Growth Estimates for m − th , m ≥ 1, Derivatives of Algebraic Polynomials in Domains with Piecewise Quasismooth Boundaries Having Zero Angles in Weighted Lebesgue Spaces

1. Introduction and Definitions

One of the fundamental problems in the theory of analytic functions of a complex variable is determining the behavior of the modulus of an analytic function as its domain of analyticity expands. This problem becomes particularly relevant in the case of algebraic polynomials. Similar studies arise naturally in various areas of analysis, including inverse problems in polynomial approximation theory. In this paper, we continue our investigation of estimates for the growth of the modulus of arbitrary algebraic polynomials at points lying outside the closure of a given domain. We simultaneously obtain corresponding estimates on the closure of the domain. The combination of these estimates provides a comprehensive description of the growth behavior of the modulus of the polynomials under consideration on the entire complex plane.

Before discussing the main results, we introduce the necessary notation and preliminary definitions.

Let $\mathbb{C}$ be a complex plane; $\overline{\mathbb{C}}:=\mathbb{C}\cup \left\{\infty \right\};G$ $\subset \mathbb{C}$ be a bounded Jordan domain with boundary $L:=\partial G$; $\mathsf{\Omega }:=\overline{\mathbb{C}}\setminus \overline{G}=extL$. For $t\in \mathbb{C}$ and $\delta >0,$ let us set: $\Delta (t,\delta ):=\left\{w\in \mathbb{C}:\left|w-t\right|>\delta \right\};$ $\Delta :=\Delta (0,1)$. Consider the univalent conformal map $\mathsf{\Phi }:\mathsf{\Omega }\to \Delta$ normalized at infinity by $\mathsf{\Phi }(\infty )=\infty$ and ${lim}_{z\to \infty }{\displaystyle \frac{\mathsf{\Phi }\left(z\right)}{z}}>0$. Let $\mathsf{\Psi }$ denote the inverse function of $\mathsf{\Phi }$.

For each $\delta \ge 0$, we define the exterior level curves associated with L by

$$L_{1+\delta }:=\left\{z:\left|\mathsf{\Phi }\left(z\right)\right|=1+\delta \right\},L_1\equiv L,$$

and we set $G_{1+\delta }:=int{L}_{1+\delta },{\mathsf{\Omega }}_{1+\delta }:=ext{L}_{1+\delta }.$ Throughout the paper,

$${\wp }_n:=\left\{P_n\left(z\right)={\displaystyle \sum _{k=0}^n}{a}_k{z}^k:a_k\in \mathbb{C}\right\},$$

denotes the class of all algebraic polynomials of degree at most $n\in \mathbb{N}$. For a set $M\subset \mathbb{C}$, denote by $d(z,M):=dist(z,M)=inf\left\{\left|\zeta -z\right|:\zeta \in M\right\}$.

Let ${\left\{z_j\right\}}_{j=1}^l\in L$ be a fixed collection of distinct points. For a fixed $R_0,$ $1<R_0<\infty ,$ we consider the generalized Jacobi weight function

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

(1)

where ${\gamma }_j>-1,$ for$j=1,2,...,l,$ and the function $h_0$ satisfies $h_0\left(z\right)\ge c_0\left(L\right)>0$ for some positive constant $c_0\left(L\right)>0.$

Let G be a Jordan domain with rectifiable boundary $L=\partial G$. For $0<p\le \infty$, we introduce:

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

A fundamental result due to Bernstein and Walsh [1] states that

$${∥P_n∥}_{C\left({\overline{G}}_{\rho }\right)}\le {(1+\delta )}^n{∥P_n∥}_{C\left(\overline{G}\right)},\forall P_n\in {\wp }_n.$$

(2)

A ${\mathcal{L}}_p\left(L\right)$-version of the above inequality, valid for $p>0$, was proved in [2]:

$${∥P_n∥}_{{\mathcal{L}}_p\left(L_{1+\delta }\right)}\le {(1+\delta )}^{n+{\displaystyle \frac{1}{p}}}{∥P_n∥}_{{\mathcal{L}}_p\left(L\right)},\forall P_n\in {\wp }_n.$$

(3)

A weighted version of estimate (3) in the space ${\mathcal{L}}_p(h,L_R)$, with h is defined (1), was obtained in [3] [Lemma 2.4]:

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

(4)

Before presenting the analogous inequality (2)–(4) for the two-dimensional analogue of ${∥P_n∥}_{A_p(h,G)},$ defined for an arbitrary Jordan domain G, we first introduce the following definition:

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

where $\sigma$ denotes the two-dimensional Lebesgue measure.

We begin by introducing the following definitions.

A curve $L$ is called quasiconformal if, for arbitrary points $z_1\in L$ and $z_2\in L$ , the diameter of the shorter arc $l(z_1,z_2)$ of the curve L joining points $z_1,z_2$ satisfies the inequality ([4][p.102]):

$${\displaystyle \frac{diaml(z_1,z_2)}{\left|z_1-z_2\right|}}\le c<+\infty .$$

(5)

It should be noted that quasiconformal curves may not be rectifiable (see, for example, [4], [5] [p.104]).

Two-dimensional analogues of the inequalities (2) - (4) for a domain G bounded by an arbitrary quasiconformal curve and a weight function h defined as (1) for ${\gamma }_j>-2,$ $j=1,2,...,l,$ are presented in [6] as follows:

$${∥P_n∥}_{A_p(h,G_{1+\delta })}\le c_1{\left[1+c_2\delta \right]}^{n+{\displaystyle \frac{1}{p}}}{∥P_n∥}_{A_p(h,G)},p>0,$$

(6)

where $c_2>0$ and $c_1:=c_1(G,p,c_2)>0$ constants, independent of n and $z.$Later, [7] [Theorem1.1] established the following version of this estimate for an arbitrary Jordan domain G when$h\left(z\right)\equiv 1$:

$${∥P_n∥}_{A_p\left(G_{1+\delta }\right)}\le c_3{(1+\delta )}^{n+{\displaystyle \frac{2}{p}}}{∥P_n∥}_{A_p(G_{1+\frac{1}{n}})},\delta >{\displaystyle \frac{1}{n}},p>0,$$

where $c_3={\left({\displaystyle \frac{2}{e^p-1}}\right)}^{{\displaystyle \frac{1}{p}}}\left[1+O\left({\displaystyle \frac{1}{n}}\right)\right],n\to \infty ,$ is asymptotically sharp constant.

In [8], for domains bounded by a rectifiable quasiconformal curve, an inequality analogous to (6) was established, incorporating the location of a point exterior to the given domain, expressed in the form:

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

where $c\left(L\right)>0$ is a constant depending only on the curve$L.$

Let $S\subset \mathbb{C}$be a rectifiable Jordan curve or arc, and$z=z\left(s\right),s\in \left[0,\left|S\right|\right],\left|S\right|:=mesS,$ be the natural parametrization of the curve S. Let $z_{1,}$ $z_2$ be arbitrary points on S, and let $l(z_1,z_2)$ denote the subarc of S joining $z_1$ and $z_2$ with smaller diameter. We denote by $\left|l(z_1,z_2)\right|$ the linear measure (length) of this arc. Following [9] [p.163], we say that a bounded Jordan curve S is a $\lambda$-quasismooth (in the sense of Lavrentiev) curve if there exists a constant $\lambda =\lambda \left(S\right)\ge 1$ such that for every pair $z_1,z_2\in S$,

$$\left|l(z_1,z_2)\right|\le \lambda \left|z_1-z_2\right|,z_1,z_2\in S.$$

The domain G is called a $\lambda -$quasismooth domain, if $L=\partial G$ is a $\lambda -$quasismooth curve. Any subarc of a $\lambda$-quasismooth curve is referred to as a $\lambda$-quasismooth arc. We denote the collection of such curves and arcs by $QS\left(\lambda \right)$, and we say that a Jordan domain $G\in QS\left(\lambda \right)$ if $\partial G\in QS\left(\lambda \right)$, for $\lambda \ge 1$. In addition, we write $L\in QS$ (or $G\in QS$) if $L\in QS\left(\lambda \right)$ (or $G\in QS\left(\lambda \right)$) for some $\lambda \ge 1$. In [10] (for $m=0$ and when L is a$\lambda -$quasismooth curve) and [11] (for $m\ge 0$ and for more general class of curves, which also includes $\lambda -$quasismooth curves, using recursive formulas) the authors investigated pointwise and uniform estimates for $\left|P_n^{\left(m\right)}\left(z\right)\right|,$ $m\ge 0,$ in unbounded domain $\mathsf{\Omega }$ and on the closure of the bounded domain G and the following type of estimate was obtained:

$$\left|P_n^{\left(m\right)}\left(z\right)\right|\le c_4{∥P_n∥}_p\left\{\begin{array}{cc}\hfill ll{\nu }_n,& z\in \overline{G},\\ \hfill {\eta }_n,& z\in \mathsf{\Omega },\end{array}\right.$$

(7)

where $c_4=c_4(L,p,m,\gamma )>0$ is a constant independent of $n,h$, $P_n,$ ${\nu }_n={\nu }_n(L,h,p)>0$ and ${\eta }_n={\eta }_n(L,h,p,z)\to \infty$, as $n\to \infty$, are constants depending on the properties of L and h.

Note that, $\lambda -$quasismooth curves do not have any cusps, i.e., points with interior or exterior zero angles on the boundary curve. Now, we introduce domains with piecewise quasismooth boundaries that admit a finite number of interior and exterior zero angles.

We now introduce a new class of domains whose boundaries are piecewise quasismooth. At boundary points, such domains may contain a finite number of both interior and exterior cusps relative to the given domain.

Let L be a bounded Jordan curve or arc and let $z\in L$. We say that L is locally $\lambda$-quasismooth at z if there exists a closed subarc $\ell \subset L$ containing z such that every open subarc of ℓ containing z is $\lambda$-quasismooth. By the “three-point” criterion [4] [p. 100], [12], every quasismooth curve is quasiconformal.

Throughout this paper, $c,c_0,c_1,c_2,...$ are positive and ${\epsilon }_0,{\epsilon }_1,{\epsilon }_2,...$ are sufficiently small positive constants (generally, different in different relations), which depend on L in general and, on parameters inessential for the argument; otherwise, the dependence will be explicitly stated. For any $k\ge 0$ and $m>k,$ notation $i=\overline{k,m}$ means $i=k,k+1,...,m$. Furthermore, we assume that the points ${\left\{z_i\right\}}_{i=1}^l\in L$ introduced in (1) and Definition 1 coincide. Without loss of generality, we further suppose that the points ${\left\{z_i\right\}}_{i=0}^l$ are arranged along the curve L in the positive orientation, so that $G$possesses interior zero angles at the points ${\left\{z_i\right\}}_{i=1}^{l_1},$ when $l_1\ge 1$ and exterior zero angles at the points ${\left\{z_i\right\}}_{i=l_1+1}^l,$ when $l\ge l_1+1$.

For each $j=1,2,\dots$ and for sufficiently small ${\epsilon }_1>0$, let $f_j,g_j:[0,{\epsilon }_1]\to \mathbb{R}$ be twice continuously differentiable functions satisfying $f_j\left(0\right)=g_j\left(0\right)=0$ and $f_j^{\left(k\right)}\left(x\right)>0$, $g_j^{\left(k\right)}\left(x\right)>0$ for $x>0$ and $k=0,1,2$.

Definition 1. 

[10] We say that a Jordan domain $G\in PQS\left(\lambda ;f_i{,g}_j\right)$, $\lambda \ge 1$, $f_i=f_i\left(x\right),i=\overline{1,l_1},$ $g_j=g_j\left(x\right),j=\overline{l_1+1,l}$, if $L:=\partial G$ $={\displaystyle \bigcup _{j=0}^l}{L}_j$is the union of the finite number of λ-quasismooth arcs $L_j$, connecting at the points ${\left\{z_j\right\}}_{j=0}^l\in L,$ and such that L is a locally λ-quasismooth arc at the $z_0\in L\setminus {\left\{z_j\right\}}_{j=1}^l$ and, in the $\left(x,y\right)$ local coordinate system with the origin at the $z_j$, $1\le j\le l,$ and axes parallel to the main coordinate system $XOY$, the following conditions are satisfied:

$a)$ for every $z_j\in L,j=\overline{1,l_1},l_1\le l,$

$$\begin{array}{cc}\hfill \left\{z=x+iy:\left|z\right|\le {\epsilon }_1,c_1{f}_i\left(x\right)\le y\le c_2{f}_i\left(x\right)\right\}& \subset \overline{G},\hfill \\ \hfill \left\{z=x+iy:\left|z\right|\le {\epsilon }_1,\left|y\right|\ge {\epsilon }_{2}x\right\}& \subset \overline{\mathsf{\Omega }};\hfill \end{array}$$

$b)$ for every $z_j\in L,j=\overline{l_1+1,l},$

$$\begin{array}{cc}\hfill \left\{z=x+iy:\left|z\right|<{\epsilon }_3,c_3{g}_j\left(x\right)\le y\le c_4{g}_j\left(x\right),0\le x\le {\epsilon }_3\right\}& \subset \overline{\mathsf{\Omega }},\hfill \\ \hfill \left\{z=x+iy:\left|z\right|<{\epsilon }_3,\left|y\right|\ge {\epsilon }_{4}x,0\le x\le {\epsilon }_3\right\}& \subset \overline{G},\hfill \end{array}$$

for some constants $-\infty <c_1<c_2<\infty$, $-\infty <c_3<c_4<\infty$, ${\epsilon }_i>0$, $i=\overline{1,4}$.

