overline{1,l},p\ge 1$$

(19)

and, consequently,

$${∥P_n^{\left(m\right)}∥}_{\infty }\le c_5{n}^{\left({\displaystyle \frac{2+{\gamma }^{\ast }}{p}}+m\right)(2-{\tilde{\nu }}^{\ast })}{∥P_n∥}_p,p>0,$$

(20)

where ${\gamma }^{\ast }$ and ${\tilde{\nu }}^{\ast }$ are defined as in (13) and (14), respectively.

Remark 1. 

The estimations (19) and (20) are sharp.

According to (4) (applied to the polynomial $Q_{n-m}\left(z\right):=P_n^{\left(m\right)}\left(z\right)$), the estimate (20) also holds for the points $z\in {\overline{G}}_R$ with a different constant. Therefore, by combining estimate (20) (for $z\in {\overline{G}}_R$) with estimates (17) and (18), we will obtain an estimate on the growth of $|P_n^{\left(m\right)}\left(z\right)|$ in the whole complex plane:

Theorem 4. 

Let $p\ge 2;$ $G\in PAC({\nu }_1,...,{\nu }_l),$ $0<{\nu }_j\le 1, j=\overline{1,l},$ and $h\left(z\right)$ be defined by (1). Then, for any $P_n\in {\wp }_n,n\in \mathbb{N},$ $j=\overline{1,l}$ and every $m=1,2,...,$

$$\left|P_n^{\left(m\right)}\left(z\right)\right|\le c_7{∥P_n∥}_p\left\{\begin{array}{cc}{n}^{\left({\displaystyle \frac{2+{\gamma }^{\ast }}{p}}+m\right)(2-{\tilde{\nu }}^{\ast })},& z\in {\overline{G}}_R,\\ {\displaystyle \frac{{\left|\Phi \left(z\right)\right|}^{n-m+1}}{d(z,\Gamma )}}{J}_{n,p}^1\left(m\right),& z\in {\Omega }_R,\end{array}\right.$$

where $c_7=c_7(G,\gamma ,m,p)>0$ is a constant independent from n and $z;$ $J_{n,p}^1\left(m\right)$ defined as in Theorem 1 for all $z\in {\Omega }_R.$

Theorem 5. 

Let $1\le p<2;$ $G\in PAC({\nu }_1,...,{\nu }_l),$ $0<{\nu }_j\le 1, j=\overline{1,l},$ and $h\left(z\right)$ be defined by (1). Then, for any $P_n\in {\wp }_n,n\in \mathbb{N},$ $j=\overline{1,l}$ and every $m=1,2,...,$

$$\left|P_n^{\left(m\right)}\left(z\right)\right|\le c_8{∥P_n∥}_p\left\{\begin{array}{cc}{n}^{\left({\displaystyle \frac{2+{\gamma }^{\ast }}{p}}+m\right)(2-{\tilde{\nu }}^{\ast })},& z\in {\overline{G}}_R,\\ {\displaystyle \frac{{\left|\Phi \left(z\right)\right|}^{n-m+1}}{d(z,\Gamma )}}{J}_{n,p}^2\left(m\right),& z\in {\Omega }_R,\end{array}\right.$$

where $c_8=c_8(G,\gamma ,m,p)>0$ is a constant independent from n and $z;$ $J_{n,p}^2\left(m\right)$ defined as in Theorem 2 for all $z\in {\Omega }_R.$

4. Some Auxiliary Results

Throughout this paper, we denote “$a⪯b$” and “$a\asymp b$” as being equivalent to $a\le cb$ and $c_1^{-1}a\le b\le c_{1}a,$ respectively, for some constants c and $c_1$.

Lemma 1 

([9]). Let G be a quasidisk, $z_1\in \Gamma ,$ $z_2,z_3\in \Omega \cap \{z:\left|z-z_1\right|⪯d(z_1,{\Gamma }_{r_0})\};$ $w_j=\Phi \left(z_j\right),$ $j=1,2,3.$ Then

(a) 

The statements $\left|z_1-z_2\right|⪯\left|z_1-z_3\right|$ and $\left|w_1-w_2\right|⪯\left|w_1-w_3\right|$ are equivalent. So are $\left|z_1-z_2\right|\asymp \left|z_1-z_3\right|$ and $\left|w_1-w_2\right|\asymp \left|w_1-w_3\right|.$

(b) 

If $\left|z_1-z_2\right|⪯\left|z_1-z_3\right|,$ then

$${\left|{\displaystyle \frac{w_1-w_3}{w_1-w_2}}\right|}^{\epsilon }⪯\left|{\displaystyle \frac{z_1-z_3}{z_1-z_2}}\right|⪯{\left|{\displaystyle \frac{w_1-w_3}{w_1-w_2}}\right|}^c,$$

where $\epsilon ,$ $0<\epsilon <1,$ $c,$ $c>1,$ $0<r_0<1$ are constants, depending on $G.$

Lemma 2. 

Let $G\in PAC({\nu }_1,...,{\nu }_l),0<{\nu }_j\le 1, j=\overline{1,l}.$ Then, for arbitrary small $\epsilon >0,$ there exist numbers ${\delta }_j<{\delta }_j^{\ast }<$ diam$\left(\overline{G}\right)$, $\overline{1,l},$ such that

$$\begin{array}{ccc}\left|\Phi \left(z\right)-\Phi \left(\xi \right)\right|\hfill & \le & M_j\left\{\begin{array}{cc}\hfill {\left|z-\xi \right|}^{{\displaystyle \frac{1}{2-{\nu }_j}}},& \hfill z,\xi \in \Omega (z_j,{\delta }_j),\\ \hfill k_j(z,\xi ){\left|z-\xi \right|}^{1-\epsilon },& \hfill z,\xi \in \Omega (z_j,{\delta }_j^{\ast })\setminus \overline{\Omega (z_j,{\delta }_j)},\\ \hfill {\left|z-\xi \right|}^{1-\epsilon },& \hfill z,\xi \in \Omega \setminus \overline{\Omega (z_j,{\delta }_j^{\ast })};\end{array}\right.\hfill \\ \left|\Psi \left(w\right)-\Psi \left(\tau \right)\right|\hfill & \le & N_j{\left|w-\tau \right|}^{1-\epsilon ,},w,\tau \in {\widehat{\Delta }}_j,\hfill \end{array}$$

(21)

holds, where $M_j=M_j\left(G\right)>0$, $N_j=N_j\left(G\right)>0$ are constants independent from $z,\xi$ and $w,\tau ;$

$$k_j(z,\xi )=max\left({\left|\xi -z_j\right|}^{{\displaystyle \frac{1}{2-{\nu }_j}}-1+\epsilon };{\left|z-z_j\right|}^{{\displaystyle \frac{1}{2-{\nu }_j}}-1+\epsilon }\right).$$

Proof. 

If $\Gamma$ has no “singular” points, (i.e., $\Gamma$ is an asymptotically conformal curve (${\nu }_j=1,j=\overline{1,l}$)) then, according to ([47], p. 342), the functions $\Phi$ and $\Psi$ belong to the $Lip\alpha$ class on $\overline{\Omega }$ and $\overline{\Delta }$ for all $\alpha <1$, respectively. Therefore,

$$\left|\Phi \left(z\right)-\Phi \left(\zeta \right)\right|\le M{\left|z-\zeta \right|}^{1-\epsilon },z,\zeta \in \overline{\Omega },$$

(22)

and

$$\left|\Psi \left(w\right)-\Psi \left(\tau \right)\right|\le N{\left|w-\tau \right|}^{1-\epsilon },\left|w\right|\ge 1,\left|\tau \right|\ge 1,$$

(23)

for arbitrarily small $\epsilon >0,$ where $M,N>0$ are constants depending only on $G.$

Let $\Gamma$ have “singular” points at ${\left\{z_j\right\}}_{j=1}^l,$ i.e., $G\in PAC({\nu }_1,...,{\nu }_l),0<{\nu }_j<1,j=\overline{1,l}.$ According to [47] (p. 342), we have $\left|\psi \left(w\right)-\psi \left(w_j\right)\right|\le c_1{\left|w-w_j\right|}^{{\nu }_j},0<{\nu }_j<1,$ for $w\in$ $\overline{B\cap U_{\delta }\left(w_j\right)}$. Consequently, using ([48], Th.1), we obtain

$$\left|\Phi \left(z\right)-\Phi \left(z_j\right)\right|\le c_2{\left|z-z_j\right|}^{{\displaystyle \frac{1}{2-{\nu }_j}}},z\in \Omega (z_j,{\delta }_j).$$

(24)

From (22) and (24), we find that $G\in Q_{1-\epsilon ,{\displaystyle \frac{1}{2-{\nu }_1}},...,{\displaystyle \frac{1}{2-{\nu }_l}}},$ ${\displaystyle \frac{1}{2}}<{\displaystyle \frac{1}{2-{\nu }_j}}<1,$ $j=\overline{1,l}$ ([49], Def. 2). Then, as follows from the definition of class $Q_{1-\epsilon ,{\displaystyle \frac{1}{2-{\nu }_1}},...,{\displaystyle \frac{1}{2-{\nu }_l}}}$, there exists ${\delta }_j<{\delta }_j^{\ast }<\mathrm{diam}\left(\overline{G}\right),$ $\overline{1,l},$ and $\forall \xi ,z\in \Omega ({\zeta }_j,{\delta }_j^{\ast })\setminus \overline{\Omega ({\zeta }_j,{\delta }_j)},$ is fulfilled estimation

$$\left|\Phi \left(z\right)-\Phi \left(\xi \right)\right|\le k_j(z,\xi ){\left|z-\xi \right|}^{1-\epsilon },\forall \epsilon >0,$$

where

$$k_j(z,\xi )=c_{j}max\left({\left|\xi -z_j\right|}^{{\displaystyle \frac{1}{2-{\nu }_j}}-1+\epsilon };{\left|z-z_j\right|}^{{\displaystyle \frac{1}{2-{\nu }_j}}-1+\epsilon }\right),$$

and the constants $c_j$ are independent of z and $\xi .$ □

For $\left|{\Psi }^{\prime }\left(\tau \right)\right|$, we will use the following two-sided estimate ([50], Th.2.8):

$$\left|{\Psi }^{\prime }\left(\tau \right)\right|\asymp {\displaystyle \frac{d(\Psi \left(\tau \right),\Gamma )}{\left|\tau \right|-1}}.$$

(25)

Lemma 3. 

Let $G\in PAC({\nu }_1,...,{\nu }_l),0<{\nu }_j<2,j=\overline{1,l},$ and $h\left(z\right)$ be defined as in (1). Then, for any fixed $ϵ\in (0,1)$ there exists a level curve ${\Gamma }_{1+ϵ(R-1)}$ such that the following holds for any polynomial $P_n\left(z\right)\in {\wp }_n$, $n\in \mathbb{N}$:

$${∥P_n∥}_{{\mathcal{L}}_p\left({\displaystyle \frac{h}{\left|{\Phi }^{\prime }\right|}},{\Gamma }_{1+ϵ(R-1)}\right)}⪯n^{{\displaystyle \frac{1}{p}}}{∥P_n∥}_p,p>0.$$

(26)

Proof. 