It follows from Definition 1 that each domain $G\in PQS\left(\lambda ;f_i{,g}_j\right)$ may have $l_1$ interior and $l-l_1$ exterior zero angles (with respect to $\overline{G})$. If a domain G does not have interior zero angles $\left(l_1=0\right)$ (exterior zero angles $\left(l_1=l\right)$), then it is written as $G\in PQS\left(\lambda ;0{,g}_j\right)$ ($G\in PQS\left(\lambda ;f_i,0\right)$). If a domain G does not have such angles $\left(l=0\right)$, then G is bounded by a $\lambda -$ quasismooth circle and in this case we set $PQS\left(\lambda ,0,0\right)\equiv QS\left(\lambda \right)$.

In the work [13], for each $m\ge 1$ and for $p>1,$ the authors used a recurrence formula to investigate a similar problem for domains bounded by piecewise $\lambda -$quasismooth curve having interior and exterior zero angles and obtained an estimate of the following type:

$$\left|P_n^{\left(m\right)}\left(z\right)\right|\le {\eta }_n{∥P_n∥}_p,z\in \mathsf{\Omega },{\eta }_n={\eta }_n(L,h,p,z)\to \infty ,n\to \infty .$$

(8)

However, estimates obtained by means of the recurrence formula require lengthy and labor-intensive calculations. Moreover, this approach leads to a significant increase in the order of magnitude of the polynomial derivative estimate. In this work, bypassing the recurrence formula and using a different approach, sharper estimates of the type (7) are obtained for domains of the class $G\in PQS\left(\lambda ;f_i{,g}_j\right)$ with some given functions f and g of power type.

Analogous results of the type (8) for various norms and classes of unbounded domains were obtained by N.A. Lebedev, P.M. Tamrazov, V.K. Dzjadyk (see, for example, [14]) [pp. 418–428], [15] [p.383]), F. G. Abdullayev et al. in [16] (for domains with piecewise smooth boundary with interior zero and nonzero angles, $m=1,$ $p>1$), [17] (for domains with piecewise smooth boundary with interior angles, $m=0,$ $p>0$), [18] ($L-\partial G$ rectifiable asymptotically conformal curve, $m=0,$ $p>0$), [19] (for quasidiscs with an additional general functional condition, $m\ge 1,$ $p>1$), [20] ( $L-\partial G$ rectifiable piecewise quasicircle with cusps, $m=0,$ $p>0$), [3] ( $L-\partial G$ rectifiable piecewise Dini-smooth curve, $m=0,$ $p>0$), [10] ($L-$ $\partial G$ piecewise quasismooth curve with cusps, $m=0,$ $p>0$), [11] (for quasidisks with an additional functional condition using a recurrence formula, $m\ge 1,$ $p>1$), [21] (for $L-$piecewise smooth curve using a recurrence formula, $m=1,$ $p>1$), [22] (for $m\ge 1,$ $p\ge 1$ and quasidisks with an additional functional condition without a recurrence formula), [23] (for $m\ge 1,$ $p\ge 1;$ $L-\partial G$ piecewise quasicircle with cusps by using a recurrence formula) and also, the references cited therein.

It should be noted that most of these studies rely on a recurrence formula, in which the estimate for each derivative is obtained from the estimate of the preceding one. In the present paper, we avoid the use of the recurrence formula and derive estimates of the type (8) for each $m\ge 1$, independently of the estimates for lower-order derivatives.

To obtain global estimates in the whole complex plane, one also requires Bernstein-Markoff-Nikol’skii type inequalities for the derivatives of algebraic polynomials $P_n^{\left(m\right)}\left(z\right),$ $z\in \overline{G}$. Specifically, we consider inequalities of the form:

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

(9)

where the value ${\nu }_n:={\nu }_n(L,h,p,m)>0$ (${\nu }_n\to \infty$, $n\to \infty$ ), depends on the properties of the boundary L and the weight function h.

It should be noted that inequalities of type (9) trace back to the classical works of [24,25,26] and have subsequently been developed and generalized in numerous directions. In particular, Bernstein–Markov–Nikol’skii type inequalities for $m\ge 0$ and various function spaces have been investigated by many authors: see, for example, [14] [pp. 418–428], [27,28,29,30] [pp. 122-133], [31,32,33,34,35,36,37], as well as the references cited therein.

Among the works most closely related to the present study are [20] ( $L-$rectifiable piecewise quasicircle with cusps, $m=0,$ $p>0;$), [3] ($L-$rectifiable piecewise Dini-smooth curve, $m=0,$ $p>0$), [10] ($L-$ piecewise quasismooth curve with cusps, $m=0,$ $p>0$), [11] (for quasidisks with an additional functional condition and $m\ge 1,$ $p>1$), [22] (for quasidisks with an additional functional condition and $m\ge 1,$ $p>0$), [38] ($L-$piecewise quasicircle with cusps, $m\ge 1,$ $p>1$). Further related results may be found in the references cited therein.

Combining the interior estimates (9) with the exterior estimate (8), we obtain the following global growth estimate for the $m-th$ derivative of an algebraic polynomial on the whole complex plane:

$$\left|P_n^{\left(m\right)}\left(z\right)\right|\le c_4{∥P_n∥}_p\left\{\begin{array}{cc}{\nu }_n,& z\in {\overline{G}}_R,\\ {\eta }_n,& z\in {\mathsf{\Omega }}_R,\end{array}\right.$$

(10)

where $c_4=c_4(L,h,p,m)>0$ is a constant independent of $n,$ $P_n,$ and ${\nu }_n\to \infty$, ${\eta }_n\to \infty$ as $n\to \infty$, depending on the properties of L and h.

2. Main Results

Before stating the main results, we introduce some notation that will be used throughout the paper. To simplify the presentation of the theorems and their proofs and to avoid cumbersome formulas, we restrict our attention to domains whose boundary contains at most two zero-angle points. These zero angles may be of the same type (both interior or both exterior) or of different types (one interior and one exterior). The extension of the results obtained below to domains with an arbitrary finite number of zero-angle points is straightforward and can be carried out by analogous arguments. For the sake of simplicity, we assume that $i=1,2;l_1=1$ and $l=2.$ Thus, the domain G is assumed to possess one interior zero angle at a point $z_1,$ characterized by an " $f_1-$touching" condition with $f_1\left(x\right)=C_1{x}^{1+{\alpha }_1},$ ${\alpha }_1\ge 0,$ and one exterior zero angle at a point $z_2$ characterized by a "$g_2-$touching" condition with $g_2\left(x\right)=C_2{x}^{1+{\beta }_2},$ ${\beta }_2>0.$ Here $C_1$ and $C_2$ are finite constants satisfying $-\infty <C_1<+\infty$, $-\infty <C_2<+\infty$, where $C_1:=C_1(c_{11}^1,c_{12}^1)$ and $C_2:=C_2(c_{21}^2,c_{22}^2)$ and the parameters $c_{j,k}^i,i,j,k=1,2,$ are those introduced in Definition 1.

Throughout this paper, we assume that $m\ge 1$ and that ${\gamma }_i>-1$, ${\alpha }_i\ge 0$, and ${\beta }_i>0$ for $i=1,2$. We introduce the following quantities:

$${\gamma }^{*}:=max\{0;{\gamma }_k,k=1,2\};\mu :=2(1-{\displaystyle \frac{1}{\pi }}arcsin{\displaystyle \frac{1}{\lambda }}),1<\mu <2,\tilde{\mu }:=\left\{\begin{array}{cc}\mu ,& \mathrm{if}{\alpha }_i=0,\\ 2,& \mathrm{if}{\alpha }_i\ne 0,\end{array}\right.i=1,2.$$

2.1. Estimates for $\left|P_n^{\left(M\right)}\left(Z\right)\right|$ in Unbounded Domains

In this subsection, we consider the case where the boundary of the domain contains both an interior and exterior zero angle, and we derive estimates for $\left|P_n^{\left(m\right)}\left(z\right)\right|$ in such unbounded domains $\mathsf{\Omega }$. The following statement holds:

Theorem 1. 

Let $p\ge 1;$ $G\in PQS\left(\lambda ;f_1{,g}_2\right)$, for some $\lambda \ge 1$, $f_1\left(x\right)=C_1{x}^{1+{\alpha }_1},{\alpha }_1\ge 0$, $g_2\left(x\right)=C_2{x}^{1+{\beta }_2},$ ${\beta }_2>0$ and let $P_n\in {\wp }_n,n\in \mathbb{N}.$ Assume that the weight function h is defined by (1) with ${\gamma }_j>-1,$ $j=1,2$. Then, the following holds for every $m=0,1,...,n$:

$$\left|P_n^{\left(m\right)}\left(z\right)\right|\le c_1{\displaystyle \frac{\left|{\mathsf{\Phi }}^{n-m+1}\left(z\right)\right|}{d(z,L)}}{∥P_n∥}_p·A_{n,1}\left(m\right),z\in \mathsf{\Omega },$$

(11)

where $c_1=c_1(L,{\gamma }_i,\beta ,m,p)>0$ is a constant independent of n and $z;$ $A_{n,1}\left(m\right):=\left(A_{n,1}^1+A_{n,1}^2\right)·D_{n,1}\left(m\right),$

$$\begin{array}{cc}\hfill A_{n,1}^1& :=\left\{\begin{array}{cc}{n}^{\left({\displaystyle \frac{{\gamma }_1+1}{p}}-1\right)\tilde{\mu }},\hfill & 1\le p<{\gamma }_1+1,{\gamma }_1>0,\hfill \\ {\left(lnn\right)}^{1-{\displaystyle \frac{1}{p}}},\hfill & p={\gamma }_1+1,{\gamma }_1>0,\hfill \\ 1,\hfill & p>{\gamma }_1+1,{\gamma }_1>0,\hfill \\ 1,\hfill & p\ge 1,-1<{\gamma }_1\le 0;\hfill \end{array}\right.\hfill \\ \hfill A_{n,1}^2& :=\left\{\begin{array}{cc}{n}^{\left({\displaystyle \frac{{\gamma }_2+1}{p}}-1\right){\displaystyle \frac{\mu }{1+{\beta }_2}}},\hfill & 1\le p<{\gamma }_2+1,{\gamma }_2>0,\hfill \\ {\left(lnn\right)}^{1-{\displaystyle \frac{1}{p}}},\hfill & p={\gamma }_2+1,{\gamma }_2>0,\hfill \\ 1,\hfill & p>{\gamma }_2+1,{\gamma }_2>0,\hfill \\ 1,\hfill & p\ge 1,-1<{\gamma }_2\le 0,\hfill \end{array}\right.\hfill \\ \hfill D_{n,1}\left(m\right)& :=n^{\tilde{\mu }m}+n^{{\displaystyle \frac{\mu m}{1+{\beta }_2}}}.\hfill \end{array}$$

As established in Theorem 1, the estimate for $A_{n,1}\left(m\right)$ explicitly describes the dependence of the growth of the polynomial and its $m-th$ derivatives in the unbounded domain $\mathsf{\Omega }$ on the boundary angles and the behavior of the weight function. Consequently, it provides a quantitative characterization of the influence exerted by the domain geometry and the weight.

We next consider domains whose boundary contains only zero angles of the same type. Assume that the domain G has two interior zero angles at the boundary points $z_1$ and $z_2$, with corresponding $f_i-$touching

$$f_i\left(x\right)=C_i{x}^{1+{\alpha }_i},{\alpha }_i\ge 0,i=1,2,$$

where $C_i$ ($-\infty <C_i<+\infty ,$ $i=1,2),$ are finite constants. Collecting the terms corresponding to these two interior angles yields the following result:

Theorem 2. 

Let $p\ge 1;$ $G\in PQS\left(\lambda ;f_1{,f}_2\right)$, for some $\lambda \ge 1$, $f_i\left(x\right)=C_i{x}^{1+{\alpha }_i},{\alpha }_i\ge 0,$ $i=1,2,$ and let $P_n\in {\wp }_n,n\in \mathbb{N}.$ Assume that the weight function h is defined by (1) with ${\gamma }_j>-1,$ $j=1,2$. Then, the following holds for every $m=0,1,...,n$:

$$\left|P_n^{\left(m\right)}\left(z\right)\right|\le c_2{\displaystyle \frac{\left|{\mathsf{\Phi }}^{n-m+1}\left(z\right)\right|}{d(z,L)}}{∥P_n∥}_p·A_{n,2}\left(m\right),z\in \mathsf{\Omega },$$

(12)

where $c_2=c_2(L,{\gamma }_i,m,p)>0$ is a constant independent of n and $z;$

$$A_{n,2}\left(m\right):=\left\{\begin{array}{ccc}{n}^{\left({\displaystyle \frac{max\left\{{\gamma }_1;{\gamma }_2\right\}+1}{p}}+m-1\right)\tilde{\mu }},\hfill & 1\le p<max\left\{{\gamma }_1;{\gamma }_2\right\}+1,\hfill & {\gamma }_1>0,\\ n^{\tilde{\mu }m}{\left(lnn\right)}^{1-{\displaystyle \frac{1}{p}}},\hfill & \begin{array}{c}p={\gamma }_1+1,p\ge {\gamma }_2+1\hfill \\ \mathrm{or}p\ge {\gamma }_1+1,p={\gamma }_2+1,\hfill \end{array}\hfill & {\gamma }_1>0,\\ n^{\tilde{\mu }m},\hfill & p>max\left\{{\gamma }_1;{\gamma }_2\right\}+1,\hfill & {\gamma }_1>0,\\ n^{\tilde{\mu }m},\hfill & p\ge 1,\hfill & -1<{\gamma }_1\le 0.\end{array}\right.$$

Analogously, suppose that the domain G has two exterior zero angles at the boundary points $z_1$ and $z_2,$ with corresponding $g_i-$touching of the form

$$g_i\left(x\right)=C_i{x}^{1+{\beta }_i},{\beta }_i>0,i=1,2.$$

Then the following estimate holds:

Theorem 3. 

Let $p\ge 1;$ $G\in PQS\left(\lambda ;g_1{,g}_2\right)$, for some $\lambda \ge 1$, $g_i\left(x\right)=C_i{x}^{1+{\beta }_i},{\beta }_i>0,$ $i=1,2,$ and let $P_n\in {\wp }_n,n\in \mathbb{N}.$ Assume that the weight function h is defined by (1) with ${\gamma }_j>-1,$ $j=1,2$. Then, the following holds for every $m=0,1,...,n$:

$$\left|P_n^{\left(m\right)}\left(z\right)\right|\le c_3{\displaystyle \frac{\left|{\mathsf{\Phi }}^{n-m+1}\left(z\right)\right|}{d(z,L)}}{∥P_n∥}_p·A_{n,3}\left(m\right),z\in \mathsf{\Omega },$$

(13)

where $c_3=c_3(L,{\gamma }_i,\beta ,m,p)>0$ is a constant independent of n and $z;$

$$A_{n,3}\left(m\right):=\left\{\begin{array}{ccc}{n}^{\left({\displaystyle \frac{max\left\{{\gamma }_1;{\gamma }_2\right\}+1}{p}}+m-1\right){\displaystyle \frac{\mu }{1+min\left\{{\beta }_1;{\beta }_2\right\}}}},\hfill & 1\le p<max\left\{{\gamma }_1;{\gamma }_2\right\}+1,\hfill & {\gamma }_2>0,\\ n^{{\displaystyle \frac{\mu m}{1+min\left\{{\beta }_1;{\beta }_2\right\}}}}{\left(lnn\right)}^{1-{\displaystyle \frac{1}{p}}},\hfill & \begin{array}{c}p={\gamma }_1+1,p\ge {\gamma }_2+1\hfill \\ \mathrm{or}p\ge {\gamma }_1+1,p={\gamma }_2+1,\hfill \end{array}\hfill & {\gamma }_2>0,\\ n^{{\displaystyle \frac{\mu m}{1+min\left\{{\beta }_1;{\beta }_2\right\}}}},\hfill & p>max\left\{{\gamma }_1;{\gamma }_2\right\}+1,\hfill & {\gamma }_2>0,\\ n^{{\displaystyle \frac{\mu m}{1+min\left\{{\beta }_1;{\beta }_2\right\}}}},\hfill & p\ge 1,\hfill & -1<{\gamma }_2\le 0.\end{array}\right.$$

2.2. Estimates for $\left|P_n^{\left(M\right)}\left(Z\right)\right|$ in Bounded Domains

To obtain estimates for $\left|P_n^{\left(m\right)}\left(z\right)\right|,$ $m\ge 0,$ in the whole complex plane, we first derive the corresponding estimates in bounded domains $G\in PQS(\lambda ;f_i,g_j)$. Therefore, in this subsection we present estimates for $\left|P_n^{\left(m\right)}\left(z\right)\right|,$ $z\in \overline{G},$ in the domains specified above. The following results hold:

Theorem 4. 

Let $p\ge 1;$ $G\in PQS\left(\lambda ;f_1{,g}_2\right)$, for some $\lambda \ge 1$, $f_1\left(x\right)=C_1{x}^{1+{\alpha }_1},{\alpha }_1\ge 0,$ $g_2\left(x\right)=C_2{x}^{1+{\beta }_2},$ ${\beta }_2>0,$ and let $P_n\in {\wp }_n,n\in \mathbb{N}.$ Assume that the weight function h is defined by (1) with ${\gamma }_j>-1,$ $j=1,2$. Then, the following holds for every $m=0,1,...,n$:

$${∥P_n^{\left(m\right)}∥}_{\infty }\le c_4{E}_{n,1}\left(m\right){∥P_n∥}_p,$$

(14)

where $c_4=c_4(L,{\gamma }_i,\beta ,p)>0$ is a constant independent of n and $z;$ $E_{n,1}\left(m\right):=E_{n,1}^1\left(m\right)+E_{n,1}^2\left(m\right)$ and

$$\begin{array}{cc}\hfill E_{n,1}^1\left(m\right)& :=n^{\left({\displaystyle \frac{{\gamma }_1^{*}+1}{p}}+m\right)\tilde{\mu }};\hfill \\ \hfill E_{n,1}^2\left(m\right)& :=\left\{\begin{array}{cc}{n}^{\left({\displaystyle \frac{{\gamma }_2^{*}+1}{p}}\right){\displaystyle \frac{\mu }{1+{\beta }_2}}},\hfill & 1\le p<1+{\displaystyle \frac{({\gamma }_2^{*}+1)\mu }{1+{\beta }_2}},{\beta }_2>0,m=0,\hfill \\ {\left(nlnn\right)}^{1-{\displaystyle \frac{1}{p}}},\hfill & p=1+{\displaystyle \frac{({\gamma }_2^{*}+1)\mu }{1+{\beta }_2}},{\beta }_2>0,m=0,\hfill \\ n^{1-{\displaystyle \frac{1}{p}}},\hfill & p>1+{\displaystyle \frac{({\gamma }_2^{*}+1)\mu }{1+{\beta }_2}},{\beta }_2>0,m=0,\hfill \\ n^{\left({\displaystyle \frac{{\gamma }_2^{*}+1}{p}}+m\right){\displaystyle \frac{\mu }{1+{\beta }_2}}},\hfill & p\ge 1,{\beta }_2<m\mu -1,m\ge 1,\hfill \\ n^{\left({\displaystyle \frac{{\gamma }_2^{*}+1}{p}}+m\right){\displaystyle \frac{\mu }{1+{\beta }_2}}},\hfill & 1\le p<{\displaystyle \frac{({\gamma }_2^{*}+1)\mu +1+{\beta }_2}{1+{\beta }_2-m\mu }},{\beta }_2\ge m\mu -1,m\ge 1,\hfill \\ {\left(nlnn\right)}^{1-{\displaystyle \frac{1}{p}}},\hfill & p={\displaystyle \frac{({\gamma }_2^{*}+1)\mu +1+{\beta }_2}{1+{\beta }_2-m\mu }},{\beta }_2\ge m\mu -1,m\ge 1,\hfill \\ n^{1-{\displaystyle \frac{1}{p}}},\hfill & p>{\displaystyle \frac{({\gamma }_2^{*}+1)\mu +1+{\beta }_2}{1+{\beta }_2-m\mu }},{\beta }_2\ge m\mu -1,m\ge 1.\hfill \end{array}\right.\hfill \end{array}$$

In particular, we have:

Corollary 1. 

Let $p\ge 1;$ $G\in PQS\left(\lambda ;f_1{,g}_2\right)$, for some $\lambda \ge 1$, $f_1\left(x\right)=C_1{x}^{1+{\alpha }_1},{\alpha }_1\ge 0,$ $g_2\left(x\right)=C_2{x}^{1+{\beta }_2},{\beta }_2>0,$ and let $P_n\in {\wp }_n,n\in \mathbb{N}.$ Assume that the weight function h is defined by (1) with ${\gamma }_j>-1,$ $j=1,2$. Then, the following holds:

$${∥P_n∥}_{\infty }\le c_5{E}_{n,2}\left(0\right){∥P_n∥}_p,$$

(15)

where $c_5=c_5(L,{\gamma }_i,\beta ,p)>0$ is a constant independent of n and $z;E_{n,2}\left(0\right)$ is defined for arbitrary small $\epsilon >0$ as follows:

$$E_{n,2}\left(0\right):=\left\{\begin{array}{cc}{n}^{\left({\displaystyle \frac{{\gamma }_1^{*}+1}{p}}\right)\tilde{\mu }},\hfill & 1\le p<1+{\displaystyle \frac{({\gamma }_2^{*}+1)\mu }{1+{\beta }_2}},{\gamma }_1\ge {\displaystyle \frac{({\gamma }_2+1)\mu }{\tilde{\mu }(1+{\beta }_2)}}-1,{\gamma }_2>-1,\hfill \\ n^{\left({\displaystyle \frac{{\gamma }_2^{*}+1}{p}}\right){\displaystyle \frac{\mu }{1+{\beta }_2}}},\hfill & 1\le p<1+{\displaystyle \frac{({\gamma }_2^{*}+1)\mu }{1+{\beta }_2}},-1<{\gamma }_1<{\displaystyle \frac{({\gamma }_2+1)\mu }{\tilde{\mu }(1+{\beta }_2)}}-1,{\gamma }_2>-1,\hfill \\ n^{\left({\displaystyle \frac{{\gamma }_1^{*}+1}{p}}\right)\tilde{\mu }},\hfill & p=1+{\displaystyle \frac{({\gamma }_2^{*}+1)\mu }{1+{\beta }_2}},{\gamma }_1\ge {\displaystyle \frac{({\gamma }_2+1)\mu }{\tilde{\mu }(1+{\beta }_2)}}-1-\epsilon ,{\gamma }_2>-1,\hfill \\ {\left(nlnn\right)}^{1-{\displaystyle \frac{1}{p}}},\hfill & p=1+{\displaystyle \frac{({\gamma }_2^{*}+1)\mu }{1+{\beta }_2}},-1<{\gamma }_1<{\displaystyle \frac{({\gamma }_2+1)\mu }{\tilde{\mu }(1+{\beta }_2)}}-1-\epsilon ,{\gamma }_2>-1,\hfill \\ n^{\left({\displaystyle \frac{{\gamma }_1^{*}+1}{p}}\right)\tilde{\mu }},\hfill & p>1+{\displaystyle \frac{({\gamma }_2^{*}+1)\mu }{1+{\beta }_2}},{\gamma }_1\ge {\displaystyle \frac{({\gamma }_2+1)\mu }{\tilde{\mu }(1+{\beta }_2)}}-1,{\gamma }_2>-1,\hfill \\ n^{1-{\displaystyle \frac{1}{p}}},\hfill & p>1+{\displaystyle \frac{({\gamma }_2^{*}+1)\mu }{1+{\beta }_2}},-1<{\gamma }_1<{\displaystyle \frac{({\gamma }_2+1)\mu }{\tilde{\mu }(1+{\beta }_2)}}-1,{\gamma }_2>-1.\hfill \end{array}\right.$$

Corollary 2. 

Let $p\ge 1;$ $G\in PQS\left(\lambda ;f_1{,g}_2\right)$, for some $\lambda \ge 1$, $f_1\left(x\right)=C_1{x}^{1+{\alpha }_1},{\alpha }_1\ge 0,$ $g_2\left(x\right)=C_2{x}^{1+{\beta }_2},{\beta }_2>0,$ and let $P_n\in {\wp }_n,n\in \mathbb{N}.$ Assume that the weight function h be given by (1) with ${\gamma }_j>-1,$ $j=1,2$. Then, the following holds for every $m=1,...,n$:

$${∥P_n^{\left(m\right)}∥}_{\infty }\le c_6{E}_{n,3}\left(m\right){∥P_n∥}_p,$$

(16)

where $c_6=c_6(L,{\gamma }_i,\beta ,p)>0$ is a constant independent of n and $z;E_{n,3}\left(m\right)$ is defined as