In ([15], Lemma 2.4), it was proved that if $\Gamma$ is a K-quasiconformal curve, then, for any fixed $\epsilon \in (0,1)$ and $R=1+{\displaystyle \frac{{\epsilon }_0}{n}}$, there exists a level curve ${\Gamma }_{1+\epsilon (R-1)}$ such that for any polynomial $P_n\left(z\right)\in {\wp }_n$, $n\in \mathbb{N},$ the following estimate holds:

$${∥P_n∥}_{{\mathcal{L}}_p\left({\displaystyle \frac{h}{\left|{\Phi }^{\prime }\right|}},{\Gamma }_{1+\epsilon (R-1)}\right)}⪯n^{{\displaystyle \frac{1}{p}}}{∥P_n∥}_p,p>0.$$

(27)

It is sufficient to demonstrate that if $\partial G\in PAC({\nu }_1,\dots ,{\nu }_l)$, where $0<{\nu }_j<2$ for $j=\overline{1,l}$, the boundary $\partial G$ is quasiconformal. As established in (11) and (12), any asymptotically conformal curve, i.e., the case where ${\nu }_j=1$ for all $j=\overline{1,l}$, is quasiconformal. We now proceed to verify that a piecewise asymptotically conformal curve (with $0<{\nu }_j<2$ and ${\nu }_j\ne 1$ for all $j=\overline{1,l}$) that does not exhibit cusps is also quasiconformal.

Let S represent a piecewise asymptotically conformal curve, which for simplicity is assumed to have a single “nonzero” angle ${\nu }_1\pi$, where $0<{\nu }_1<2$, at the point $z_1\in S$. If the subarc $S({\zeta }_2,{\zeta }_3)$ of S, with endpoints ${\zeta }_2$ and ${\zeta }_3$, lies entirely on one side of $z_1$, the quasiconformality of the curve S is immediately apparent, as S is asymptotically conformal.

Now, suppose that the arc $S({\zeta }_2,{\zeta }_3)$ is such that its endpoints ${\zeta }_2$ and ${\zeta }_3$ lie on opposite sides of $z_1$, i.e., $z_1\in S({\zeta }_2,{\zeta }_3)$. By connecting the points $z_1$, ${\zeta }_2$, and ${\zeta }_3$, we form a triangle $\Lambda (z_1,{\zeta }_2,{\zeta }_3)$ with vertices at $z_1$, ${\zeta }_2$, and ${\zeta }_3$. The angles at these vertices are ${\nu }_j\pi$, with $0<{\nu }_j<2$. Applying the Law of Sines, we obtain

$${\displaystyle \frac{\left|{\zeta }_2-{\zeta }_3\right|}{sin{\nu }_1\pi }}={\displaystyle \frac{\left|{\zeta }_2-z_1\right|}{sin{\nu }_3\pi }}={\displaystyle \frac{\left|{\zeta }_3-z_1\right|}{sin{\nu }_2\pi }}.$$

Then, we obtain

$$1\le {\displaystyle \frac{\left|{\zeta }_2-z_1\right|+\left|{\zeta }_3-z_1\right|}{\left|{\zeta }_2-{\zeta }_3\right|}}={\displaystyle \frac{\frac{sin{\nu }_3\pi }{sin{\nu }_1\pi }\left|{\zeta }_2-{\zeta }_3\right|+\frac{sin{\nu }_2\pi }{sin{\nu }_1\pi }\left|{\zeta }_2-{\zeta }_3\right|}{\left|{\zeta }_2-{\zeta }_3\right|}}={\displaystyle \frac{sin{\nu }_3\pi +sin{\nu }_2\pi }{sin{\nu }_1\pi }}\le c<+\infty .$$

Therefore, according to (11), the curve $S$ is quasiconformal and (26) follows from (27). □

Let ${\left\{z_j\right\}}_{j=1}^l$ be a fixed system of the points on L and the weight function $h\left(z\right)$ defined as in (1). The following result is the integral analog of the familiar lemma of Bernstein–Walsh ([1], p. 101) for the $A_p(h,G)$ -norm.

Lemma 4 

([51]). Let $\Gamma =\partial G\in Q\left(\kappa \right)$ for some $0\le \kappa <1$ and $P_n\left(z\right),$ $deg{P}_n\le n,$ $n=1,2,...,$ is an arbitrary polynomial and the weight function $h\left(z\right)$ satisfy the condition (1). Then, for any $R>1,$ $p>0$ and $n=1,2,...$

$${∥P_n∥}_{A_p(h,G_R)}\le c_1{\left(1+c_2(R-1)\right)}^{n+{\displaystyle \frac{1}{p}}}{∥P_n∥}_{A_p(h,G)},$$

where $c_1,c_2$ are constants, independent of n and $R.$

Lemma 5 

([3], Lemma 2.4). Let $\Gamma =\partial G$ be a rectifiable Jordan curve and $P_n\left(z\right),$ $deg{P}_n\le n,$ $n=1,2,...,$ be arbitrary polynomial and the weight function $h\left(z\right)$ satisfy the condition (1) with ${\gamma }_j>-1,$ for all $j=\overline{1,l}$. Then, for any $R>1,$ $p>0$ and $n=1,2,...$

$${∥P_n∥}_{{\mathcal{L}}_p(h,{\Gamma }_R)}\le R^{n+{\displaystyle \frac{1+{\gamma }^{\ast }}{p}}}{∥P_n∥}_{{\mathcal{L}}_p(h,\Gamma )}.$$

5. Proof of Theorems

Proofs of Theorem 1 and Theorem 2. 

The proofs of Theorem 1 and Theorem 2 will be carried out together, emphasizing the different cases respect to p. Let $G\in PAC({\nu }_1,\cdots ,{\nu }_l)$, for some $0<{\nu }_j\le 1$, $j=1,\cdots ,l$, and let $R=1+{\epsilon }_0/n,R_1:=1+(R-1)/3,R_2:=1+2(R-1)/3.$ For any $n\ge m\ge 1$ and $z\in \Omega ,$ let us define:

$$T_{n,m}\left(z\right):={\displaystyle \frac{P_n^{\left(m\right)}\left(z\right)}{{\Phi }^{n-m+1}\left(z\right)}}.$$

Clearly, $T_{n,m}\left(z\right)$ is analytic in $\Omega ,$ continuous on $\overline{\Omega }$ and $H_{n,m}\left(\infty \right)=0.$ Cauchy integral representation for unbounded regions gives:

$$T_{n,m}\left(z\right)=-{\displaystyle \frac{1}{2\pi i}}\underset{{\Gamma }_{R_1}}{\int }{T}_{n,m}\left(\zeta \right){\displaystyle \frac{d\zeta }{\zeta -z}},z\in {\Omega }_{R_1}.$$

Moving on to the modules and taking into account that $\left|{\Phi }^{n-m+1}\left(\zeta \right)\right|=R_1^{n-m+1}>1,$ we get:

$$\left|P_n^{\left(m\right)}\left(z\right)\right|\le {\displaystyle \frac{\left|{\Phi }^{n-m+1}\left(z\right)\right|}{2\pi }}\underset{{\Gamma }_{R_1}}{\int }\left|{\displaystyle \frac{P_n^{\left(m\right)}\left(\zeta \right)}{{\Phi }^{n-m+1}\left(\zeta \right)}}\right|{\displaystyle \frac{\left|d\zeta \right|}{\left|\zeta -z\right|}}\le {\displaystyle \frac{\left|{\Phi }^{n-m+1}\left(z\right)\right|}{2\pi d(z,{\Gamma }_{R_1})}}\underset{{\Gamma }_{R_1}}{\int }\left|P_n^{\left(m\right)}\left(\zeta \right)\right|\left|d\zeta \right|,$$

and consequently,

$$\left|P_n^{\left(m\right)}\left(z\right)\right|⪯{\displaystyle \frac{\left|{\Phi }^{n-m+1}\left(z\right)\right|}{d(z,{\Gamma }_{R_1})}}\underset{{\Gamma }_{R_1}}{\int }\left|P_n^{\left(m\right)}\left(\zeta \right)\right|\left|d\zeta \right|.$$

(28)

Let us write the Cauchy integral representation for the integrand function $P_n^{\left(m\right)}\left(\zeta \right)$:

$$P_n^{\left(m\right)}\left(\zeta \right)={\displaystyle \frac{m!}{2\pi i}}\underset{{\Gamma }_{R_2}}{\int }{P}_n\left(t\right){\displaystyle \frac{dt}{{\left(t-\zeta \right)}^{m+1}}},\zeta \in G_{R_2}.$$

Let $\zeta \in {\Gamma }_{R_1}$. Substituting $P_n^{\left(m\right)}\left(\zeta \right)$ in (28), we have

$$\begin{array}{ccc}\left|P_n^{\left(m\right)}\left(z\right)\right|\hfill & ⪯& {\displaystyle \frac{\left|{\Phi }^{n-m+1}\left(z\right)\right|}{d(z,{\Gamma }_{R_1})}}·\underset{{\Gamma }_{R_1}}{\int }\left(\underset{{\Gamma }_{R_2}}{\int }\left|P_n\left(t\right)\right|{\displaystyle \frac{\left|dt\right|}{{\left|t-\zeta \right|}^{m+1}}}\right)\left|d\zeta \right|\hfill \\ & ⪯& {\displaystyle \frac{\left|{\Phi }^{n-m+1}\left(z\right)\right|}{d(z,{\Gamma }_{R_1})}}·\underset{t\in {\Gamma }_{R_2}}{sup}\left(\underset{{\Gamma }_{R_1}}{\int }{\displaystyle \frac{\left|d\zeta \right|}{{\left|t-\zeta \right|}^{m+1}}}\right)·\underset{{\Gamma }_{R_2}}{\int }\left|P_n\left(t\right)\right|\left|dt\right|.\hfill \end{array}$$

(29)

Denote by

$$J_{n,m}\left(t\right):=\underset{{\Gamma }_{R_1}}{\int }{\displaystyle \frac{\left|d\zeta \right|}{{\left|t-\zeta \right|}^{m+1}}};Y_n:=\underset{{\Gamma }_{R_2}}{\int }\left|P_n\left(t\right)\right|\left|dt\right|,$$

and estimate these integrals separately. After replacing the variable $\tau =\Phi \left(\zeta \right),$ $\zeta \in {\Gamma }_{R_1};$ $w=\Phi \left(t\right)$, $t\in {\Gamma }_{R_2}$ and applying (25), we have

$$J_{n,m}\left(t\right)=\underset{{\Gamma }_{R_1}}{\int }{\displaystyle \frac{\left|d\zeta \right|}{{\left|\zeta -t\right|}^{m+1}}}=\underset{\left|\tau \right|=R_1}{\int }{\displaystyle \frac{\left|{\Psi }^{\prime }\left(\tau \right)\right|\left|d\tau \right|}{{\left|\Psi \left(\tau \right)-\Psi \left(w\right)\right|}^{m+1}}}⪯n\underset{\left|\tau \right|=R_1}{\int }{\displaystyle \frac{d(\Psi \left(\tau \right),\Gamma )\left|d\tau \right|}{{\left|\Psi \left(\tau \right)-\Psi \left(w\right)\right|}^{m+1}}}.$$

Since a curve $\Gamma$ is $K_1-$quasiconformal, for some $K_1>1$, then all level curves ${\Gamma }_t,$ $t>1,$ are also $c\left(K_1\right)-$quasiconformal, for some $c\left(K_1\right)>1,$ where $c\left(K_1\right)$ depends only on $K_1.$ Taking this into account and setting $z_1=\zeta \in {\Gamma }_{R_1};$ $z_2\in \Gamma$ where $\left|\zeta -z_2\right|=d(\Psi \left(\tau \right),\Gamma );$ $z_3:=t\in {\Gamma }_{R_2}$ and $w_j:=\Phi \left(z_j\right),j=1,2,3,$ applying Lemmas 1 and 2, we obtain

$$\left|{\displaystyle \frac{\zeta -t}{\zeta -z_2}}\right|⪰{\left|{\displaystyle \frac{\tau -w}{\tau -w_2}}\right|}^{c_1}⪰{\left({\displaystyle \frac{R_2-1}{R_1-1}}\right)}^{c_1}⪰1.$$