$$E_{n,3}\left(m\right):=\left\{\begin{array}{cc}{n}^{\left({\displaystyle \frac{{\gamma }_1^{*}+1}{p}}+m\right)\tilde{\mu }},\hfill & p\ge 1,{\gamma }_1\ge {\tilde{\gamma }}_1,{\gamma }_2>{\tilde{\gamma }}_2,{\beta }_2<m\mu -1,\hfill \\ n^{\left({\displaystyle \frac{{\gamma }_1^{*}+1}{p}}+m\right)\tilde{\mu }},\hfill & p\ge 1,{\gamma }_1>-1,-1<{\gamma }_2\le {\tilde{\gamma }}_2,{\beta }_2<m\mu -1,\hfill \\ n^{\left({\displaystyle \frac{{\gamma }_2^{*}+1}{p}}+m\right){\displaystyle \frac{\mu }{1+{\beta }_2}}},\hfill & p\ge 1,-1<{\gamma }_1<{\tilde{\gamma }}_1,{\gamma }_2>{\tilde{\gamma }}_2,{\beta }_2<m\mu -1,\hfill \\ n^{\left({\displaystyle \frac{{\gamma }_1^{*}+1}{p}}+m\right)\tilde{\mu }},\hfill & 1\le p<{\displaystyle \frac{({\gamma }_2+1)\mu +1+{\beta }_2}{1+{\beta }_2-m\mu }},{\gamma }_1\ge {\tilde{\gamma }}_1,{\gamma }_2>{\tilde{\gamma }}_2,{\beta }_2\ge m\mu -1,\hfill \\ n^{\left({\displaystyle \frac{{\gamma }_1^{*}+1}{p}}+m\right)\tilde{\mu }},\hfill & 1\le p<{\displaystyle \frac{({\gamma }_2+1)\mu +1+{\beta }_2}{1+\beta -m\mu }},{\gamma }_1>-1,-1<{\gamma }_2\le {\tilde{\gamma }}_2,{\beta }_2\ge m\mu -1,\hfill \\ n^{\left({\displaystyle \frac{{\gamma }_2^{*}+1}{p}}+m\right){\displaystyle \frac{\mu }{1+{\beta }_2}}},\hfill & 1\le p<{\displaystyle \frac{({\gamma }_2+1)\mu +1+{\beta }_2}{1+{\beta }_2-m\mu }},0<{\gamma }_1<{\tilde{\gamma }}_1,{\gamma }_2>{\tilde{\gamma }}_2,{\beta }_2\ge m\mu -1,\hfill \\ n^{\left({\displaystyle \frac{{\gamma }_1^{*}+1}{p}}+m\right)\tilde{\mu }},\hfill & p\ge {\displaystyle \frac{({\gamma }_2^{*}+1)\mu +1+{\beta }_2}{1+{\beta }_2-m\mu }},{\gamma }_1>-1,{\gamma }_2>-1,{\beta }_2\ge m\mu -1,\hfill \end{array}\right.$$

and ${\tilde{\gamma }}_1:={\displaystyle \frac{({\gamma }_2+pm+1)\mu }{\tilde{\mu }\left(1+{\beta }_2\right)}}-(pm+1);$ ${\tilde{\gamma }}_2:=\left[{\displaystyle \frac{\tilde{\mu }\left(1+{\beta }_2\right)}{\mu }}-1\right]pm-1.$

Theorem 5. 

Let $p\ge 1;$ $G\in PQS\left(\lambda ;f_1{,f}_2\right)$, for some $\lambda \ge 1$, $f_i\left(x\right)=C_i{x}^{1+{\alpha }_i},{\alpha }_i\ge 0,$ $i=1,2,$ and let $P_n\in {\wp }_n,n\in \mathbb{N}.$ Assume that the weight function h is defined by (1) with ${\gamma }_j>-1,$ $j=1,2$. Then, the following holds for every $m=0,1,...,n$:

$${∥P_n^{\left(m\right)}∥}_{\infty }\le c_7{E}_{n,4}\left(m\right){∥P_n∥}_p,$$

(17)

where $c_7=c_7(L,{\gamma }_i,\beta ,p)>0$ is a constant independent of n and $z;$

$$E_{n,4}\left(m\right):=n^{\left({\displaystyle \frac{max\left\{{\gamma }_1^{*};{\gamma }_1^{*}\right\}+1}{p}}+m\right)\tilde{\mu }}.$$

Theorem 6. 

Let $p\ge 1;$ $G\in PQS\left(\lambda ;g_1{,g}_2\right)$, for some $\lambda \ge 1$, $g_i\left(x\right)=C_i{x}^{1+{\beta }_i},{\beta }_i>0,$ $i=1,2,$ and let $P_n\in {\wp }_n,n\in \mathbb{N}.$ Assume that the weight function h is defined by (1) with ${\gamma }_j>-1,$ $j=1,2$. Then, following holds for every $m=0,1,...,n$:

$${∥P_n^{\left(m\right)}∥}_{\infty }\le c_8{E}_{n,5}\left(m\right){∥P_n∥}_p,$$

(18)

where $c_8=c_8(L,{\gamma }_i,\beta ,p)>0$ is a constant independent of n and $z;$

$$E_{n,5}\left(m\right):=n^{\left({\displaystyle \frac{max\left\{{\gamma }_1^{*};{\gamma }_1^{*}\right\}+1}{p}}\right){\displaystyle \frac{\mu }{1+min\left\{{\beta }_1;{\beta }_2\right\}}}}.$$

Theorem 7. 

Let $p\ge 1;$ $G\in PQS\left(\lambda ;f_1{,g}_2\right)$, for some $\lambda \ge 1$, $f_1\left(x\right)=c_1{x}^{1+{\alpha }_1},{\alpha }_1\ge 0,$ $g_2\left(x\right)=c_2{x}^{1+{\beta }_2},{\beta }_2>0,$ and let $P_n\in {\wp }_n,n\in \mathbb{N}.$ Assume that the weight function h is defined by (1) with ${\gamma }_j>-1,$ $j=1,2$. Then, the following holds for every $m=0,1,...,n$:

$$\left|P_n^{\left(m\right)}\left(z_j\right)\right|\le c_9{n}^{\left({\displaystyle \frac{{\gamma }_j+1}{p}}+m\right){\omega }_j}{∥P_n∥}_p,j=1,2,$$

(19)

where a constant $c_9=c_9(L,\gamma ,m,p)>0$ independent of n and $z;$ ${\gamma }^{*}$; ${\nu }_j$ is defined as follows:

$${\omega }_j:=\left\{\begin{array}{cc}\tilde{\mu },& j=1,\\ {\displaystyle \frac{\mu }{1+{\beta }_2}},& j=2.\end{array}\right.$$

2.2.1. 2.1. Sharpness of the Estimates

Remark 1. 

[37] [Remark 2.16] Estimates (14) - (19) are sharp in the order sense for some specially chosen domain and weight function.

2.3. Estimates for $\left|P_n^{\left(M\right)}\left(Z\right)\right|$ in Whole Complex Plane

In this subsection, we combine the results obtained in the two previous subsections for infinite and finite domains, respectively, and gain insight into the behavior of derivatives of polynomials in the entire complex plane. Hence, combining estimations (14)–(18) with the results of Theorems 1, 2 and 3, we obtain growth estimate for $|P_n^{\left(m\right)}\left(z\right)|$ in the whole complex plane.

Theorem 8. 

Let $p\ge 1;$ $G\in PQS\left(\lambda ;f_1{,g}_2\right)$, for some $\lambda \ge 1$, $f_1\left(x\right)=C_1{x}^{1+{\alpha }_1},{\alpha }_1\ge 0$, $g_2\left(x\right)=C_2{x}^{1+{\beta }_2},{\beta }_2>0,$ and let $P_n\in {\wp }_n,n\in \mathbb{N}.$ Assume that the weight function h is defined by (1) with ${\gamma }_j>-1,$ $j=1,2$. Then, the following holds for every $m=0,1,...,n$:

$$\left|P_n^{\left(m\right)}\left(z\right)\right|\le c_7{∥P_n∥}_p\left\{\begin{array}{cc}{E}_{n,1}\left(m\right),& z\in \overline{G},\\ {\displaystyle \frac{\left|{\mathsf{\Phi }}^{n-m+1}\left(z\right)\right|}{d(z,L)}}{∥P_n∥}_p·A_{n,1}\left(m\right),& z\in \mathsf{\Omega },\end{array}\right.$$

where $c_7=c_7(L,{\gamma }_i,\beta ,m,p)>0$ is a constant independent of n and $z;$ $E_{n,1}\left(m\right)$ and ${\tilde{A}}_{n,1}\left(m\right)$ are defined as in Theorem 4 for all $z\in \overline{G}$ and in Theorem 1 for all $z\in \mathsf{\Omega }$, respectively.

In the special case where the domain G has identical zero angles at both boundary points $z_1$ and $z_2$ namely, $f_i\left(x\right)=C_i{x}^{1+{\alpha }_i}$, or $g_i\left(x\right)=C_i{x}^{1+{\beta }_i},$ $i=1,2$, results similar to Corollaries 1 and 3 follow directly from the above estimates together with Theorems 2 and 3, respectively. In particular, a result analogous to Corollary 3 can also be obtained for domains that have only interior or only exterior zero angles.

As a consequence of Theorem 8 and Theorem 1 for $m=0,$ we obtain the following corollary:

Corollary 3. 

Under the assumptions of Theorem8, the following holds:

$$\left|P_n\left(z\right)\right|\le c_8{∥P_n∥}_p\left\{\begin{array}{cc}{E}_{n,2}\left(0\right),& z\in \overline{G},\\ {\displaystyle \frac{\left|{\mathsf{\Phi }}^{n+1}\left(z\right)\right|}{d(z,L)}}{∥P_n∥}_p·A_{n,1}\left(0\right),& z\in \mathsf{\Omega },\end{array}\right.$$

where $c_8=c_8(L,{\gamma }_i,\beta ,p)>0$ is a constant independent of n and $z;$ $E_{n,2}\left(0\right)$ and $A_{n,1}\left(0\right)$ are defined as in (15) and Theorem 1, respectively.

Corollary 4. 

Under the assumptions of Theorem8, the following holds for every $m=1,2,...,n$:

$$\left|P_n^{\left(m\right)}\left(z\right)\right|\le c_8{∥P_n∥}_p\left\{\begin{array}{cc}{E}_{n,3}\left(m\right),& z\in \overline{G},\\ {\displaystyle \frac{\left|{\mathsf{\Phi }}^{n+1-m}\left(z\right)\right|}{d(z,L)}}{∥P_n∥}_p·A_{n,2}\left(m\right),& z\in \mathsf{\Omega },\end{array}\right.$$

where $c_8=c_8(L,{\gamma }_i,\beta ,p)>0$ is a constant independent of n and $z;$ $E_{n,2}\left(m\right)$ and $A_{n,2}\left(m\right)$ are defined as in (16) and Theorem 1, respectively.

Corollary 5. 

Under the assumptions of Theorem2, the following holds for every $m=0,1,2,...,n$:

$$\left|P_n^{\left(m\right)}\left(z\right)\right|\le c_8{∥P_n∥}_p\left\{\begin{array}{cc}{E}_{n,4}\left(m\right),& z\in \overline{G},\\ {\displaystyle \frac{\left|{\mathsf{\Phi }}^{n+1-m}\left(z\right)\right|}{d(z,L)}}{∥P_n∥}_p·A_{n,2}\left(m\right),& z\in \mathsf{\Omega },\end{array}\right.$$

where $c_8=c_8(L,{\gamma }_i,p)>0$ is a constant independent of n and $z;$ $E_{n,4}\left(m\right)$ and $A_{n,2}\left(m\right)$ are defined as in Theorem 5 and Theorem 2, respectively.

Corollary 6. 

Under the assumptions of Theorem3, the following holds for every $m=0,1,2,...,n$:

$$\left|P_n^{\left(m\right)}\left(z\right)\right|\le c_8{∥P_n∥}_p\left\{\begin{array}{cc}{E}_{n,5}\left(m\right),& z\in \overline{G},\\ {\displaystyle \frac{\left|{\mathsf{\Phi }}^{n+1-m}\left(z\right)\right|}{d(z,L)}}{∥P_n∥}_p·A_{n,3}\left(m\right),& z\in \mathsf{\Omega },\end{array}\right.$$

where $c_8=c_8(L,{\gamma }_i,\beta ,p)>0$ is a constant independent of n and $z;$ $E_{n,5}\left(m\right)$ and $A_{n,3}\left(m\right)$ are defined as in Theorem 6 and Theorem 3, respectively.

3. Some Auxiliary Results

For $a>0$ and $b>0$, the notation $``a⪯b``$ means that $a\le cb$ for some constant $c>0$, while $``a\asymp b``$ means that $c_{1}a\le b\le c_{2}a$ for some constants $c_1,c_2>0.$

Lemma 1. 

[39] Let G be a quasidisk, $z_1\in L,$ $z_2,z_3\in \mathsf{\Omega }\cap \{z:\left|z-z_1\right|⪯d(z_1,L_{r_0})\};$ $w_j=\mathsf{\Phi }\left(z_j\right),$ $j=1,2,3.$ 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. Therefore, $\left|z_1-z_2\right|\asymp \left|z_1-z_3\right|$ and $\left|w_1-w_2\right|\asymp \left|w_1-w_3\right|$ also are equivalent.

b)

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

$${\left|{\displaystyle \frac{w_1-w_3}{w_1-w_2}}\right|}^{c_1}⪯\left|{\displaystyle \frac{z_1-z_3}{z_1-z_2}}\right|⪯{\left|{\displaystyle \frac{w_1-w_3}{w_1-w_2}}\right|}^{c_2},$$

where $0<r_0<1$ is a constant, depending on $G.$

Corollary 7. 

Under the conditions of Lemma 1, we have:

$${\left|w_1-w_2\right|}^{c_1}⪯\left|z_1-z_2\right|⪯{\left|w_1-w_2\right|}^{\epsilon },$$

where $\epsilon =\epsilon \left(G\right)<1.$

Lemma 2. 