Then,

$$\left|\Psi \left(\tau \right)-\Psi \left(w\right)\right|⪰d(\Psi \left(\tau \right),\Gamma ).$$

(30)

So, applying Lemma 2, we obtain

$$J_{n,m}\left(t\right)⪯n\underset{\left|\tau \right|=R_1}{\int }{\displaystyle \frac{\left|d\tau \right|}{{\left|\Psi \left(\tau \right)-\Psi \left(w\right)\right|}^m}}.$$

(31)

For simplicity of calculations, we can restrict ourselves to only one point on the boundary $z_1$, where the weight function and the boundary curve of which have singularity, i.e., let $h\left(z\right)$ be defined as in (1) for $l=1$ and ${\nu }_1\le 1,$ and we put $\gamma :={\gamma }_1;$ $\nu :={\nu }_1.$

Let the numbers ${\delta }_1,{\delta }_1^{\ast }$, with $0<{\delta }_1<{\delta }_1^{\ast }<\mathrm{diam}\left(\overline{G}\right),$ be chosen as in Lemma 2. For any $t\ge 1,$ we set

$$\begin{array}{ccc}{\Gamma }_{t,1}^1\hfill & :& ={\Gamma }_t^1\cap \overline{\Omega (z_1,{\delta }_1)},\hfill \\ {\Gamma }_{t,2}^1\hfill & :& ={\Gamma }_t^1\cap \overline{(\Omega (z_1,{\delta }_1^{\ast })\setminus \Omega (z_1,{\delta }_1))},\hfill \\ {\Gamma }_{t,3}^1\hfill & :& ={\Gamma }_t\setminus ({\Gamma }_{t,1}^1\cup {\Gamma }_{t,2}^1);F_{t,i}^1:=\Phi \left({\Gamma }_{t,i}^1\right).\hfill \end{array}$$

According to the this notations, from (31), we obtain

$$J_{n,m}\left(t\right)⪯n\underset{\left|\tau \right|=R_1}{\int }{\displaystyle \frac{\left|d\tau \right|}{{\left|\Psi \left(\tau \right)-\Psi \left(w\right)\right|}^m}}=n\sum _{j=1}^3\underset{F_{R_1,j}^1}{\int }{\displaystyle \frac{\left|d\tau \right|}{{\left|\Psi \left(\tau \right)-\Psi \left(w\right)\right|}^m}}$$

(32)

and

$$\begin{array}{ccc}\underset{F_{R_1,1}^1}{\int }{\displaystyle \frac{\left|d\tau \right|}{{\left|\Psi \left(\tau \right)-\Psi \left(w\right)\right|}^m}}\hfill & ⪯& \underset{F_{R_1,1}^1}{\int }{\displaystyle \frac{\left|d\tau \right|}{{\left|\tau -w\right|}^{m(2-\nu )}}}⪯n^{m(2-\nu )-1};\hfill \\ \underset{F_{R_1,3}^1}{\int }{\displaystyle \frac{\left|d\tau \right|}{{\left|\Psi \left(\tau \right)-\Psi \left(w\right)\right|}^m}}\hfill & ⪯& \underset{F_{R_1,3}^1}{\int }{\displaystyle \frac{\left|d\tau \right|}{{\left|\tau -w\right|}^{m(1+\epsilon )}}}⪯n^{m(1+\epsilon )-1};\hfill \end{array}$$

(33)

For $\zeta \in {\Gamma }_{R_1,2}^1$, we obtain

$$\begin{array}{ccc}{\left|\zeta -t\right|}^{1-\epsilon }\hfill & ⪰& max\left\{\left|{\left|\zeta -z_1\right|}^{1-{\displaystyle \frac{1}{2-\nu }}-\epsilon }\right|,\left|{\left|t-z_1\right|}^{1-{\displaystyle \frac{1}{2-\nu }}-\epsilon }\right|\right\}\left|\tau -w\right|\hfill \\ & ⪰& {\left({\delta }_1\right)}^{1-{\displaystyle \frac{1}{2-\nu }}-\epsilon }\left|\tau -w\right|⪰\left|\tau -w\right|,\hfill \end{array}$$

(34)

and so,

$$\underset{F_{R_1,2}^1}{\int }{\displaystyle \frac{\left|d\tau \right|}{{\left|\Psi \left(\tau \right)-\Psi \left(w\right)\right|}^m}}⪯\underset{\Phi \left({\Gamma }_{R_1,2}^1\right)}{\int }{\displaystyle \frac{\left|d\tau \right|}{{\left|\tau -w\right|}^{m(1+\epsilon )}}}⪯n^{m(1+\epsilon )-1}.$$

(35)

Therefore, combining estimates (32)–(35), we have

$$J_{n,m}\left(t\right)⪯n^{m{\nu }^{\ast }}$$

for the integral $J_{n,m}\left(t\right)$. To estimate $Y_n$, let us first assume that $p>1$. By substituting $t=\Psi \left(w\right)$, multiplying both the numerator and denominator of the integrand by $h^{{\displaystyle \frac{1}{p}}}$, and applying Hölder’s inequality, we obtain the following result:

$$\begin{array}{ccc}{Y}_n\hfill & =& \underset{{\Gamma }_{R_2}}{\int }\left|P_n\left(t\right)\right|\left|dt\right|=\underset{\left|w\right|=R_2}{\int }\left|P_n\left(\Psi \left(w\right)\right)\right|\left|{\Psi }^{\prime }\left(w\right)\right|\left|dw\right|\hfill \\ & =& \underset{\left|w\right|=R_2}{\int }{\displaystyle \frac{h^{\frac{1}{p}}(\Psi \left(w\right))\left|P_n\left(\Psi \left(w\right)\right)\right|{\left|{\Psi }^{\prime }\left(w\right)\right|}^{2/p}{\left|{\Psi }^{\prime }\left(w\right)\right|}^{1-\frac{2}{p}}\left|dw\right|}{h^{\frac{1}{p}}(\Psi \left(w\right))}}\hfill \end{array}$$

(36)

$$\begin{array}{ccc}& \le & {\left(\underset{\left|w\right|=R_2}{\int }h(\Psi \left(w\right)){\left|P_n\left(\Psi \left(w\right)\right)\right|}^p{\left|{\Psi }^{\prime }\left(w\right)\right|}^2\left|dw\right|\right)}^{{\displaystyle \frac{1}{p}}}\times {\left(\underset{\left|w\right|=R_2}{\int }{\displaystyle \frac{{\left|{\Psi }^{\prime }\left(w\right)\right|}^{q(1-\frac{2}{p})}\left|dw\right|}{h^{\frac{q}{p}}(\Psi \left(w\right))}}\right)}^{{\displaystyle \frac{1}{q}}}\hfill \\ & =& {\left(\underset{{\Gamma }_{R_2}}{\int }{\displaystyle \frac{h\left(z\right)}{\left|{\Phi }^{\prime }\left(z\right)\right|}}{\left|P_n\left(z\right)\right|}^p\left|dz\right|\right)}^{{\displaystyle \frac{1}{p}}}{\left(\underset{\left|w\right|=R_2}{\int }{\displaystyle \frac{{\left|{\Psi }^{\prime }\left(w\right)\right|}^{q(1-\frac{2}{p})}\left|dw\right|}{h^{\frac{q}{p}}(\Psi \left(w\right))}}\right)}^{{\displaystyle \frac{1}{q}}}\hfill \\ & =& :Y_{n,1}·Y_{n,2},{\displaystyle \frac{1}{p}}+{\displaystyle \frac{1}{q}}=1.\hfill \end{array}$$

Applying Lemma 3, for the integral $Y_{n,1}$, we have

$$Y_{n,1}⪯n^{{\displaystyle \frac{1}{p}}}·{∥P_n∥}_p.$$

(37)

According to (25), for the integral $Y_{n,2},$ we find

$$\begin{array}{ccc}{\left(Y_{n,2}\right)}^q\hfill & =& \underset{\left|w\right|=R_2}{\int }{\displaystyle \frac{{\left|{\Psi }^{\prime }\left(w\right)\right|}^{2-q}\left|dw\right|}{h^{q-1}(\Psi \left(w\right))}}\asymp \underset{\left|w\right|=R_2}{\int }{\left({\displaystyle \frac{d(\Psi \left(w\right),\Gamma )}{\left|w\right|-1}}\right)}^{2-q}{\displaystyle \frac{\left|dw\right|}{{\left|\Psi \left(w\right)-\Psi \left(w_1\right)\right|}^{\gamma (q-1)}}}\hfill \\ & ⪯& n^{2-q}\underset{\left|w\right|=R_2}{\int }{\displaystyle \frac{d^{2-q}(\Psi \left(w\right),\Gamma )\left|dw\right|}{{\left|\Psi \left(w\right)-\Psi \left(w_1\right)\right|}^{\gamma (q-1)}}}=:n^{2-q}I\left(F_{R_2}\right),\hfill \end{array}$$

(38)

where $F_{R_2}:=\Phi \left({\Gamma }_{R_2}\right)$. To estimate the integral

$$I\left(F_{R_2}\right):=\left\{\begin{array}{cc}\underset{F_{R_2}}{\int }{\displaystyle \frac{d^{2-q}(\Psi \left(w\right),\Gamma )\left|dw\right|}{{\left|\Psi \left(w\right)-\Psi \left(w_1\right)\right|}^{\gamma (q-1)}}},& \mathrm{for}q\le 2,\\ \underset{F_{R_2}}{\int }{\displaystyle \frac{\left|dw\right|}{d^{q-2}(\Psi \left(w\right),\Gamma ){\left|\Psi \left(w\right)-\Psi \left(w_1\right)\right|}^{\gamma (q-1)}}},& \mathrm{for}q>2,\end{array}\right.$$

(39)

we set:

$$\begin{array}{ccc}{E}_{R_2}^{11}\hfill & :& =\left\{w:w\in F_{R_2},\left|w-w_1\right|<{\delta }_1\right\},\hfill \\ E_{R_2}^{12}\hfill & :& =\left\{w:w\in F_{R_2},{\delta }_1\le \left|w-w_1\right|<{\delta }_1^{\ast }\right\},\hfill \\ E_{R_2}^{13}\hfill & :& =\left\{w:w\in F_{R_2},{\delta }_1^{\ast }\le \left|w-w_1\right|<\mathrm{diam}\left(\overline{G}\right)\right\}.\hfill \end{array}$$

(40)

Clearly, $F_{R_2}=\bigcup _{k=1}^3{E}_{R_2}^{1k}.$

Given the possible values $q(q>2$ and $q\le 2)$ and $\gamma (-2<\gamma <0$ and $\gamma \ge 0)$, we will consider the cases separately.

Case 1.

Let $1<q\le 2(p\ge 2).$ Then,

$$I\left(E_{R_2}^{1k}\right)=\underset{E_{R_2}^{1k}}{\int }{\displaystyle \frac{d^{2-q}(\Psi \left(w\right),\Gamma )\left|dw\right|}{{\left|\Psi \left(w\right)-\Psi \left(w_1\right)\right|}^{\gamma (q-1)}}}\left|dw\right|.$$

(41)

1.1.

Let $\gamma \ge 0$. By applying Lemma 2 and considering (30), we derive the following result:

1.1.1.

$$\begin{array}{ccc}I\left(E_{R_2}^{11}\right)\hfill & ⪯& \underset{E_{R_2}^{11}}{\int }{\displaystyle \frac{\left|dw\right|}{{\left|\Psi \left(w\right)-\Psi \left(w_1\right)\right|}^{\gamma (q-1)-\left(2-q\right)}}}⪯\underset{E_{R_2}^{11}}{\int }{\displaystyle \frac{\left|dw\right|}{{\left|w-w_1\right|}^{\left[\gamma (q-1)-(2-q)\right](2-\nu )}}}\hfill \\ & ⪯& \left\{\begin{array}{cc}{n}^{\left[\gamma (q-1)-(2-q)\right](2-\nu )-1},& \left[\gamma (q-1)-(2-q)\right](2-\nu )>1,\\ lnn,& \left[\gamma (q-1)-(2-q)\right](2-\nu )=1,\\ 1,& \left[\gamma (q-1)-(2-q)\right](2-\nu )<1,\end{array}\right.\hfill \end{array}$$

(42)

for $0<\nu <1,$ and

$$\begin{array}{ccc}I\left(E_{R_2}^{11}\right)\hfill & ⪯& \underset{E_{R_2}^{11}}{\int }{\displaystyle \frac{\left|dw\right|}{{\left|\Psi \left(w\right)-\Psi \left(w_1\right)\right|}^{\gamma (q-1)-\left(2-q\right)}}}⪯\underset{E_{R_2}^{11}}{\int }{\displaystyle \frac{\left|dw\right|}{{\left|w-w_1\right|}^{\left[\gamma (q-1)-(2-q)\right](1+\epsilon )}}}\hfill \\ & ⪯& \left\{\begin{array}{cc}{n}^{\left[\gamma (q-1)-(2-q)\right](2-\nu )-1},& \left[\gamma (q-1)-(2-q)\right](1+\epsilon )>1,\\ lnn,& \left[\gamma (q-1)-(2-q)\right](1+\epsilon )=1,\\ 1,& \left[\gamma (q-1)-(2-q)\right](1+\epsilon )<1,\end{array}\right.\forall \epsilon >0,\hfill \end{array}$$

for $\nu =1.$

1.1.2.

Using (40), for the integral $I\left(E_{R_2}^{12}\right),$ we have:

$$I\left(E_{R_2}^{12}\right)⪯\underset{E_{R_2}^{12}}{\int }{\displaystyle \frac{\left|dw\right|}{{\left|\Psi \left(w\right)-\Psi \left(w_1\right)\right|}^{\gamma (q-1)-\left(2-q\right)}}}⪯\underset{E_{R_2}^{12}}{\int }{\displaystyle \frac{\left|dw\right|}{{\left({\delta }_1\right)}^{\left[\gamma (q-1)-(2-q)\right]}}}⪯1.$$

(43)

1.1.3.

For w that $\left|w-w_1\right|>{\delta }_1^{\ast },$ according to Lemma 1, we obtain: $\left|\Psi \left(w\right)-\Psi \left(w_1\right)\right|⪰1$. Then, using Lemma 2, we have

$$\begin{array}{ccc}I\left(E_{R_2}^{13}\right)\hfill & ⪯& \underset{E_{R_2}^{13}}{\int }{\displaystyle \frac{d^{2-q}(\Psi \left(w\right),\Gamma )}{{\left|\Psi \left(w\right)-\Psi \left(w_1\right)\right|}^{\gamma (q-1)}}}\left|dw\right|⪯\underset{E_{R_2}^{13}}{\int }{\displaystyle \frac{d^{2-q}(\Psi \left(w\right),\Gamma )}{{\left({\delta }_1^{\ast }\right)}^{\gamma (q-1)}}}\left|dw\right|\hfill \\ & ⪯& \underset{E_{R_2}^{13}}{\int }{d}^{2-q}(\Psi \left(w\right),\Gamma )\left|dw\right|⪯{\left({\displaystyle \frac{1}{n}}\right)}^{\left(2-q\right)\left(1+\epsilon \right)}\underset{E_{R_2}^{13}}{\int }\left|dw\right|⪯1.\hfill \end{array}$$

(44)

Combining (38), (39) and (42)–(44), for $p\ge 2,\gamma \ge 0$, we obtain:

$${\left(Y_{n,2}\right)}^q=n^{2-q}I\left(F_{R_2}\right)⪯\left\{\begin{array}{cc}{n}^{\left[\gamma (q-1)-(2-q)\right]{\nu }^{\ast }-q+1},& \left[\gamma (q-1)-(2-q)\right]{\nu }^{\ast }>1,\\ n^{2-q}lnn,& \left[\gamma (q-1)-(2-q)\right]{\nu }^{\ast }=1,\\ n^{2-q},& \left[\gamma (q-1)-(2-q)\right]{\nu }^{\ast }<1.\end{array}\right.$$

$$Y_{n,2}⪯\left\{\begin{array}{ccc}{n}^{\left({\displaystyle \frac{\gamma +2}{p}}-1\right){\nu }^{\ast }-{\displaystyle \frac{1}{p}}},& 2\le p<1+{\displaystyle \frac{{\nu }^{\ast }}{{\nu }^{\ast }+1}}\left(\gamma +1\right),& \gamma >{\displaystyle \frac{1}{{\nu }^{\ast }}},\\ n^{1-{\displaystyle \frac{2}{p}}}{\left(lnn\right)}^{1-{\displaystyle \frac{1}{p}}},& p=1+{\displaystyle \frac{{\nu }^{\ast }}{{\nu }^{\ast }+1}}\left(\gamma +1\right),& \gamma >{\displaystyle \frac{1}{{\nu }^{\ast }}},\\ n^{1-{\displaystyle \frac{2}{p}}},& p>1+{\displaystyle \frac{{\nu }^{\ast }}{{\nu }^{\ast }+1}}\left(\gamma +1\right),& \gamma >{\displaystyle \frac{1}{{\nu }^{\ast }}},\\ n^{1-{\displaystyle \frac{2}{p}}},& p\ge 2,& 0\le \gamma \le {\displaystyle \frac{1}{{\nu }^{\ast }}}.\end{array}\right.$$

(45)

Therefore, from (36), (37) and (45) for integral $Y_n,$ in case of $p\ge 2,$ $\gamma \ge 0$, we obtain:

$$Y_n⪯\left\{\begin{array}{ccc}{n}^{\left({\displaystyle \frac{\gamma +2}{p}}+m-1\right){\nu }^{\ast }},& 2\le p<1+{\displaystyle \frac{{\nu }^{\ast }}{{\nu }^{\ast }+1}}\left(\gamma +1\right),& \gamma >{\displaystyle \frac{1}{{\nu }^{\ast }}},\\ n^{m{\nu }^{\ast }}{\left(nlnn\right)}^{1-{\displaystyle \frac{1}{p}}},& p=1+{\displaystyle \frac{{\nu }^{\ast }}{{\nu }^{\ast }+1}}\left(\gamma +1\right),& \gamma >{\displaystyle \frac{1}{{\nu }^{\ast }}},\\ n^{m{\nu }^{\ast }+1-{\displaystyle \frac{1}{p}}},& p>1+{\displaystyle \frac{{\nu }^{\ast }}{{\nu }^{\ast }+1}}\left(\gamma +1\right),& \gamma >{\displaystyle \frac{1}{{\nu }^{\ast }}},\\ n^{m{\nu }^{\ast }+1-{\displaystyle \frac{1}{p}}},& p\ge 2,& 0\le \gamma \le {\displaystyle \frac{1}{{\nu }^{\ast }}}.\end{array}\right.$$

(46)

Combining (29) and (31) with (46), we obtain:

$$\left|P_n^{\left(m\right)}\left(z\right)\right|⪯{\displaystyle \frac{\left|{\Phi }^{n-m+1}\left(z\right)\right|}{d(z,{\Gamma }_{R_1})}}{∥P_n∥}_p\left\{\begin{array}{ccc}{n}^{\left({\displaystyle \frac{\gamma +2}{p}}+m-1\right){\nu }^{\ast }},& 2\le p<1+{\displaystyle \frac{{\nu }^{\ast }}{{\nu }^{\ast }+1}}\left(\gamma +1\right),& \gamma >{\displaystyle \frac{1}{{\nu }^{\ast }}},\\ n^{m{\nu }^{\ast }}{\left(nlnn\right)}^{1-{\displaystyle \frac{1}{p}}},& p=1+{\displaystyle \frac{{\nu }^{\ast }}{{\nu }^{\ast }+1}}\left(\gamma +1\right),& \gamma >{\displaystyle \frac{1}{{\nu }^{\ast }}},\\ n^{m{\nu }^{\ast }+1-{\displaystyle \frac{1}{p}}},& p>1+{\displaystyle \frac{{\nu }^{\ast }}{{\nu }^{\ast }+1}}\left(\gamma +1\right),& \gamma >{\displaystyle \frac{1}{{\nu }^{\ast }}},\\ n^{m{\nu }^{\ast }+1-{\displaystyle \frac{1}{p}}},& p\ge 2,& 0\le \gamma \le {\displaystyle \frac{1}{{\nu }^{\ast }}}.\end{array}\right.$$

(47)

1.2.

Let $\gamma <0.$ For any $k=1,2,3,$ in this case we have

$$I\left(E_{R_2}^{1k}\right)⪯\underset{E_{R_2}^{1k}}{\int }{\left|\Psi \left(w\right)-\Psi \left(w_1\right)\right|}^{(-\gamma )(q-1)}{d}^{2-q}(\Psi \left(w\right),\Gamma )\left|dw\right|⪯\underset{E_{R_2}^{1k}}{\int }\left|dw\right|⪯1.$$

(48)

Combining (37), (38) and (48) for $\gamma <0$, we have

$$Y_n⪯{∥P_n∥}_p·n^{1-{\displaystyle \frac{1}{p}}}.$$

(49)

and

$$\left|P_n^{\left(m\right)}\left(z\right)\right|⪯{\displaystyle \frac{\left|{\Phi }^{n-m+1}\left(z\right)\right|}{d(z,{\Gamma }_{R_1})}}{∥P_n∥}_p·n^{m{\nu }^{\ast }+1-{\displaystyle \frac{1}{p}}}.$$

Therefore, combining (29), (47) and (49), for arbitrary $\gamma >-2,$ each $m=1,2,...,$ and $p\ge 2,$ we obtain

$$\left|P_n^{\left(m\right)}\left(z\right)\right|⪯{\displaystyle \frac{\left|{\Phi }^{n-m+1}\left(z\right)\right|}{d(z,{\Gamma }_{R_1})}}{∥P_n∥}_p\left\{\begin{array}{ccc}{n}^{\left({\displaystyle \frac{\gamma +2}{p}}+m-1\right){\nu }^{\ast }},& 2\le p<1+{\displaystyle \frac{{\nu }^{\ast }}{{\nu }^{\ast }+1}}\left(\gamma +1\right),& \gamma >{\displaystyle \frac{1}{{\nu }^{\ast }}},\\ n^{m{\nu }^{\ast }}{\left(nlnn\right)}^{1-{\displaystyle \frac{1}{p}}},& p=1+{\displaystyle \frac{{\nu }^{\ast }}{{\nu }^{\ast }+1}}\left(\gamma +1\right),& \gamma >{\displaystyle \frac{1}{{\nu }^{\ast }}},\\ n^{m{\nu }^{\ast }+1-{\displaystyle \frac{1}{p}}},& p>1+{\displaystyle \frac{{\nu }^{\ast }}{{\nu }^{\ast }+1}}\left(\gamma +1\right),& \gamma >{\displaystyle \frac{1}{{\nu }^{\ast }}},\\ n^{m{\nu }^{\ast }+1-{\displaystyle \frac{1}{p}}},& p\ge 2,& -2<\gamma \le {\displaystyle \frac{1}{{\nu }^{\ast }}}.\end{array}\right.$$

(50)

Case 2.