[40,41] Let L$\in QS\left(\lambda \right)$ for some $\lambda \ge 1.$ Then

$$\left|\mathsf{\Psi }\left(w_1\right)-\mathsf{\Psi }\left(w_2\right)\right|⪰{\left|w_1-w_2\right|}^{\mu },$$

for all $w_1,w_2\in \overline{\Delta },$ where $\mu :=2(1-{\displaystyle \frac{1}{\pi }}arcsin{\displaystyle \frac{1}{\lambda }}).$

This follows from a suitable result for mappings $f\in \Sigma \left(\kappa \right)$ established in [42] [p.287], together with an estimate for ${\mathsf{\Psi }}^{\prime }$ obtained in [43] [Th.2.8]:

$$d(\mathsf{\Psi }\left(\tau \right),L)\asymp \left|{\mathsf{\Psi }}^{\prime }\left(\tau \right)\right|\left(\left|\tau \right|-1\right).$$

(20)

Let ${\left\{z_j\right\}}_{j=1}^l\in L$ be fixed system, and let h be the weight function defined by (1) with ${\gamma }_j>-1,$ $j=\overline{1,l}$.

Lemma 3. 

[44] Let $L=\partial G$ be a rectifiable Jordan curve, and let $P_n\left(z\right),$ $deg{P}_n\le n,$ $n=1,2,...,$ be an arbitrary polynomial. Assume that the weight function $h\left(z\right)$ satisfies the condition (1). Then for any $R>1,$ $p>0$ and $n=1,2,...$

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

4. Proofs of Theorems

Proof of 1. 

Let $G\in PQS\left(\lambda ;f_1{,g}_2\right)$, for some $\lambda \ge 1$, $f_1\left(x\right)=C_1{x}^{1+{\alpha }_1},$ ${\alpha }_1\ge 0,$ and $g_2\left(x\right)=C_2{x}^{1+{\beta }_2},{\beta }_2>0.$ Let $R=1+{\displaystyle \frac{{\epsilon }_0}{n}},$ $0<{\epsilon }_0<{\displaystyle \frac{1}{2}}$. For each $n\in \mathbb{N}$ and m with $1\le m<n$, define:

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

The function $H_{n,m}\left(z\right)$ is analytic in $\mathsf{\Omega },$ continuous on $\overline{\mathsf{\Omega }}$ and satisfies $H_{n,m}\left(\infty \right)=0,$ then the Cauchy integral formula for unbounded domain give:

$$H_{n,m}\left(z\right)=-{\displaystyle \frac{1}{2\pi i}}\underset{L}{\int }{H}_{n,m}\left(\zeta \right){\displaystyle \frac{d\zeta }{\zeta -z}},z\in \mathsf{\Omega }.$$

Then,

$$\left|{\displaystyle \frac{P_n^{\left(m\right)}\left(z\right)}{{\mathsf{\Phi }}^{n-m+1}\left(z\right)}}\right|\le {\displaystyle \frac{1}{2\pi }}\underset{L}{\int }\left|{\displaystyle \frac{P_n^{\left(m\right)}\left(\zeta \right)}{{\mathsf{\Phi }}^{n-m+1}\left(\zeta \right)}}\right|{\displaystyle \frac{\left|d\zeta \right|}{\left|\zeta -z\right|}}.$$

Since $\left|{\mathsf{\Phi }}^{n-m+1}\left(\zeta \right)\right|=1,$ for all $\zeta \in L$, it follows that:

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

(21)

Applying the Cauchy integral representation for the derivatives $P_n^{\left(m\right)}\left(\zeta \right)$, we get:

$$P_n^{\left(m\right)}\left(\zeta \right)={\displaystyle \frac{1}{2\pi i}}\underset{L_R}{\int }{\displaystyle \frac{P_n\left(t\right)}{{\left(t-\zeta \right)}^{m+1}}}dt,\zeta \in G_R.$$

For $\zeta \in L,$ substituting $P_n^{\left(m\right)}\left(\zeta \right)$ into (21) yields:

$$\begin{array}{cc}\hfill \left|P_n^{\left(m\right)}\left(z\right)\right|& ⪯{\displaystyle \frac{\left|{\mathsf{\Phi }}^{n-m+1}\left(z\right)\right|}{d(z,L)}}·\underset{L}{\int }\left|{\displaystyle \frac{1}{2\pi i}}\underset{L_R}{\int }{\displaystyle \frac{P_n\left(t\right)}{{\left(t-\zeta \right)}^{m+1}}}dt\right|\left|d\zeta \right|\hfill \\ \hfill & ⪯{\displaystyle \frac{\left|{\mathsf{\Phi }}^{n-m+1}\left(z\right)\right|}{d(z,L)}}·\underset{L}{\int }\left(\underset{L_R}{\int }{\displaystyle \frac{\left|P_n\left(t\right)\right|}{{\left|t-\zeta \right|}^{m+1}}}\left|dt\right|\right)\left|d\zeta \right|.\hfill \end{array}$$

(22)

$$\left|P_n^{\left(m\right)}\left(z\right)\right|⪯{\displaystyle \frac{\left|{\mathsf{\Phi }}^{n-m+1}\left(z\right)\right|}{d(z,L)}}·\left[\underset{L_R}{\int }\left|P_n\left(t\right)\right|\left|dt\right|\right]·\underset{t\in L_R}{sup}\left(\underset{L}{\int }{\displaystyle \frac{\left|d\zeta \right|}{{\left|t-\zeta \right|}^{m+1}}}\right).$$

(23)

Denote by

$$A_n:=\underset{L_R}{\int }\left|P_n\left(t\right)\right|\left|dt\right|;B_{n,m}\left(t\right):=\underset{L}{\int }{\displaystyle \frac{\left|d\zeta \right|}{{\left|t-\zeta \right|}^{m+1}}};$$

(24)

and estimate these integrals separately.

We begin with an estimate of the integral $A_n$. Assuming that $p>1$, we multiply both the numerator and denominator of the integrand by $h^{{\displaystyle \frac{1}{p}}}\left(\zeta \right)$ and then apply Hölder’s inequality to obtain:

$$\begin{array}{cc}\hfill A_n& =\underset{L_R}{\int }\left|P_n\left(t\right)\right|\left|dt\right|=\underset{L_R}{\int }{\displaystyle \frac{h^{\frac{1}{p}}\left(t\right)\left|P_n\left(t\right)\right|}{h^{\frac{1}{p}}\left(t\right)}}\left|dt\right|\hfill \\ \hfill & \le {\left(\underset{L_R}{\int }h\left(t\right){\left|P_n\left(t\right)\right|}^p\left|dt\right|\right)}^{{\displaystyle \frac{1}{p}}}\times {\left(\underset{L_R}{\int }{\displaystyle \frac{\left|dt\right|}{h^{\frac{q}{p}}\left(t\right)}}\right)}^{{\displaystyle \frac{1}{q}}}\hfill \\ \hfill & ⪯{∥P_n∥}_{{\mathcal{L}}_p(h,L_R)}{\left(\underset{L_R}{\int }{\displaystyle \frac{\left|dt\right|}{h^{q-1}\left(t\right)}}\right)}^{{\displaystyle \frac{1}{q}}},{\displaystyle \frac{1}{p}}+{\displaystyle \frac{1}{q}}=1.\hfill \end{array}$$

(25)

According to Lemma 3, we get:

$$\begin{array}{cc}\hfill A_n& ⪯{∥P_n∥}_p{\left(\underset{L_R}{\int }{\displaystyle \frac{\left|dt\right|}{h^{q-1}\left(t\right)}}\right)}^{{\displaystyle \frac{1}{q}}}\hfill \\ \hfill & ⪯{∥P_n∥}_p\left[{\left(\underset{L_R^1}{\int }{\displaystyle \frac{\left|d\zeta \right|}{{\left|\zeta -z_1\right|}^{(q-1){\gamma }_1}}}\right)}^{{\displaystyle \frac{1}{q}}}+{\left(\underset{L_R^2}{\int }{\displaystyle \frac{\left|d\zeta \right|}{{\left|\zeta -z_2\right|}^{(q-1){\gamma }_2}}}\right)}^{{\displaystyle \frac{1}{q}}}\right]=:{∥P_n∥}_p\left(J_R^1+J_R^2\right),\hfill \end{array}$$

(26)

since the points $z_1,$ $z_2\in L$are distinct, and we begin to estimate this integrals.

To simplify the calculations, we put $z_1=-1,$ $z_2=1;$ $(-1,1)\subset G$ and let local coordinate axis in Definition 1 be parallel to natural axis $OX$ and $OY$ in the coordinate system $XOY$; $L_R=L_R^{+}\cup L_R^{-},$ where $L_R^{+}:=\left\{z\in L_R:\mathrm{Im}z\ge 0\right\},$ $L_R^{-}:=\left\{z\in L_R:\mathrm{Im}z<0\right\}.$ Moreover, let $z_R^{\pm }\in \mathsf{\Psi }\left(w_R^{\pm }\right),$ where $w_R^{\pm }:=\left\{w=R·e^{i\theta }:\theta ={\displaystyle \frac{{\phi }_1\pm {\phi }_2}{2}}\right\},$ $z_{R,i}\in \mathsf{\Psi }\left(R\mathsf{\Phi }\left(z_i\right)\right),$ and $L_{R,i}^{\pm }(z_{R,i},z_R^{\pm })$ denote the arcs, connected the points $z_{R,i}$ which $z_R^{\pm },$ respectively; $\left|L_{R,i}^{\pm }\right|:=mes{L}_{R,i}^{\pm }(z_{R,i},z_R^{\pm }),$ $i=1,2.$ Then, from (26), we have

$$A_n⪯{∥P_n∥}_p\left\{{\left[J_R^1\right]}^{{\displaystyle \frac{1}{q}}}+{\left[J_R^2\right]}^{{\displaystyle \frac{1}{q}}}\right\}.$$

(27)

Now, for estimates the integrals $J_R^1$ and $J_R^2$, let us introduce some notation. For any $\rho \ge 1$, we denote:

$$\begin{array}{cc}\hfill d_{i,R}& :=d(z_i,L_R);\hfill \\ \hfill E_{\rho ,1}^{1,\pm }& :=\left\{\zeta \in L_{\rho }^1:\left|\zeta -z_1\right|<c_1{d}_{1,R}\right\},\hfill \\ \hfill & E_{\rho ,2}^{1,\pm }:=\left\{\zeta \in L_{\rho }^1:c_1{d}_{1,R}\le \left|\zeta -z_1\right|\le \left|L_{\rho ,1}^{\pm }\right|\right\},\hfill \\ \hfill E_{\rho ,1}^{2,\pm }& :=\left\{\zeta \in L_{\rho }^2:\left|\zeta -z_2\right|<c_2{d}_{2,R}\right\},\hfill \\ \hfill & E_{\rho ,2}^{2,\pm }:=\left\{\zeta \in L_{\rho }^2:c_2{d}_{2,R}\le \left|\zeta -z_2\right|\le \left|L_{\rho ,2}^{\pm }\right|\right\};\hfill \\ \hfill {\Im }_{\rho ,k}^{i,\pm }& :=\underset{E_{\rho ,k}^{i,\pm }}{\int }{\displaystyle \frac{\left|d\zeta \right|}{{\left|\zeta -z_i\right|}^{{\gamma }_i(q-1)}}},i,k=1,2.\hfill \end{array}$$

(28)

Then, using these notations and for $\rho =R$, from (27) we get:

$$\begin{array}{cc}\hfill A_n& ⪯{∥P_n∥}_p{\displaystyle \sum _{i=1}^2}{\left[{\Im }_{R,1}^{i,\pm }+{\Im }_{R,2}^{i,\pm }\right]}^{{\displaystyle \frac{1}{q}}},i=1,2.\hfill \end{array}$$

(29)

Therefore, we can start to estimate the integrals ${\Im }_{R,k}^{i,\pm }$for each $i,k=1,2.$

1.1. Let ${\gamma }_1,{\gamma }_2>0.$ Then, we get:

$$\begin{array}{cc}\hfill {\Im }_{R,1}^{1,\pm }& ⪯{\displaystyle \underset{0}{\overset{c_1{d}_{1,R}}{\int }}}{\displaystyle \frac{ds}{s^{{\gamma }_1(q-1)}}}⪯{\left({\displaystyle \frac{1}{d_{1,R}}}\right)}^{{\gamma }_1(q-1)}·\mathrm{mes}\left(E_{R,1}^{1,\pm }\right)⪯{\left({\displaystyle \frac{1}{d_{1,R}}}\right)}^{{\gamma }_1(q-1)-1};\hfill \\ \hfill {\Im }_{R,2}^{1,\pm }& ⪯{\displaystyle \underset{c_1{d}_{1,R}}{\overset{\left|L_{R,1}^{\pm }\right|}{\int }}}{\displaystyle \frac{ds}{s^{{\gamma }_1(q-1)}}}⪯\left\{\begin{array}{cc}{\left({\displaystyle \frac{1}{d_{1,R}}}\right)}^{{\gamma }_1(q-1)-1},\hfill & {\gamma }_1(q-1)>1,\hfill \\ ln\left({\displaystyle \frac{1}{d_{1,R}}}\right),\hfill & {\gamma }_1(q-1)=1,\hfill \\ 1,\hfill & {\gamma }_1(q-1)<1;\hfill \end{array}\right.\hfill \\ \hfill {\Im }_{R,1}^{1,\pm }+{\Im }_{R,2}^{1,\pm }& ⪯{\left({\displaystyle \frac{1}{d_{1,R}}}\right)}^{{\gamma }_1(q-1)-1}+\left\{\begin{array}{cc}{\left({\displaystyle \frac{1}{d_{1,R}}}\right)}^{{\gamma }_1(q-1)-1},\hfill & {\gamma }_1(q-1)>1,\hfill \\ ln\left({\displaystyle \frac{1}{d_{1,R}}}\right),\hfill & {\gamma }_1(q-1)=1,\hfill \\ 1,\hfill & {\gamma }_1(q-1)<1,\hfill \end{array}\right.\hfill \\ \hfill & ⪯\left\{\begin{array}{cc}{\left({\displaystyle \frac{1}{d_{1,R}}}\right)}^{{\gamma }_1(q-1)-1},\hfill & {\gamma }_1(q-1)>1,\hfill \\ ln\left({\displaystyle \frac{1}{d_{1,R}}}\right),\hfill & {\gamma }_1(q-1)=1,\hfill \\ 1,\hfill & {\gamma }_1(q-1)<1.\hfill \end{array}\right.\hfill \end{array}$$