Let $q>2(1<p<2).$ In this case for $Y_{n,2},$ similarly to (38), we have

$${\left(Y_{n,2}\right)}^q=\underset{\left|w\right|=R_2}{\int }{\displaystyle \frac{\left|dw\right|}{{\left|{\Psi }^{\prime }\left(w\right)\right|}^{q-2}{h}^{q-1}(\Psi \left(w\right))}}\asymp n^{2-q}\underset{\left|w\right|=R_2}{\int }{\displaystyle \frac{\left|dw\right|}{d^{q-2}(\Psi \left(w\right),\Gamma ){\left|\Psi \left(w\right)-\Psi \left(w_1\right)\right|}^{\gamma (q-1)}}},$$

(51)

and consequently, we must evaluate the integral

$$I\left(F_{R_2}\right)=\underset{E_{R_2}^{1k}}{\int }{\displaystyle \frac{\left|dw\right|}{d^{q-2}(\Psi \left(w\right),\Gamma ){\left|\Psi \left(w\right)-\Psi \left(w_1\right)\right|}^{\gamma (q-1)}}}$$

(52)

for each $k=1,2,3.$

2.1.

If $\gamma \ge 0$,applying the Lemmas 1 and 2 and (25), we obtain the following cases:

2.1.1.

For $w\in E_{R_2}^{11}$, let point $z^{\ast }$ represent one of the points on $\Gamma$ such that $d(\Psi \left(w\right),\Gamma )=\left|\Psi \left(w\right)-z^{\ast }\right|$ and let $w^{\ast }=\Phi \left(z^{\ast }\right).$ Then, $\left|\Psi \left(w\right)-\Psi \left(w_1\right)\right|\ge \left|\Psi \left(w\right)-\Psi \left(w^{\ast }\right)\right|$ and consequently, according to the Lemma 2, we find

$$\begin{array}{ccc}I\left(E_{R_2}^{11}\right)\hfill & ⪯& \underset{E_{R_2}^{11}}{\int }{\displaystyle \frac{\left|dw\right|}{d^{q-2}(\Psi \left(w\right),\Gamma ){\left|\Psi \left(w\right)-\Psi \left(w_1\right)\right|}^{\gamma (q-1)}}}⪯\underset{E_{R_2}^{11}}{\int }{\displaystyle \frac{\left|dw\right|}{{\left|\Psi \left(w\right)-\Psi \left(w^{\ast }\right)\right|}^{\gamma (q-1)+q-2}}}\hfill \\ & ⪯& \left\{\begin{array}{cc}{n}^{\left[\gamma (q-1)+q-2\right](2-\nu )-1},& \left[\gamma (q-1)+q-2\right]>{\displaystyle \frac{1}{2-\nu }};\\ lnn,& \left[\gamma (q-1)+q-2\right]={\displaystyle \frac{1}{2-\nu }},\\ 1,& \left[\gamma (q-1)+q-2\right]<{\displaystyle \frac{1}{2-\nu }},\end{array}\right.\hfill \end{array}$$

(53)

for $0<\nu <1,$ and

$$\begin{array}{ccc}I\left(E_{R_2}^{11}\right)\hfill & ⪯& \underset{E_{R_2}^{11}}{\int }{\displaystyle \frac{\left|dw\right|}{d^{q-2}(\Psi \left(w\right),\Gamma ){\left|\Psi \left(w\right)-\Psi \left(w_1\right)\right|}^{\gamma (q-1)}}}⪯\underset{E_{R_2}^{11}}{\int }{\displaystyle \frac{\left|dw\right|}{{\left|\Psi \left(w\right)-\Psi \left(w^{\ast }\right)\right|}^{\gamma (q-1)+q-2}}}\hfill \\ & ⪯& \left\{\begin{array}{cc}{n}^{\left[\gamma (q-1)+q-2\right](2-\nu )-1},& \left[\gamma (q-1)+q-2\right]>{\displaystyle \frac{1}{2-\nu }};\\ lnn,& \left[\gamma (q-1)+q-2\right]={\displaystyle \frac{1}{2-\nu }},\\ 1,& \left[\gamma (q-1)+q-2\right]<{\displaystyle \frac{1}{2-\nu }},\end{array}\right.\forall \epsilon >0,\hfill \end{array}$$

for $\nu =1.$

2.1.2.

For $w\in E_{R_2}^{12},$

$$\begin{array}{ccc}I\left(E_{R_2}^{12}\right)\hfill & ⪯& \underset{E_{R_2}^{12}}{\int }{\displaystyle \frac{\left|dw\right|}{d^{q-2}(\Psi \left(w\right),\Gamma ){\left|\Psi \left(w\right)-\Psi \left(w_1\right)\right|}^{\gamma (q-1)}}}⪯{\left({\displaystyle \frac{1}{{\delta }_1}}\right)}^{\gamma (q-1)}\underset{E_{R_2}^{12}}{\int }{\displaystyle \frac{\left|dw\right|}{d^{q-2}(\Psi \left(w\right),\Gamma )}}\hfill \\ & ⪯& \underset{E_{R_2}^{12}}{\int }{\displaystyle \frac{\left|dw\right|}{{\left|w-w^{\ast }\right|}^{\left(q-2\right)(1+\epsilon )}}}⪯n^{(q-2)(1+\epsilon )},\forall \epsilon >0.\hfill \end{array}$$

2.1.3.

For $w\in E_{R_2}^{13}$,

$$\begin{array}{ccc}I\left(E_{R_2}^{13}\right)\hfill & ⪯& {\left({\displaystyle \frac{1}{{\delta }_1^{\ast }}}\right)}^{\gamma (q-1)}\underset{E_{R_2}^{13}}{\int }{\displaystyle \frac{\left|dw\right|}{{\left|\Psi \left(w\right)-\Psi \left(w^{\ast }\right)\right|}^{q-2}}}⪯\underset{E_{R_2}^{13}}{\int }{\displaystyle \frac{\left|dw\right|}{{\left|w-w^{\ast }\right|}^{\left(q-2\right)(1+\epsilon )}}}\hfill \\ & ⪯& n^{(q-2)(1+\epsilon )},\forall \epsilon >0.\hfill \end{array}$$

(54)

Combining (52)–(54) for $\gamma \ge 0,1<p<2,$ we have:

$${\left(Y_{n,2}\right)}^q⪯\left\{\begin{array}{cc}{n}^{\left[\gamma (q-1)+q-2\right](2-\nu )-1},& \left[\gamma (q-1)+q-2\right]>{\displaystyle \frac{1}{2-\nu }};\\ lnn,& \left[\gamma (q-1)+q-2\right]={\displaystyle \frac{1}{2-\nu }},\\ 1,& \left[\gamma (q-1)+q-2\right]<{\displaystyle \frac{1}{2-\nu }}.\end{array}\right.+n^{(q-2)(1+\epsilon )}+n^{(q-2)(1+\epsilon )};$$

$$Y_{n,2}⪯\left\{\begin{array}{ccc}{n}^{\left({\displaystyle \frac{\gamma +2}{p}}-1\right){\nu }^{\ast }-{\displaystyle \frac{1}{p}}},& 1<p<1+{\displaystyle \frac{{\nu }^{\ast }}{{\nu }^{\ast }+1}}(\gamma +1),& 0\le \gamma \le {\displaystyle \frac{1}{{\nu }^{\ast }}},\\ n^{1-{\displaystyle \frac{2}{p}}}{\left(lnn\right)}^{1-{\displaystyle \frac{1}{p}}},& p=1+{\displaystyle \frac{{\nu }^{\ast }}{{\nu }^{\ast }+1}}(\gamma +1),& 0\le \gamma \le {\displaystyle \frac{1}{{\nu }^{\ast }}},\\ n^{1-{\displaystyle \frac{2}{p}}},& 1+{\displaystyle \frac{{\nu }^{\ast }}{{\nu }^{\ast }+1}}(\gamma +1)<p<2,& 0\le \gamma \le {\displaystyle \frac{1}{{\nu }^{\ast }}},\\ n^{\left({\displaystyle \frac{\gamma +2}{p}}-1\right){\nu }^{\ast }-{\displaystyle \frac{1}{p}}},& 1<p<2,& \gamma >{\displaystyle \frac{1}{{\nu }^{\ast }}}.\end{array}\right.$$

Therefore, for integral $Y_n,$ from (36), (37) and (45), (46) for $1<p<2,$ $\gamma \ge 0$, we obtain

$$Y_n⪯\left\{\begin{array}{ccc}{n}^{\left({\displaystyle \frac{\gamma +2}{p}}-1\right){\nu }^{\ast }},& 1<p<1+{\displaystyle \frac{{\nu }^{\ast }}{{\nu }^{\ast }+1}}(\gamma +1),& 0\le \gamma \le {\displaystyle \frac{1}{{\nu }^{\ast }}},\\ n^{1-{\displaystyle \frac{1}{p}}}{\left(lnn\right)}^{1-{\displaystyle \frac{1}{p}}},& p=1+{\displaystyle \frac{{\nu }^{\ast }}{{\nu }^{\ast }+1}}(\gamma +1),& 0\le \gamma \le {\displaystyle \frac{1}{{\nu }^{\ast }}},\\ n^{1-{\displaystyle \frac{1}{p}}},& 1+{\displaystyle \frac{{\nu }^{\ast }}{{\nu }^{\ast }+1}}(\gamma +1)<p<2,& 0\le \gamma \le {\displaystyle \frac{1}{{\nu }^{\ast }}},\\ n^{\left({\displaystyle \frac{\gamma +2}{p}}-1\right){\nu }^{\ast }},& 1<p<2,& \gamma >{\displaystyle \frac{1}{{\nu }^{\ast }}}.\end{array}\right.$$

(55)

Combining (29), (31), (32) with (55) and (55), correspondingly, we have

$$\left|P_n^{\left(m\right)}\left(z\right)\right|⪯{\displaystyle \frac{\left|{\Phi }^{n-m+1}\left(z\right)\right|}{d(z,{\Gamma }_{R_1})}}{∥P_n∥}_p\left\{\begin{array}{ccc}{n}^{\left({\displaystyle \frac{\gamma +2}{p}}+m-1\right){\nu }^{\ast }},& 1<p<1+{\displaystyle \frac{{\nu }^{\ast }}{{\nu }^{\ast }+1}}(\gamma +1),& 0\le \gamma \le {\displaystyle \frac{1}{{\nu }^{\ast }}},\\ n^{m{\nu }^{\ast }}{\left(nlnn\right)}^{1-{\displaystyle \frac{1}{p}}},& p=1+{\displaystyle \frac{{\nu }^{\ast }}{{\nu }^{\ast }+1}}(\gamma +1),& 0\le \gamma \le {\displaystyle \frac{1}{{\nu }^{\ast }}},\\ n^{m{\nu }^{\ast }+1-{\displaystyle \frac{1}{p}}},& 1+{\displaystyle \frac{{\nu }^{\ast }}{{\nu }^{\ast }+1}}(\gamma +1)<p<2,& 0\le \gamma \le {\displaystyle \frac{1}{{\nu }^{\ast }}},\\ n^{\left({\displaystyle \frac{\gamma +2}{p}}+m-1\right){\nu }^{\ast }},& 1<p<2,& \gamma >{\displaystyle \frac{1}{{\nu }^{\ast }}}.\end{array}\right.$$

(56)

2.2.