(30)

Similarly, for the $J_R^2$ in the neighborhood of the point $z_2$, we have:

$$\begin{array}{cc}\hfill {\Im }_{R,1}^{2,\pm }& ⪯{\displaystyle \underset{0}{\overset{c_2{d}_{2,R}}{\int }}}{\displaystyle \frac{ds}{s^{{\gamma }_2(q-1)}}}⪯{\left({\displaystyle \frac{1}{d_{2,R}}}\right)}^{{\gamma }_2(q-1)}·\mathrm{mes}\left(E_{R,1}^{2,\pm }\right)⪯{\left({\displaystyle \frac{1}{d_{2,R}}}\right)}^{{\gamma }_2(q-1)-1};\hfill \\ \hfill {\Im }_{R,2}^{2,\pm }& ⪯{\displaystyle \underset{c_2{d}_{2,R}}{\overset{\left|L_{R,2}^{\pm }\right|}{\int }}}{\displaystyle \frac{ds}{s^{{\gamma }_2(q-1)}}}⪯\left\{\begin{array}{cc}{\left({\displaystyle \frac{1}{d_{2,R}}}\right)}^{{\gamma }_2(q-1)-1},\hfill & {\gamma }_2(q-1)>1,\hfill \\ ln\left({\displaystyle \frac{1}{d_{2,R}}}\right),\hfill & {\gamma }_2(q-1)=1,\hfill \\ 1,\hfill & {\gamma }_2(q-1)<1;\hfill \end{array}\right.\hfill \end{array}$$

(31)

$$\begin{array}{cc}\hfill {\Im }_{R,1}^{2,\pm }+{\Im }_{R,1}^{2,\pm }& ⪯{\left({\displaystyle \frac{1}{d_{2,R}}}\right)}^{{\gamma }_2(q-1)-1}+\left\{\begin{array}{cc}{\left({\displaystyle \frac{1}{d_{2,R}}}\right)}^{{\gamma }_2(q-1)-1},\hfill & {\gamma }_2(q-1)>1,\hfill \\ ln\left({\displaystyle \frac{1}{d_{2,R}}}\right),\hfill & {\gamma }_2(q-1)=1,\hfill \\ 1,\hfill & {\gamma }_2(q-1)<1,\hfill \end{array}\right.\hfill \\ \hfill & ⪯\left\{\begin{array}{cc}{\left({\displaystyle \frac{1}{d_{2,R}}}\right)}^{{\gamma }_2(q-1)-1},\hfill & {\gamma }_2(q-1)>1,\hfill \\ ln\left({\displaystyle \frac{1}{d_{2,R}}}\right),\hfill & {\gamma }_2(q-1)=1,\hfill \\ 1,\hfill & {\gamma }_2(q-1)<1.\hfill \end{array}\right.\hfill \end{array}$$

1.2. Let ${\gamma }_1,{\gamma }_2\le 0.$Then, analogously to the (30) and (31), we obtain:

$$\begin{array}{cc}\hfill {\Im }_{R,1}^{1,\pm }& =\underset{E_{R,1}^{1,\pm }}{\int }{\left|\zeta -z_1\right|}^{-{\gamma }_1(q-1)}\left|d\zeta \right|⪯{\left(\mathrm{diam}{L}_R\right)}^{-{\gamma }_1(q-1)}\mathrm{mes}{E}_{R,1}^{1,\pm }⪯1;\hfill \\ \hfill {\Im }_{R,2}^{1,\pm }& ⪯\underset{E_{R,2}^{1,\pm }}{\int }{\left|\zeta -z_1\right|}^{-{\gamma }_1(q-1)}\left|d\zeta \right|⪯{\displaystyle \underset{c_1{d}_{1,R}}{\overset{\left|L_1^{\pm }\right|}{\int }}}{s}^{-{\gamma }_1(q-1)}ds⪯1;\hfill \end{array}$$

(32)

and

$$\begin{array}{cc}\hfill {\Im }_{R,1}^{2,\pm }& ⪯\underset{E_{R,1}^{2,\pm }}{\int }{\left|\zeta -z_2\right|}^{-{\gamma }_2(q-1)}\left|d\zeta \right|⪯{\left(\mathrm{diam}{L}_R\right)}^{(-{\gamma }_2)(q-1)}\mathrm{mes}{E}_{R,1}^{2,\pm }⪯1;\hfill \\ \hfill {\Im }_{R,2}^{2,\pm }& ⪯\underset{E_{R,2}^{2,\pm }}{\int }{\left|\zeta -z_2\right|}^{-{\gamma }_2(q-1)}\left|d\zeta \right|⪯{\displaystyle \underset{c_2{d}_{2,R}}{\overset{\left|L_2^{\pm }\right|}{\int }}}ds⪯1.\hfill \end{array}$$

In this case, combining (29)–(32), we get:

$$A_n⪯{∥P_n∥}_p\left\{\begin{array}{cc}{\left({\displaystyle \frac{1}{d_{1,R}}}\right)}^{{\displaystyle \frac{({\gamma }_1+1)}{p}}-1}+{\left({\displaystyle \frac{1}{d_{2,R}}}\right)}^{{\displaystyle \frac{({\gamma }_2+1)}{p}}-1},& {\gamma }_1>p-1,{\gamma }_2>p-1,\\ {\left(ln\left({\displaystyle \frac{1}{d_{1,R}}}\right)\right)}^{1-{\displaystyle \frac{1}{p}}}+{\left(ln\left({\displaystyle \frac{1}{d_{2,R}}}\right)\right)}^{1-{\displaystyle \frac{1}{p}}},& {\gamma }_1=p-1,{\gamma }_2=p-1,\\ 1,& -1<{\gamma }_1,{\gamma }_2<p-1.\end{array}\right.$$

(33)

According to Lemma 2, 43] [p.61] and [45] [p.10], we obtain:

$$d_{1,R}⪰\left\{\begin{array}{cc}\hfill ll{n}^{-\mu },& if{\alpha }_1=0;\\ \hfill n^{-2},& if{\alpha }_1\ne 0.\end{array}\right.$$

(34)

Now, we define:$z_R\in L_R:$ $d_{2,R}=\left|z_2-z_R\right|;{\zeta }^{\pm }\in L^{\pm }:$ $d(z_R,L^2\cap L^{\pm }):=d(z_R,L^{+});$ $z_2^{\pm }$ $:=\zeta \in L^2:\left|\zeta -z_2\right|=c_2{d}_{2,R}$ for the estimate $d_{2,R}$. Then, we have

$$d_R^{\pm }:=d(z_R,L^2\cap L^{\pm })\asymp \left|z_R-z_2^{\pm }\right|\asymp d_{2,R}^{1+{\beta }_2},$$

from Lemma 1. Hence $d_{2,R}={\left(d_R^{\pm }\right)}^{{\displaystyle \frac{1}{1+{\beta }_2}}}.$ Further, according to Lemma 2 and [45] [Corollary 2], we get $d_R^{\pm }⪰n^{-\mu }$ and

$$d_{2,R}⪰n^{{\displaystyle \frac{-\mu }{1+{\beta }_2}}}.$$

(35)

Therefore, taking into account these estimates in (33), we obtain:

$$A_n⪯{∥P_n∥}_p\left\{\begin{array}{cc}{n}^{\left({\displaystyle \frac{{\gamma }_1+1}{p}}-1\right)\tilde{\mu }}+n^{\left({\displaystyle \frac{{\gamma }_2+1}{p}}-1\right){\displaystyle \frac{\mu }{1+{\beta }_2}}},& {\gamma }_1,{\gamma }_2>p-1,\\ {\left(lnn\right)}^{1-{\displaystyle \frac{1}{p}}},& \left\{\begin{array}{c}{\gamma }_1=p-1,-1<{\gamma }_2\le p-1,\\ \mathrm{or}-1<{\gamma }_1\le p-1,{\gamma }_2=p-1,\end{array}\right.\\ 1,& -1<{\gamma }_1,{\gamma }_2<p-1.\end{array}\right.$$

(36)

We now estimate estimate $B_{n,m}\left(\rho \right).$ Using notations (28) and setting $\rho =1$, we get:

$$B_{n,m}\left(t\right)=\underset{L}{\int }{\displaystyle \frac{\left|d\zeta \right|}{{\left|t-\zeta \right|}^{m+1}}}=:{\displaystyle \sum _{i=1}^2}\left[{\Im }_{1,1}^{i,\pm }+{\Im }_{1,2}^{i,\pm }\right],$$

(37)

where

$${\Im }_{1,k}^{i,\pm }:=\underset{E_{1,k}^{i,\pm }}{\int }{\displaystyle \frac{\left|d\zeta \right|}{{\left|\zeta -z_i\right|}^{m+1}}};i,k=1,2.$$

We have:

$$\begin{array}{cc}\hfill {\Im }_{1,1}^{1,\pm }& =\underset{E_{1,1}^{1,\pm }}{\int }{\displaystyle \frac{\left|d\zeta \right|}{{\left|\zeta -z_1\right|}^{m+1}}}⪯{\displaystyle \underset{0}{\overset{c_1{d}_{1,R}}{\int }}}{\displaystyle \frac{ds}{s^{m+1}}}⪯{\left({\displaystyle \frac{1}{d_{1,R}}}\right)}^{m+1}·\mathrm{mes}\left(E_{1,1}^{1,\pm }\right)⪯{\left({\displaystyle \frac{1}{d_{1,R}}}\right)}^m,\hfill \\ \hfill {\Im }_{1,2}^{1,\pm }& =\underset{E_{1,2}^{1,\pm }}{\int }{\displaystyle \frac{\left|d\zeta \right|}{{\left|\zeta -z_1\right|}^{m+1}}}⪯{\displaystyle \underset{c_1{d}_{1,R}}{\overset{\left|L_{1,1}^{\pm }\right|}{\int }}}{\displaystyle \frac{ds}{s^{m+1}}}⪯{\left({\displaystyle \frac{1}{d_{1,R}}}\right)}^m,\hfill \\ \hfill {\Im }_{1,1}^{1,\pm }+{\Im }_{1,2}^{1,\pm }& ⪯{\left({\displaystyle \frac{1}{d_{1,R}}}\right)}^m;\hfill \end{array}$$

(38)

and

$$\begin{array}{cc}\hfill {\Im }_{1,1}^{2,\pm }& =\underset{E_{1,1}^{2,\pm }}{\int }{\displaystyle \frac{\left|d\zeta \right|}{{\left|\zeta -z_2\right|}^{m+1}}}⪯{\displaystyle \underset{0}{\overset{c_2{d}_{2,R}}{\int }}}{\displaystyle \frac{ds}{s^{m+1}}}⪯{\left({\displaystyle \frac{1}{d_{2,R}}}\right)}^{m+1}·\mathrm{mes}\left(E_{1,1}^{2,\pm }\right)⪯{\left({\displaystyle \frac{1}{d_{2,R}}}\right)}^m,\hfill \\ \hfill {\Im }_{1,2}^{2,\pm }& =\underset{E_{1,2}^{2,\pm }}{\int }{\displaystyle \frac{\left|d\zeta \right|}{{\left|\zeta -z_2\right|}^{m+1}}}⪯{\displaystyle \underset{c_2{d}_{2,R}}{\overset{\left|L_{1,2}^{\pm }\right|}{\int }}}{\displaystyle \frac{ds}{s^{m+1}}}⪯{\left({\displaystyle \frac{1}{d_{2,R}}}\right)}^m,\hfill \\ \hfill {\Im }_{1,1}^{2,\pm }+{\Im }_{1,2}^{2,\pm }& ⪯{\left({\displaystyle \frac{1}{d_{2,R}}}\right)}^m.\hfill \end{array}$$

(39)

From (37), (38) and (39), we get:

$$B_{n,m}\left(t\right)⪯{\left({\displaystyle \frac{1}{d_{1,R}}}\right)}^m+{\left({\displaystyle \frac{1}{d_{2,R}}}\right)}^m.$$

Therefore, using estimates (34) and (35), we obtain:

$$B_{n,m}\left(t\right)⪯n^{\tilde{\mu }m}+n^{{\displaystyle \frac{-\mu }{1+{\beta }_2}}m}.$$

(40)