Let $\gamma <0.$ For any $k=1,2,3,$ according to Lemma 1, we have

$$\begin{array}{ccc}I\left(E_{R_2}^{1k}\right)\hfill & ⪯& \underset{E_{R_2}^{1k}}{\int }{\displaystyle \frac{{\left|\Psi \left(w\right)-\Psi \left(w_1\right)\right|}^{(-\gamma )(q-1)}}{d^{q-2}(\Psi \left(w\right),\Gamma )}}\left|dw\right|⪯\underset{E_{R_2}^{1k}}{\int }{\displaystyle \frac{\left|dw\right|}{d^{q-2}(\Psi \left(w\right),\Gamma )}}\hfill \\ & ⪯& \underset{E_{R_2}^{1k}}{\int }{\displaystyle \frac{\left|dw\right|}{{\left|w-w^{\ast }\right|}^{(q-2){\nu }^{\ast }}}}⪯\left\{\begin{array}{cc}{n}^{\left(q-2\right)(2-\nu )-1},& q-2>{\displaystyle \frac{1}{{\nu }^{\ast }}};\\ lnn,& q-2={\displaystyle \frac{1}{{\nu }^{\ast }}},\\ 1,& q-2<{\displaystyle \frac{1}{{\nu }^{\ast }}}.\end{array}\right.\hfill \end{array}$$

So, for $\gamma <0,$ combining (36), (37), and the last estimate, we have

$$Y_n⪯{∥P_n∥}_p\left\{\begin{array}{cc}{n}^{\left({\displaystyle \frac{2}{p}}-1\right){\nu }^{\ast }},& 1<p<1+{\displaystyle \frac{{\nu }^{\ast }}{{\nu }^{\ast }+1}},\\ n^{1-{\displaystyle \frac{1}{p}}}{\left(lnn\right)}^{1-{\displaystyle \frac{1}{p}}},& p=1+{\displaystyle \frac{{\nu }^{\ast }}{{\nu }^{\ast }+1}},\\ n^{1-{\displaystyle \frac{1}{p}}},& 1+{\displaystyle \frac{{\nu }^{\ast }}{{\nu }^{\ast }+1}}<p<2.\end{array}\right.$$

(57)

Therefore, for any $\gamma \ge -2,1<$ $p<2,$ from (56) and (57), we obtain

$$\left|P_n^{\left(m\right)}\left(z\right)\right|⪯{\displaystyle \frac{\left|{\Phi }^{n-m+1}\left(z\right)\right|}{d(z,{\Gamma }_{R_1})}}{∥P_n∥}_p\left\{\begin{array}{ccc}{n}^{\left({\displaystyle \frac{\gamma +2}{p}}+m-1\right){\nu }^{\ast }},& 1<p<1+{\displaystyle \frac{{\nu }^{\ast }}{{\nu }^{\ast }+1}}(\gamma +1),& 0\le \gamma \le {\displaystyle \frac{1}{{\nu }^{\ast }}},\\ n^{m{\nu }^{\ast }}{\left(nlnn\right)}^{1-{\displaystyle \frac{1}{p}}},& p=1+{\displaystyle \frac{{\nu }^{\ast }}{{\nu }^{\ast }+1}}(\gamma +1),& 0\le \gamma \le {\displaystyle \frac{1}{{\nu }^{\ast }}},\\ n^{m{\nu }^{\ast }+1-{\displaystyle \frac{1}{p}}},& 1+{\displaystyle \frac{{\nu }^{\ast }}{{\nu }^{\ast }+1}}(\gamma +1)<p<2,& 0\le \gamma \le {\displaystyle \frac{1}{{\nu }^{\ast }}},\\ n^{\left({\displaystyle \frac{\gamma +2}{p}}+m-1\right){\nu }^{\ast }},& 1<p<2,& \gamma >{\displaystyle \frac{1}{{\nu }^{\ast }}},\end{array}\right.$$

if $\gamma \ge 0,$ and

$$\left|P_n^{\left(m\right)}\left(z\right)\right|⪯{\displaystyle \frac{\left|{\Phi }^{n-m+1}\left(z\right)\right|}{d(z,{\Gamma }_{R_1})}}{∥P_n∥}_p\left\{\begin{array}{cc}{n}^{\left({\displaystyle \frac{2}{p}}+m-1\right){\nu }^{\ast }},& 1<p<1+{\displaystyle \frac{{\nu }^{\ast }}{{\nu }^{\ast }+1}},\\ n^{m{\nu }^{\ast }+1-{\displaystyle \frac{1}{p}}}{\left(lnn\right)}^{1-{\displaystyle \frac{1}{p}}},& p=1+{\displaystyle \frac{{\nu }^{\ast }}{{\nu }^{\ast }+1}},\\ n^{m{\nu }^{\ast }+1-{\displaystyle \frac{1}{p}}},& 1+{\displaystyle \frac{{\nu }^{\ast }}{{\nu }^{\ast }+1}}<p<2,\end{array}\right.$$

if $\gamma <0$. Therefore, the proofs of Theorems 1 and 2 for $p>1$ are completed.

We now address the case where $p=1$. By multiplying both the numerator and denominator of the integrand in the inner integral by h, as specified in (29), and applying Lemma 5, we derive the result

$$\begin{array}{ccc}\left|P_n^{\left(m\right)}\left(z\right)\right|\hfill & ⪯& {\displaystyle \frac{\left|{\Phi }^{n-m+1}\left(z\right)\right|}{d(z,{\Gamma }_{R_1})}}·\underset{t\in {\Gamma }_{R_2}}{sup}\left({\displaystyle \frac{1}{h\left(t\right)}}\underset{{\Gamma }_{R_1}}{\int }{\displaystyle \frac{\left|d\zeta \right|}{{\left|t-\zeta \right|}^{m+1}}}\right)\left(\underset{{\Gamma }_{R_2}}{\int }h\left(t\right)\left|P_n\left(t\right)\right|\left|dt\right|\right)\hfill \\ & ⪯& {\displaystyle \frac{\left|{\Phi }^{n-m+1}\left(z\right)\right|}{d(z,{\Gamma }_{R_1})}}{∥P_n∥}_{{\mathcal{L}}_1(h,{\Gamma }_{R_2})}·\underset{t\in {\Gamma }_{R_2}}{sup}\left({\displaystyle \frac{1}{{\left|t-z_1\right|}^{\gamma }}}\underset{{\Gamma }_{R_1}}{\int }{\displaystyle \frac{\left|d\zeta \right|}{{\left|t-\zeta \right|}^{m+1}}}\right).\hfill \end{array}$$

(58)

After replacing the variable $t=\Psi \left(w\right)$ and using the Lemma 2, we obtain

$$\begin{array}{ccc}{\displaystyle \frac{1}{{\left|t-z_1\right|}^{\gamma }}}\hfill & =& {\displaystyle \frac{1}{{\left|\Psi \left(w\right)-\Psi \left(w_1\right)\right|}^{\gamma }}}=\left\{\begin{array}{cc}{\displaystyle \frac{1}{{\left|w-w_1\right|}^{\gamma (2-\nu )}}},& w\in E_{R_2}^{11},\\ {\displaystyle \frac{1}{{\left|w-w_1\right|}^{\gamma (1+\epsilon )}}},& w\in E_{R_2}^{1k},k=1,2,\forall \epsilon >0,\\ 1,& w\in E_{R_2}^{1k},k=1,2,3,\end{array}\right.\hfill \\ & ⪯& \left\{\begin{array}{ccc}{n}^{\gamma {\nu }^{\ast }},& w\in E_{R_2}^{11},& \gamma \ge 0,\\ n^{\gamma (1+\epsilon )},& w\in E_{R_2}^{1k},k=1,2,\forall \epsilon >0,& \gamma \ge 0,\\ 1,& w\in E_{R_2}^{1k},k=1,2,3,& \gamma <0.\end{array}\right.\hfill \\ & ⪯& \left\{\begin{array}{ccc}{n}^{\gamma {\nu }^{\ast }},& w\in F_{R_1},& \gamma \ge 0,\\ 1,& w\in E_{R_2}^{1k},k=1,2,3,& \gamma <0.\end{array}\right.\hfill \end{array}$$

(59)

Next, after replacing the variable $t=\Psi \left(w\right)$ and according (25), we have:

$$\underset{{\Gamma }_{R_1}}{\int }{\displaystyle \frac{\left|d\zeta \right|}{{\left|t-\zeta \right|}^{m+1}}}=\underset{\left|\tau \right|=R_1}{\int }{\displaystyle \frac{\left|{\Psi }^{\prime }\left(\tau \right)\right|\left|d\tau \right|}{{\left|\Psi \left(\tau \right)-\Psi \left(w\right)\right|}^{m+1}}}⪯n\underset{\left|\tau \right|=R_1}{\int }{\displaystyle \frac{\left|d\tau \right|}{{\left|\Psi \left(\tau \right)-\Psi \left(w\right)\right|}^m}}=:n·J\left(F_{R_1}\right).$$

(60)

The estimation of the integral $J\left(F_{R_1}\right)$ follows a procedure similar to that of $I\left(F_{R_2}\right)$ from (39), by partitioning $\Phi \left(F_{R_1}\right)$ into segments analogous to the partition of $\Phi \left(F_{R_2}\right)$, as described in Formula (40). After applying the relevant estimates for the integrals $J\left(F_{R_1}\right)$ based on this division, we obtain

$$J\left(E_{R_1}^{11}\right)=\underset{E_{R_1}^{11}}{\int }{\displaystyle \frac{\left|d\tau \right|}{{\left|\Psi \left(\tau \right)-\Psi \left(w\right)\right|}^m}}⪯\underset{E_{R_1}^{11}}{\int }{\displaystyle \frac{\left|d\tau \right|}{{\left|\tau -w\right|}^{m(2-\nu )}}}⪯n^{m{\nu }^{\ast }-1};$$

(61)

$$J\left(E_{R_1}^{1k}\right)=\underset{E_{R_1}^{1k}}{\int }{\displaystyle \frac{\left|d\tau \right|}{{\left|\Psi \left(\tau \right)-\Psi \left(w\right)\right|}^m}}⪯\underset{E_{R_1}^{1k}}{\int }{\displaystyle \frac{\left|d\tau \right|}{{\left|\tau -w\right|}^{m(1+\epsilon )}}}⪯n^{m(1+\epsilon )-1},k=2,3,\forall \epsilon >0.$$

Then,

$$J\left(F_{R_1}\right)⪯n^{m{\nu }^{\ast }-1}.$$

(62)

According to Lemmas 1, 3, and (25), we find

$$\underset{{\Gamma }_{R_2}}{\int }h\left(t\right)\left|P_n\left(t\right)\right|\left|dt\right|\le \underset{t\in {\Gamma }_{R_2}}{sup}\left|{\Phi }^{\prime }\left(t\right)\right|\underset{{\Gamma }_{R_2}}{\int }{\displaystyle \frac{h\left(t\right)}{\left|{\Phi }^{\prime }\left(t\right)\right|}}\left|P_n\left(t\right)\right|\left|dt\right|⪯n^{{\nu }^{\ast }}{∥P_n∥}_1.$$

(63)

Combining estimates (58), (59), (60)–(63), we find:

$$\left|P_n^{\left(m\right)}\left(z\right)\right|⪯{\displaystyle \frac{\left|{\Phi }^{n-m+1}\left(z\right)\right|}{d(z,{\Gamma }_{R_1})}}{n}^{({\gamma }^{\ast }+m+1){\nu }^{\ast }}{∥P_n∥}_1.$$