Combining estimates (23), (36) and (40), we have:

$$\begin{array}{cc}\hfill \left|P_n^{\left(m\right)}\left(z\right)\right|& ⪯{\displaystyle \frac{\left|{\mathsf{\Phi }}^{n-m+1}\left(z\right)\right|}{d(z,L)}}{∥P_n∥}_p\left[n^{\tilde{\mu }m}+n^{{\displaystyle \frac{-\mu }{1+{\beta }_2}}m}\right]\hfill \\ \hfill & \times \left[\left\{\begin{array}{cc}{n}^{\left({\displaystyle \frac{{\gamma }_1+1}{p}}-1\right)\tilde{\mu }}+n^{\left({\displaystyle \frac{{\gamma }_2+1}{p}}-1\right){\displaystyle \frac{\mu }{1+{\beta }_2}}},\hfill & {\gamma }_1,{\gamma }_2>p-1,\hfill \\ {\left(lnn\right)}^{1-{\displaystyle \frac{1}{p}}},\hfill & \left\{\begin{array}{c}{\gamma }_1=p-1,-1<{\gamma }_2\le p-1\hfill \\ \mathrm{or}-1<{\gamma }_1\le p-1,{\gamma }_2=p-1,\hfill \end{array}\right.\hfill \\ 1,\hfill & -1<{\gamma }_1,{\gamma }_2<p-1.\hfill \end{array}\right.\right]\hfill \end{array}$$

Now suppose that $p=1.$

After multiplying the integrand of the second integral on the right-hand side of inequality (22) by $h\left(t\right)$ and applying Lemma lem3, we obtain:

$$\begin{array}{cc}\hfill \left|P_n^{\left(m\right)}\left(z\right)\right|& ⪯{\displaystyle \frac{\left|{\mathsf{\Phi }}^{n-m+1}\left(z\right)\right|}{d(z,L)}}·\left[\underset{L_R}{\int }h\left(t\right)\left|P_n\left(t\right)\right|\left|dt\right|\right]·\underset{t\in L_R}{sup}\left(\underset{L}{\int }{\displaystyle \frac{\left|d\zeta \right|}{h\left(t\right){\left|\zeta -t\right|}^{m+1}}}\right)\hfill \\ \hfill & ⪯{\displaystyle \frac{\left|{\mathsf{\Phi }}^{n-m+1}\left(z\right)\right|}{d(z,L)}}·{∥P_n∥}_1·\underset{t\in L_R}{sup}\left(\underset{L}{\int }{\displaystyle \frac{\left|d\zeta \right|}{h\left(t\right){\left|\zeta -t\right|}^{m+1}}}\right).\hfill \end{array}$$

(41)

Denote by

$${\tilde{B}}_{n,m}\left(t\right):={\displaystyle \frac{1}{h\left(t\right)}}\underset{L}{\int }{\displaystyle \frac{\left|d\zeta \right|}{{\left|\zeta -t\right|}^{m+1}}}={\displaystyle \frac{1}{h\left(t\right)}}{B}_{n,m}\left(t\right),$$

and estimate this integral. We have:

$${\displaystyle \frac{1}{h\left(t\right)}}⪯{\displaystyle \frac{1}{{\left|t-z_1\right|}^{{\gamma }_1}{\left|t-z_2\right|}^{{\gamma }_2}}}⪯\left\{\begin{array}{cc}{n}^{\tilde{\mu }{\gamma }_1^{*}},& t\in E_{1,R}^{1,\pm }\cup E_{2,R}^{1,\pm },\\ n^{\mu },& t\in L_R\setminus {\displaystyle \bigcup _{i,j=1}^2}{E}_{j,R}^{i,\pm },\\ n^{{\displaystyle \frac{\mu {\gamma }_2^{*}}{1+{\beta }_2}}},& t\in E_{1,R}^{2,\pm }\cup E_{2,R}^{2,\pm },\end{array}\right.$$

and, so, from (40), we get:

$$\begin{array}{cc}\hfill {\tilde{B}}_{n,m}\left(t\right)& ⪯\left[n^{\tilde{\mu }m}+n^{{\displaystyle \frac{-\mu }{1+{\beta }_2}}m}\right]\left\{\begin{array}{cc}{n}^{\tilde{\mu }{\gamma }_1^{*}},\hfill & t\in E_{1,R}^{1,\pm }\cup E_{2,R}^{1,\pm },\hfill \\ n^{\mu },\hfill & t\in L_R\setminus {\displaystyle \bigcup _{i,j=1}^2}{E}_{j,R}^{i,\pm },\hfill \\ n^{{\displaystyle \frac{{\gamma }_2^{*}}{1+{\beta }_2}}\mu },\hfill & t\in E_{1,R}^{2,\pm }\cup E_{2,R}^{2,\pm },\hfill \end{array}\right.\hfill \\ \hfill & ⪯\left[n^{\tilde{\mu }m}+n^{{\displaystyle \frac{m}{1+{\beta }_2}}\mu }\right]\left[n^{\tilde{\mu }{\gamma }_1^{*}}+n^{{\displaystyle \frac{{\gamma }_2^{*}}{1+{\beta }_2}}\mu }\right]\hfill \\ \hfill & =n^{({\gamma }_1^{*}+m)\tilde{\mu }}+n^{{\displaystyle \frac{{\gamma }_2^{*}+m}{1+{\beta }_2}}\mu }.\hfill \end{array}$$

Therefore, we complete the proof of Theorem 1.

□

Proof of Theorem 4. 

The proof for $p>1$ was proved in [13] [Theorem 2.7]. We note that in the conditions of Theorem 2.7 [13] $p>0$ was incorrectly written and should have been written- $p>1$, although in subsequent corollaries and in the proof of the theorem it is correctly indicated that $p>1$. Also, on the left-hand side of the estimate is written ${∥P_n∥}_{\infty }$, while should have been written-${∥P_n^{\left(m\right)}∥}_{\infty }.$

We now prove that this is also true for $p=1$. Let the domain $G\in PQS\left(\lambda ;f_i{,g}_i\right)$, for some $\lambda \ge 1$, $f_1\left(x\right)=c_1{x}^{1+{\alpha }_1},{\alpha }_1\ge 0,$ and $g_2\left(x\right)=c_2{x}^{1+{\beta }_2},{\beta }_2>0.$ Assume that the weight function h is defined by (1), where ${\gamma }_j>-1,$ for $j=1,2$. The Cauchy integral representation for $m-th$ derivatives on $G_R$ yields:

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

Then,

$$\left|P_n^{\left(m\right)}\left(z\right)\right|\le {\displaystyle \frac{1}{2\pi }}\underset{L_R}{\int }\left|P_n\left(\zeta \right)\right|{\displaystyle \frac{\left|d\zeta \right|}{{\left|\zeta -z\right|}^{m+1}}},z\in G_R.$$

Let $z\in L$ is arbitrary fixed. According to Lemma 3, we get:

$$\begin{array}{cc}\hfill \left|P_n^{\left(m\right)}\left(z\right)\right|& \le {\displaystyle \frac{1}{2\pi }}\underset{L_R}{\int }\left|P_n\left(\zeta \right)\right|{\displaystyle \frac{\left|d\zeta \right|}{{\left|\zeta -z\right|}^{m+1}}}\le \underset{z\in L}{sup}\left({\displaystyle \frac{1}{h\left(\zeta \right){\left|\zeta -z\right|}^{m+1}}}\right)·\underset{L_R}{\int }h\left(\zeta \right)\left|P_n\left(\zeta \right)\right|\left|d\zeta \right|\hfill \\ \hfill & ={∥P_n∥}_{{\mathcal{L}}_1(h,L_R)}·\underset{z\in L}{sup}\left({\displaystyle \frac{1}{h\left(\zeta \right){\left|\zeta -z\right|}^{m+1}}}\right)⪯{∥P_n∥}_1·\underset{z\in L}{sup}{\tilde{D}}_m\left(\zeta \right),\hfill \end{array}$$

(42)

where

$${\tilde{D}}_m\left(\zeta \right):={\displaystyle \frac{1}{h\left(\zeta \right){\left|\zeta -z\right|}^{m+1}}},z\in L,\zeta \in L_R.$$

(43)

It remains for us to estimate of ${\tilde{D}}_m\left(\zeta \right).$ According to (1) for $j=1,2$, we have:

$${\tilde{D}}_m\left(\zeta \right)⪯{\displaystyle \frac{1}{{\left|\zeta -z_1\right|}^{{\gamma }_1}{\left|\zeta -z_2\right|}^{{\gamma }_2}{\left|\zeta -z\right|}^{m+1}}}=:{\displaystyle \bigcup _{k=0}^2}{\tilde{D}}_{n,m}^k,$$

(44)

where ${\tilde{D}}_{n,m}^k\left(\zeta \right):=\left\{\zeta :\zeta \in L_R^k,k=0,1,2\right\}.$

a) Let $\zeta \in L_R^0.$ Then:

$${\tilde{D}}_{n,m}^0={\displaystyle \frac{1}{{\left|\zeta -z_1\right|}^{{\gamma }_1}{\left|\zeta -z_2\right|}^{{\gamma }_2}{\left|\zeta -z\right|}^{m+1}}}⪯{\displaystyle \frac{1}{{\left|\zeta -z\right|}^{m+1}}}⪯{\displaystyle \frac{1}{d^{m+1}(z,L_R)}}⪯n^{\left(m+1\right)\mu };$$

(45)

b) For the $\zeta \in L_R^1,$ we get:

$$\begin{array}{cc}\hfill {\tilde{D}}_{n,m}^1& ={\displaystyle \frac{1}{{\left|\zeta -z_1\right|}^{{\gamma }_1}{\left|\zeta -z_2\right|}^{{\gamma }_2}{\left|\zeta -z\right|}^{m+1}}}⪯{\displaystyle \frac{1}{{\left|\zeta -z_1\right|}^{{\gamma }_1}{\left|\zeta -z\right|}^{m+1}}}\hfill \\ \hfill & ⪯{\displaystyle \frac{1}{d_{1,R}^{{\gamma }_1+m}}}⪯n^{({\gamma }_1+m+1)\tilde{\mu }},\mathrm{if}{\gamma }_1>0;\hfill \\ \hfill {\tilde{D}}_{n,m}^1& ={\displaystyle \frac{{\left|\zeta -z_1\right|}^{-{\gamma }_1}}{{\left|\zeta -z_2\right|}^{{\gamma }_2}{\left|\zeta -z\right|}^{m+1}}}⪯{\displaystyle \frac{1}{{\left|\zeta -z\right|}^{m+1}}}⪯{\displaystyle \frac{1}{d_{1,R}^{m+1}}}⪯n^{(m+1)\tilde{\mu }},\mathrm{if}{\gamma }_1\le 0.\hfill \end{array}$$

(46)

c) Let us set: ${\left(E_{2,R}^{1,\pm }\right)}_1:=\left\{\zeta \in L_R^2:\left|\zeta -z\right|\ge \left|\zeta -z_2\right|\right\},{\left(E_{2,R}^{1,\pm }\right)}_2:=E_{2;R}^{1,\pm }\setminus {\left(E_{2,R}^{1,\pm }\right)}_1.$ Then:

$$\begin{array}{cc}\hfill {\tilde{D}}_{n,m}^2& ⪯{\displaystyle \frac{1}{{\left|\zeta -z_2\right|}^{{\gamma }_2+m+1}}}⪯{\displaystyle \frac{1}{d_{2,R}^{{\gamma }_2+m+1}}}⪯n^{\left({\gamma }_2+m+1\right){\displaystyle \frac{\mu }{1+{\alpha }_2}}},z\in {\left(E_{2,R}^{1,\pm }\right)}_1,\mathrm{if}{\gamma }_2>0;\hfill \\ \hfill {\tilde{D}}_{n,m}^2& ⪯{\displaystyle \frac{1}{{\left|\zeta -z\right|}^{{\gamma }_2+m+1}}}⪯{\displaystyle \frac{1}{d_{2,R}^{{\gamma }_2+m+1}}}⪯n^{\left({\gamma }_2+m+1\right){\displaystyle \frac{\mu }{1+{\alpha }_2}}},z\in {\left(E_{2,R}^{1,\pm }\right)}_2,\mathrm{if}{\gamma }_2>0;\hfill \\ \hfill {\tilde{D}}_{n,m}^2& ={\displaystyle \frac{{\left|\zeta -z_2\right|}^{-{\gamma }_2}}{{\left|\zeta -z_2\right|}^{m+1}}}⪯{\displaystyle \frac{1}{d_{2,R}^{m+1}}}⪯n^{{\displaystyle \frac{m+1}{1+{\alpha }_2}}\mu },z\in {\left(E_{2,R}^{1,\pm }\right)}_1,\mathrm{if}{\gamma }_2\le 0;\hfill \\ \hfill {\tilde{D}}_{n,m}^2& ={\displaystyle \frac{{\left|\zeta -z_2\right|}^{-{\gamma }_2}}{{\left|\zeta -z\right|}^{m+1}}}⪯{\displaystyle \frac{1}{d_{2,R}^{m+1}}}⪯n^{{\displaystyle \frac{m+1}{1+{\alpha }_2}}\mu },z\in {\left(E_{2,R}^{1,\pm }\right)}_2,\mathrm{if}{\gamma }_2\le 0;\hfill \end{array}$$

(47)

Combining estimates (43)-(47), we get:

$${\tilde{D}}_m\left(\zeta \right)⪯n^{({\gamma }_1^{*}+m+1)\tilde{\mu }}+n^{\left({\gamma }_2^{*}+m+1\right){\displaystyle \frac{\mu }{1+{\alpha }_2}}}.$$

Therefore, we complete the proof for any $p=1$. □

Proof of Theorem 3.3. 