Finally, to complete the proof of the theorems, let us show how the distance of point z from curve ${\Gamma }_{R_1}$, obtained in the expressions in the proof of the theorem, is replaced by the distance of this point from the boundary of the region. That is, let us show that $d(z,{\Gamma }_{R_1})⪰d(z,\Gamma )$ holds for all $z\in {\Omega }_R$. For this let us give some notations. For $0<{\delta }_j<{\delta }_0:={\displaystyle \frac{1}{4}}min\left\{\left|z_i-z_j\right|:i,j=1,2,...,l,i\ne j\right\}$, let $\Omega (z_j,{\delta }_j):=\Omega \cap \left\{z:\left|z-z_j\right|\le {\delta }_j\right\};$ $\delta :={\displaystyle \underset{1\le j\le l}{min}}{\delta }_j$; For $\Gamma =\partial G$ we set: $U_{\infty }(\Gamma ,\delta ):=\bigcup _{\zeta \in \Gamma }U(\zeta ,\delta )-$infinite open cover of the curve $\Gamma$; $U(\Gamma ,\delta ):={\displaystyle \bigcup _{j=1}^N}{U}_j(\Gamma ,\delta )\subset U_{\infty }(\Gamma ,\delta )-$finite open cover of the curve $L;$ $\Omega \left(\delta \right):=\Omega (\Gamma ,\delta ):=\Omega \cap U_N(\Gamma ,\delta ).$ Now, for the points $z\notin \Omega ({\Gamma }_{R_1},d({\Gamma }_{R_1},{\Gamma }_R)),$ we have: $d(z,{\Gamma }_{R_1})⪰\delta ⪰d(z,\Gamma ).$ Next, let $z\in \Omega ({\Gamma }_{R_1},d({\Gamma }_{R_1},{\Gamma }_R)).$ Denote by ${\xi }_1\in {\Gamma }_{R_1}$ the point such that $d(z,{\Gamma }_{R_1})=\left|z-{\xi }_1\right|,$ and point ${\xi }_2\in \Gamma$, such that $d(z,\Gamma )=\left|z-{\xi }_2\right|,$ and for $w=\Phi \left(z\right),t_1=\Phi \left({\xi }_1\right),t_2=\Phi \left({\xi }_2\right),$ we have: $\left|w-w_1\right|\ge \left|\left|w-w_2\right|-\left|w_2-w_1\right|\right|\ge \left|\left|w-w_2\right|-{\displaystyle \frac{1}{2}}\left|w-w_2\right|\right|\ge {\displaystyle \frac{1}{2}}\left|w-w_2\right|.$ Then, according to Lemma 1, we obtain that $d(z,{\Gamma }_{R_1})⪰d(z,\Gamma ).$ Thus, the proofs of Theorems 1 and 2 for any $p\ge 1$ are completed. □

Proof of Theorem 3. 

Let us give one fact from our joint work ([32], Th.1.1). Let $p>0.$ Suppose that $G\in PAC({\nu }_1,...,{\nu }_l)$ for some $0<{\nu }_1,...,{\nu }_l\le 1;$ $h\left(z\right)$ defined as in (1). Then, for any $P_n\in {\wp }_n,n\in \mathbb{N},$ and arbitrarily small $\epsilon >0,$ there exists $c_1=c_1(G,p,{\gamma }_j)>0$ such that

$${∥P_n∥}_{\infty }\le c_1{n}^{{\displaystyle \frac{2+{\gamma }^{\ast }}{p}}\left(2-{\tilde{\nu }}^{\ast }\right)}{∥P_n∥}_p,$$

(64)

where ${\gamma }^{\ast }:=max\left\{0,{\gamma }_i\right\}$ and ${\tilde{\nu }}^{\ast }:=min\left\{{\nu }_j^{\ast }\right\},$ ${\nu }_j^{\ast }:=\left\{\begin{array}{cc}2-{\nu }_j,& \mathrm{if}0<{\nu }_j<1,\\ 1+\epsilon ,& \mathrm{if}{\nu }_j=1,\end{array}\right.$ for all $j=\overline{1,l}.$

Note that in ([32], Th.1.1), on the right-hand side of (64), the multiplier is incorrectly indicated as ${(n+1)}^{{\displaystyle \frac{2+{\gamma }^{\ast }}{p}}\left(2-{\nu }^{\ast }\right)+\epsilon }.$ This technical error is evident from the proof of the theorem ([32], Th.1.1; see, p. 150, line 8 from the bottom).

Let z be an arbitrary fixed point on $\Gamma$ and $U(z,d(z,{\Gamma }_R)):=\left\{t:\left|t-z\right|<d(z,{\Gamma }_R)\right\}$. By Cauchy integral formula for derivatives, we have

$$P_n^{\left(m\right)}\left(z\right)={\displaystyle \frac{m!}{2\pi i}}\underset{\partial U(z,d(z,{\Gamma }_R))}{\int }{\displaystyle \frac{P_n\left(\zeta \right)}{{(\zeta -z)}^{m+1}}}d\zeta ,m=0,1,2,...$$

Then,

$$\begin{array}{ccc}\left|P_n^{\left(m\right)}\left(z\right)\right|\hfill & \le & {\displaystyle \frac{m!}{2\pi }}\underset{z\in \partial U(z,d(z,{\Gamma }_R))}{max}\left|P_n\left(\zeta \right)\right|\underset{\partial U(z,d(z,{\Gamma }_R))}{\int }{\displaystyle \frac{\left|d\zeta \right|}{{\left|\zeta -z\right|}^{m+1}}}\hfill \\ & ⪯& \underset{\zeta \in {\overline{G}}_R}{max}\left|P_n\left(\zeta \right)\right|{\displaystyle \frac{1}{d^{m+1}(z,{\Gamma }_R)}}·2\pi d(z,{\Gamma }_R)⪯\underset{\zeta \in {\overline{G}}_R}{max}\left|P_n\left(\zeta \right)\right|{\displaystyle \frac{1}{d^m(z,{\Gamma }_R)}}.\hfill \end{array}$$

(65)

Let $z_R^1$ denote one of the points on ${\Gamma }_R$ such that $d(z,{\Gamma }_R)=\left|z-z_R^1\right|$ and let $w=\Phi \left(z\right),$ $w_R^1=\Phi \left(z_R^1\right)$. Subsequently, based on Lemmas 1 and 2, we derive

$$\begin{array}{ccc}\left|w-w_R^1\right|\hfill & =& \left|\Phi \left(z\right)-\Phi \left(z_R^1\right)\right|⪯d^{{\displaystyle \frac{1}{2-\nu }}}(z,{\Gamma }_R),z\in {\Gamma }_{1,1}^1,\hfill \\ \left|w-w_R^1\right|\hfill & =& \left|\Phi \left(z\right)-\Phi \left(z_R^1\right)\right|⪯d^{1-\epsilon }(z,{\Gamma }_R),z\in {\Gamma }_{1,k}^1,k=2,3,\hfill \end{array}$$

for arbitrary small $\epsilon >0.$ Therefore,

$$d(z,{\Gamma }_R)⪰\left\{\begin{array}{c}{\left|w-w_R^1\right|}^{2-\nu }\\ {\left|w-w_R^1\right|}^{1+\epsilon }\end{array}\right.⪰\left\{\begin{array}{cc}{\left({\displaystyle \frac{1}{n}}\right)}^{2-\nu },& z\in {\Gamma }_{1,1}^1,\\ {\left({\displaystyle \frac{1}{n}}\right)}^{1+\epsilon },& z\in {\Gamma }_{1,k}^1,k=2,3,\end{array}\right.\forall \epsilon >0.$$

Considering this estimation and utilizing (4) along with (64), we obtain the result

$$\left|P_n^{\left(m\right)}\left(z\right)\right|⪯n^{{\displaystyle \frac{2+{\gamma }^{\ast }}{p}}\left(2-{\tilde{\nu }}^{\ast }\right)}{∥P_n∥}_p·n^{m(2-{\tilde{\nu }}^{\ast })}⪯n^{\left({\displaystyle \frac{2+{\gamma }^{\ast }}{p}}+m\right)(2-{\tilde{\nu }}^{\ast })}{∥P_n∥}_p.$$

from (65).

Now, we will begin to prove (19). By the Cauchy integral formula for derivatives, we have

$$P_n^{\left(m\right)}\left(z\right)={\displaystyle \frac{m!}{2\pi i}}\underset{{\Gamma }_R}{\int }{\displaystyle \frac{P_n\left(\zeta \right)}{{(\zeta -z)}^{m+1}}}d\zeta ,m=0,1,2,...,z\in G_R.$$

(66)

As emphasized above, we can prove the theorem for an arbitrary point $z_j$. For simplicity, we assume $j=1$, and we will provide the proof for the point $z_1$. By substituting $t=\Psi \left(w\right)$, multiplying both the numerator and denominator of the integrand by $h^{{\displaystyle \frac{1}{p}}}$, and utilizing Hölder’s inequality, we derive the result

$$\begin{array}{ccc}\left|P_n^{\left(m\right)}\left(z_1\right)\right|\hfill & \le & {\displaystyle \frac{m!}{2\pi }}\underset{{\Gamma }_R}{\int }{\displaystyle \frac{\left|P_n\left(\zeta \right)\right|\left|d\zeta \right|}{{\left|\zeta -z_1\right|}^{m+1}}}\hfill \\ & =& {\displaystyle \frac{m!}{2\pi }}\underset{\left|w\right|=R}{\int }{\displaystyle \frac{h^{\frac{1}{p}}(\Psi \left(\tau \right))\left|P_n\left(\Psi \left(\tau \right)\right)\right|{\left|{\Psi }^{\prime }\left(\tau \right)\right|}^{2/p}{\left|{\Psi }^{\prime }\left(\tau \right)\right|}^{1-\frac{2}{p}}\left|d\tau \right|}{h^{\frac{1}{p}}(\Psi \left(\tau \right)){\left|\Psi \left(\tau \right)-\Psi \left(w_1\right)\right|}^{m+1}}}\hfill \\ & ⪯& {\left(\underset{\left|w\right|=R}{\int }h(\Psi \left(\tau \right)){\left|P_n\left(\Psi \left(\tau \right)\right)\right|}^p{\left|{\Psi }^{\prime }\left(\tau \right)\right|}^2\left|d\tau \right|\right)}^{{\displaystyle \frac{1}{p}}}\hfill \\ & & \times {\left(\underset{\left|w\right|=R}{\int }{\displaystyle \frac{{\left|{\Psi }^{\prime }\left(\tau \right)\right|}^{q(1-\frac{2}{p})}\left|d\tau \right|}{h^{\frac{q}{p}}(\Psi \left(\tau \right)){\left|\Psi \left(\tau \right)-\Psi \left(w_1\right)\right|}^{q(m+1)}}}\right)}^{{\displaystyle \frac{1}{q}}}\hfill \\ & =& {\left(\underset{{\Gamma }_R}{\int }{\displaystyle \frac{h\left(\zeta \right)}{\left|{\Phi }^{\prime }\left(\zeta \right)\right|}}{\left|P_n\left(\zeta \right)\right|}^p\left|d\zeta \right|\right)}^{{\displaystyle \frac{1}{p}}}{\left(\underset{\left|w\right|=R}{\int }{\displaystyle \frac{{\left|{\Psi }^{\prime }\left(w\right)\right|}^{q(1-\frac{2}{p})}\left|dw\right|}{{\left|\Psi \left(\tau \right)-\Psi \left(w_1\right)\right|}^{{\gamma }_1(q-1)+q(m+1)}}}\right)}^{{\displaystyle \frac{1}{q}}}\hfill \\ & =& :D_{n,1}·D_{n,2},{\displaystyle \frac{1}{p}}+{\displaystyle \frac{1}{q}}=1.\hfill \end{array}$$

(67)

Applying Lemma 3, for the integral $D_{n,1}$, we have

$$D_{n,1}⪯n^{{\displaystyle \frac{1}{p}}}·{∥P_n∥}_p.$$

(68)

For the integral $D_{n,2},$ using (25), we find

$$\begin{array}{ccc}{\left(D_{n,2}\right)}^q\hfill & =& \underset{\left|w\right|=R}{\int }{\displaystyle \frac{{\left|{\Psi }^{\prime }\left(w\right)\right|}^{2-q}\left|dw\right|}{{\left|\Psi \left(\tau \right)-\Psi \left(w_1\right)\right|}^{{\gamma }_1(q-1)+q(m+1)}}}\hfill \\ & \asymp & \underset{\left|w\right|=R}{\int }{\left({\displaystyle \frac{d(\Psi \left(w\right),\Gamma )}{\left|w\right|-1}}\right)}^{2-q}{\displaystyle \frac{\left|dw\right|}{{\left|\Psi \left(\tau \right)-\Psi \left(w_1\right)\right|}^{{\gamma }_1(q-1)+q(m+1)}}}\hfill \\ & ⪯& n^{2-q}\underset{\left|w\right|=R}{\int }{\displaystyle \frac{d^{2-q}(\Psi \left(w\right),\Gamma )\left|dw\right|}{{\left|\Psi \left(\tau \right)-\Psi \left(w_1\right)\right|}^{{\gamma }_1(q-1)+q(m+1)}}}=:n^{2-q}I\left(F_R\right),\hfill \end{array}$$

(69)

where $F_R:=\Phi \left({\Gamma }_R\right)$. For the estimation of the integral $I\left(F_R\right),$ we set

• $E_R^{11}:=\left\{w:w\in F_R,\left|w-w_1\right|<{\delta }_1\right\},$
• $E_R^{12}:=\left\{w:w\in F_R,{\delta }_1\le \left|w-w_1\right|<{\delta }_1^{\ast }\right\},$
• $E_R^{13}:=\left\{w:w\in F_R,{\delta }_1^{\ast }\le \left|w-w_1\right|<\mathrm{diam}\left(\overline{G}\right)\right\};\Phi \left({\Gamma }_R\right)=\bigcup _{k=1}^3{E}_R^{1k}.$

Since for any $q>1,$ ${\gamma }_1>-2$ and $m\ge 1$, ${\gamma }_1(q-1)+q(m+1)-(2-q)=({\gamma }_1+2)(q-1)+qm>1,$ then for the integral $I\left(F_R\right),$ we obtain

$$\begin{array}{ccc}I\left(E_R^{11}\right)\hfill & ⪯& \underset{E_{R_2}^{11}}{\int }{\displaystyle \frac{\left|dw\right|}{{\left|\Psi \left(w\right)-\Psi \left(w_1\right)\right|}^{{\gamma }_1(q-1)+q(m+1)-\left(2-q\right)}}}\hfill \\ & ⪯& \underset{E_{R_2}^{11}}{\int }{\displaystyle \frac{\left|dw\right|}{{\left|w-w_1\right|}^{\left[{\gamma }_1(q-1)+q(m+1)-(2-q)\right](2-{\nu }_1)}}}⪯n^{\left[{\gamma }_1(q-1)+q(m+1)-(2-q)\right](2-{\nu }_1)-1},\hfill \end{array}$$

(70)

for $0<{\nu }_1<1,$ and

$$\begin{array}{ccc}I\left(E_R^{11}\right)\hfill & ⪯& \underset{E_R^{11}}{\int }{\displaystyle \frac{\left|dw\right|}{{\left|\Psi \left(w\right)-\Psi \left(w_1\right)\right|}^{\gamma (q-1)+q(m+1)-\left(2-q\right)}}}⪯\underset{E_R^{11}}{\int }{\displaystyle \frac{\left|dw\right|}{{\left|w-w_1\right|}^{\left[\gamma (q-1)+q(m+1)-(2-q)\right](1+\epsilon )}}}\hfill \\ & ⪯& n^{\left[\gamma (q-1)+q(m+1)-(2-q)\right]-1+\epsilon },\forall \epsilon >0,\hfill \end{array}$$

for ${\nu }_1=1;$

$$I\left(E_R^{12}\right)⪯\underset{E_R^{12}}{\int }{\displaystyle \frac{d^{2-q}(\Psi \left(w\right),\Gamma )\left|dw\right|}{{\left|\Psi \left(w\right)-\Psi \left(w_1\right)\right|}^{\gamma (q-1)+q(m+1)}}}⪯\underset{E_R^{12}}{\int }{\displaystyle \frac{\left|dw\right|}{{\left({\delta }_1\right)}^{\gamma (q-1)+q(m+1)}}}⪯1;$$

(71)

$$I\left(E_R^{13}\right)⪯\underset{E_R^{13}}{\int }{\displaystyle \frac{d^{2-q}(\Psi \left(w\right),\Gamma )\left|dw\right|}{{\left|\Psi \left(w\right)-\Psi \left(w_1\right)\right|}^{\gamma (q-1)+q(m+1)}}}⪯\underset{E_R^{13}}{\int }{\displaystyle \frac{\left|dw\right|}{{\left({\delta }_1^{\ast }\right)}^{\gamma (q-1)+q(m+1)}}}⪯1.$$

(72)

Combining (69)–(72), for $p>1,{\gamma }_1>-2$, we obtain

$$D_{n,2}⪯n^{\left({\displaystyle \frac{{\gamma }_1+2}{p}}+m\right){\nu }_1^{\ast }-{\displaystyle \frac{1}{p}}}.$$

(73)

Thus, by combining (67) and (68) with (73), for the case where $p>1$ and $\gamma >-2$, we obtain the result

$$\left|P_n^{\left(m\right)}\left(z_1\right)\right|⪯n^{\left({\displaystyle \frac{{\gamma }_1+2}{p}}+m\right){\nu }_1^{\ast }}{∥P_n∥}_p.$$

(74)

Let us show that it is also true for $p=1$. To do this, we write the integral representation (66) for $z_1$, by multiplying both the numerator and denominator of the integrand by h, and subsequently applying (25) to transition to the moduli in both components, we arrive at the result

$$\begin{array}{ccc}\left|P_n^{\left(m\right)}\left(z_1\right)\right|\hfill & ⪯& \underset{{\Gamma }_R}{\int }{\displaystyle \frac{h\left(\zeta \right)\left|P_n\left(\zeta \right)\right|\left|d\zeta \right|}{{\left|\zeta -z_1\right|}^{{\gamma }_1}{\left|\zeta -z_1\right|}^{m+1}}}=\underset{{\Gamma }_R}{\int }{\displaystyle \frac{\frac{h\left(\zeta \right)}{\left|{\Phi }^{\prime }\left(\zeta \right)\right|}\left|{\Phi }^{\prime }\left(\zeta \right)\right|\left|P_n\left(\zeta \right)\right|\left|d\zeta \right|}{{\left|\zeta -z_1\right|}^{{\gamma }_1}{\left|\zeta -z_1\right|}^{m+1}}}\hfill \\ & ⪯& \underset{\zeta \in {\Gamma }_R}{sup}\left({\displaystyle \frac{\left|{\Phi }^{\prime }\left(\zeta \right)\right|}{{\left|\zeta -z_1\right|}^{{\gamma }_1+m+1}}}\right)\underset{{\Gamma }_R}{\int }{\displaystyle \frac{h\left(\zeta \right)}{\left|{\Phi }^{\prime }\left(\zeta \right)\right|}}\left|P_n\left(\zeta \right)\right|\left|d\zeta \right|\hfill \\ & ⪯& \underset{\zeta \in {\Gamma }_R}{sup}\left({\displaystyle \frac{\left|\Phi \left(\zeta \right)\right|-1}{d(\zeta ,\Gamma ){\left|\zeta -z_1\right|}^{{\gamma }_1+m+1}}}\right)·n·{∥P_n∥}_1\hfill \\ & =& \underset{\zeta \in {\Gamma }_R}{sup}\left({\displaystyle \frac{1}{d(\zeta ,\Gamma ){\left|\zeta -z_1\right|}^{{\gamma }_1+m+1}}}\right){∥P_n∥}_1.\hfill \end{array}$$

(75)

Let us estimate the last fractional expression on the parts ${\Gamma }_R$:

(a) For $\zeta \in E_R^{11}$:

$${\displaystyle \frac{1}{d(\zeta ,\Gamma ){\left|\zeta -z_1\right|}^{{\gamma }_1+m+1}}}⪯{\displaystyle \frac{1}{{\left|\tau -w_1\right|}^{\left({\gamma }_1+m+2\right)(2-{\nu }_1^{\ast })}}}⪯n^{({\gamma }_1+m+2)(2-{\nu }_1^{\ast })},$$

for $0<{\nu }_1<1,$ and

$${\displaystyle \frac{1}{d(\zeta ,\Gamma ){\left|\zeta -z_1\right|}^{{\gamma }_1+m+1}}}⪯{\displaystyle \frac{1}{{\left|\tau -w_1\right|}^{\left({\gamma }_1+m+2\right)(1+\epsilon )}}}⪯n^{({\gamma }_1+m+2)(1+\epsilon )},\forall \epsilon >0,$$

for ${\nu }_1=1;$

(b) For $\zeta \in E_R^{1k},k=2,3$:

$${\displaystyle \frac{1}{d(\zeta ,\Gamma ){\left|\zeta -z_1\right|}^{{\gamma }_1+m+1}}}⪯{\displaystyle \frac{1}{{\left({\delta }_1\right)}^{({\gamma }_1+m+1)}d(\zeta ,\Gamma )}}⪯n^{(2-{\nu }_1^{\ast })};$$

$${\displaystyle \frac{1}{d(\zeta ,\Gamma ){\left|\zeta -z_1\right|}^{{\gamma }_1+m+1}}}⪯{\displaystyle \frac{1}{{\left({\delta }_1^{\ast }\right)}^{({\gamma }_1+m+1)}d(\zeta ,\Gamma )}}⪯n^{(2-{\nu }_1^{\ast })}.$$

By substituting the derived estimates into (75), we obtain the following expression:

$$\left|P_n^{\left(m\right)}\left(z_1\right)\right|⪯n^{({\gamma }_1+m+2){\nu }_1^{\ast }}{∥P_n∥}_1,$$

and as a result, for $p\ge 1$ and $\gamma >-2$, we derive the estimate

$$\left|P_n^{\left(m\right)}\left(z_1\right)\right|⪯n^{\left({\displaystyle \frac{{\gamma }_1+2}{p}}+m\right){\nu }_1^{\ast }}{∥P_n∥}_p.$$

Since, the point $z_1$ was chosen arbitrarily, the proof of Theorem 3 is completed. □

Proof of Remark 1. 

The sharpness of the inequalities (19) and (20) can be demonstrated as follows. The sharpness for $m=0$ was established in [32]. For $m\ge 1$, the sharpness is obtained by successively applying the well-known Markov inequality. □

6. Discussion

This work presents a comprehensive treatment of the asymptotic growth of the mth-order derivatives of algebraic polynomials within the framework of weighted Bergman spaces over regions exhibiting complex geometric structures. Specifically, we have examined regions bounded by bounded piecewise asymptotically conformal curves with strictly positive interior angles—a class of regions that encapsulates both analytic complexity and geometric irregularity, beyond the scope of classical conformal or smoothly bounded regions.

The principal methodological novelty of this study lies in the integration of local analytic behavior with global geometric constraints through sharp growth estimates. By initially establishing norm estimates in unbounded subregions and subsequently extending the analysis to closure of the bounded regions, we have constructed a unified analytic apparatus that captures the interaction between derivative growth and boundary geometry. This approach facilitates a rigorous passage from localized behavior to global control, culminating in robust growth bounds across the entirety of the complex plane.

Furthermore, the analytical techniques developed here may be viewed as a generalization of traditional Bernstein- and Markov-type inequalities, adapted to the non-Euclidean metric induced by weighted Bergman structures and nontrivial region topology. These results are expected to have ramifications in a variety of contexts, including weighted potential theory, the theory of extremal functions, and approximation in non-smooth or non-convex settings. In particular, the treatment opens avenues for the analysis of sharp constants, optimal growth rates, and the spectral geometry of associated integral operators in more generalized function-theoretic environments. In this sense, the findings not only advance our understanding of polynomial growth in weighted analytic function spaces but also offer a versatile analytic framework capable of being adapted to more abstract settings involving quasiconformal structures and singular boundary behavior.