Let $G\in PQS\left(\lambda ;f_1{,g}_2\right)$, for some $\lambda \ge 1$, $f_1\left(x\right)=C_1{x}^{1+{\alpha }_1},$ ${\alpha }_1\ge 0,$ and $g_2\left(x\right)=C_2{x}^{1+{\beta }_2},{\beta }_2>0.$ Let $R=1+{\displaystyle \frac{{\epsilon }_0}{n}},0<{\epsilon }_0<{\displaystyle \frac{1}{2}}$. For $0\le m<n$ and $z\in G_R$, by Cauchy integral formulas for derivatives, we have:

$$P_n^{\left(m\right)}\left(z_j\right)={\displaystyle \frac{m!}{2\pi i}}\underset{L_R}{\int }{\displaystyle \frac{P_n\left(\zeta \right)}{{(\zeta -z_j)}^{m+1}}}d\zeta ,j=1,2.$$

(48)

First, let us $p>1.$ Multiplying the numerator and denominator of the integrand by $h^{{\displaystyle \frac{1}{p}}}$ and using Hölder’s inequality, we get:

$$\begin{array}{cc}\hfill \left|P_n^{\left(m\right)}\left(z_j\right)\right|& \le {\displaystyle \frac{m!}{2\pi }}{\left(\underset{L_R}{\int }h\left(\zeta \right){\left|P_n\left(\zeta \right)\right|}^p\left|d\zeta \right|\right)}^{1/p}·{\left(\underset{L_R}{\int }{\displaystyle \frac{\left|d\zeta \right|}{h^{q/p}\left(\zeta \right){\left|\zeta -z_j\right|}^{q(m+1)}}}\right)}^{1/q}\hfill \\ \hfill & =:M_{n,1}·M_{n,2},{\displaystyle \frac{1}{p}}+{\displaystyle \frac{1}{q}}=1.\hfill \end{array}$$

(49)

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

$$M_{n,1}⪯{∥P_n∥}_p.$$

(50)

For the integral $M_{n,2},$ using (20), we find:

$$\begin{array}{cc}\hfill {\left(M_{n,2}\right)}^q& =\underset{L_R}{\int }{\displaystyle \frac{\left|d\zeta \right|}{h^{q-1}\left(\zeta \right){\left|\zeta -z_j\right|}^{q(m+1)}}}\hfill \\ \hfill & \asymp \underset{L_R}{\int }{\displaystyle \frac{\left|d\zeta \right|}{{\left|\zeta -z_1\right|}^{(q-1){\gamma }_1}{\left|\zeta -z_2\right|}^{(q-1){\gamma }_2}{\left|\zeta -z_j\right|}^{q(m+1)}}}\hfill \\ \hfill & ⪯\underset{L_R^1}{\int }{\displaystyle \frac{\left|d\zeta \right|}{{\left|\zeta -z_1\right|}^{(q-1){\gamma }_1+q(m+1)}}}+\underset{L_R^2}{\int }{\displaystyle \frac{\left|d\zeta \right|}{{\left|\zeta -z_2\right|}^{(q-1){\gamma }_2+q(m+1)}}}=:{\tilde{J}}_R^1+{\tilde{J}}_R^2.\hfill \end{array}$$

(51)

Now, for estimates the integrals ${\tilde{J}}_R^1$ and ${\tilde{J}}_R^2$, let us use the notation (28) for $\rho =R$ and denote:

$${\tilde{J}}_{R,k}^{i,\pm }:=\underset{E_{R,k}^{i,\pm }}{\int }{\displaystyle \frac{\left|d\zeta \right|}{{\left|\zeta -z_i\right|}^{{\gamma }_i(q-1)+q(m+1)}}};i,k=1,2.$$

Then, using these notations from (51), we get:

$$\begin{array}{cc}\hfill {\left(M_{n,2}\right)}^q& ⪯{\displaystyle \sum _{i=1}^2}\left[{\tilde{J}}_{R,1}^{i,\pm }+{\tilde{J}}_{R,2}^{i,\pm }\right],i=1,2.\hfill \end{array}$$

(52)

Thus, it remains for us to estimate the integrals ${\tilde{J}}_{R,k}^{i,\pm }$for each $i,k=1,2.$ For arbitrary ${\gamma }_1,{\gamma }_2>-1,$ we get:

$$\begin{array}{c}\begin{array}{cc}{\tilde{J}}_{R,1}^{1,\pm }\hfill & ⪯{\displaystyle \underset{0}{\overset{c_1{d}_{1,R}}{\int }}}{\displaystyle \frac{ds}{s^{{\gamma }_1(q-1)+q(m+1)}}}\hfill \\ \hfill & ⪯{\left({\displaystyle \frac{1}{d_{1,R}}}\right)}^{{\gamma }_1(q-1)+q(m+1)}·\mathrm{mes}\left(E_{R,1}^{1,\pm }\right)⪯{\left({\displaystyle \frac{1}{d_{1,R}}}\right)}^{{\gamma }_1(q-1)+q(m+1)-1};\hfill \end{array}\hfill \\ \begin{array}{c}{\tilde{J}}_{R,2}^{1,\pm }⪯{\displaystyle \underset{c_1{d}_{1,R}}{\overset{\left|L_{R,1}^{\pm }\right|}{\int }}}{\displaystyle \frac{ds}{s^{{\gamma }_1(q-1)+q(m+1)}}}\hfill \\ ⪯{\left({\displaystyle \frac{1}{d_{1,R}}}\right)}^{{\gamma }_1(q-1)+q(m+1)-1};\hfill \end{array}\hfill \\ {\tilde{J}}_{R,1}^{1,\pm }+{\tilde{J}}_{R,2}^{1,\pm }⪯{\left({\displaystyle \frac{1}{d_{1,R}}}\right)}^{{\gamma }_1(q-1)+q(m+1)-1}.\hfill \end{array}$$

(53)

Similarly, for the ${\tilde{J}}_R^2$ in the neighborhood of the point $z_2$, we have:

$$\begin{array}{c}\begin{array}{cc}{\tilde{J}}_{R,1}^{2,\pm }\hfill & ⪯{\displaystyle \underset{0}{\overset{c_2{d}_{2,R}}{\int }}}{\displaystyle \frac{ds}{s^{{\gamma }_2(q-1)+q(m+1)}}}\hfill \\ \hfill & ⪯{\left({\displaystyle \frac{1}{d_{2,R}}}\right)}^{{\gamma }_2(q-1)+q(m+1)}·\mathrm{mes}\left(E_{R,1}^{2,\pm }\right)⪯{\left({\displaystyle \frac{1}{d_{2,R}}}\right)}^{{\gamma }_2(q-1)+q(m+1)-1};\hfill \end{array}\hfill \\ \begin{array}{cc}{\tilde{J}}_{R,2}^{2,\pm }\hfill & ⪯{\displaystyle \underset{c_2{d}_{2,R}}{\overset{\left|L_{R,2}^{\pm }\right|}{\int }}}{\displaystyle \frac{ds}{s^{{\gamma }_2(q-1)+q(m+1)}}}\hfill \\ \hfill & ⪯{\left({\displaystyle \frac{1}{d_{2,R}}}\right)}^{{\gamma }_2(q-1)+q(m+1)-1};\hfill \end{array}\hfill \\ {\tilde{J}}_{R,1}^{2,\pm }+{\tilde{J}}_{R,1}^{2,\pm }⪯{\left({\displaystyle \frac{1}{d_{2,R}}}\right)}^{{\gamma }_2(q-1)+q(m+1)-1}.\hfill \end{array}$$

(54)

In this case, combining (49) - (54), we get:

$$\left|P_n^{\left(m\right)}\left(z_j\right)\right|⪯{∥P_n∥}_p{\left({\displaystyle \frac{1}{d_{j,R}}}\right)}^{{\displaystyle \frac{{\gamma }_j+1}{p}}+m}.$$

To complete the proof of Theorem 7 in the case $p>1$, it remains to use estimates (34) and (35).

The proof argument for $p=1$ is identical to that used in the proof of Theorem 4 for $p=1$, upon setting $z=z_j,$ $j=1,2.$ □

Proof of Remark 1. 

The sharpness of estimates (14) - (19) follows from the fact that they combine the classical sharp Markov inequalities ${∥P_n^{\left(m\right)}∥}_{\infty }⪯n^m{∥P_n∥}_{\infty }$, $m\ge 1,$ with weighted norm inequalities of the form ${∥P_n∥}_{\infty }⪰F_n(\gamma ,p){∥P_n∥}_{\mathcal{L}(h^{*},L)},$ $p>1.$ The sharpness of the above inequalities may be verified by means of the following examples. Let $T_n\left(z\right)=1+z+...+z^n$ and consider the weights $h^{*}\left(z\right)=$ $h_0\left(z\right)$ and $h^{**}\left(z\right)={\left|z-1\right|}^{\gamma },\gamma >0$, on the unit circle $L:=\left\{z:\left|z\right|=1\right\}$. Then, for every $n\in \mathbb{N}$, there exist constants $c_3=c_3(h^{*},p)>0$ and $c_3^{\prime }=c_3^{\prime }(h^{**},p)>0,$ satisfying:

$$\begin{array}{cc}\hfill a){∥T_n∥}_{\infty }& \ge c_3{n}^{{\displaystyle \frac{1}{p}}}{∥T_n∥}_{{\mathcal{L}}_p(h^{*},L)},p>1;\hfill \\ \hfill b){∥T_n∥}_{\infty }& \ge c_3^{\prime }{n}^{{\displaystyle \frac{\gamma +1}{p}}}{∥T_n∥}_{{\mathcal{L}}_p(h^{**},L)},p>\gamma +1.\hfill \end{array}$$

Really, if $L:=\left\{z:\left|z\right|=1\right\},$ then L is quasismooth and so, $L\in PQS(1;0.0)=QS\left(1\right).$

a) $h^{*}\left(z\right)\equiv 1;$ b) $h^{**}\left(z\right)={\left|z-1\right|}^{\gamma },\gamma >0.$ Obviously,

$$\left|T_n\left(z\right)\right|\le {\displaystyle \sum _{j=0}^{n-1}}\left|z^j\right|=n,\left|z\right|=1;\left|T_n\left(1\right)\right|=n.$$

So,

$${∥T_n∥}_{\infty }=n.$$

On the other hand, according to [26] [p. 236], we have:

$${∥T_n∥}_{{\mathcal{L}}_p(h^{*},L)}\asymp n^{1-{\displaystyle \frac{1}{p}}},p>1,$$

and

$${∥T_n∥}_{{\mathcal{L}}_p(h^{**},L)}\asymp n^{1-{\displaystyle \frac{\gamma +1}{p}}},p>\gamma +1.$$

Therefore,

$$\begin{array}{cc}\hfill a){∥T_n∥}_{\infty }& =n\asymp n^{{\displaystyle \frac{1}{p}}}{∥T_n∥}_{{\mathcal{L}}_p(h^{*},L)},p>1;\hfill \\ \hfill b){∥T_n∥}_{\infty }& =n=n·n^{1-{\displaystyle \frac{\gamma +1}{p}}}·n^{{\displaystyle \frac{\gamma +1}{p}}-1}\asymp n^{{\displaystyle \frac{\gamma +1}{p}}}{∥T_n∥}_{{\mathcal{L}}_p(h^{**},L)},p>\gamma +1.\hfill \end{array}$$

□

5. Discussion

In this paper, we continue our study of the growth properties of the $m-th$ $(m\ge 1)$ derivatives of algebraic polynomials in weighted Lebesgue spaces on both bounded and unbounded domains. In our previous works and related studies (see the references in the introduction),this problem has been investigated for various classes of curves. It should be noted that in [13] a similar problem was treated using a recursive method; namely, the estimate for the m-th derivative was derived from the corresponding estimate for the $(m-1)$-th derivative. In contrast, in the present paper we avoid recursive arguments and treat all $m\ge 1$ simultaneously, obtaining stronger results than those in [13]. We explicitly demonstrate the influence of the order of tangency of boundary arcs at junction points, as well as the effect of zeros and poles of the weight function on the growth of the polynomial (see Theorems 1–3 and the corresponding corollaries).

Furthermore, we extend the range of the exponent p in the space ${\mathcal{L}}_p(h,L)$ to $p\ge 1$ in the estimates for the m-th $(m\ge 1)$ derivatives. Here, we reveal the influence of the type of zero angles on the growth behavior of polynomials and demonstrate the sharpness of the obtained estimates in several special cases. By combining the results of Theorems 1–3, Theorems 4–8, and the corresponding corollaries, we derive global estimates for the growth of polynomials in the entire complex plane. Moreover, we also establish growth estimates at the boundary points $z_j$, $j=1,2$. The obtained estimates clearly demonstrate the influence of the nature of the zero angles of tangency of the boundary arcs and the "zeros" and "poles" of the weight function on the orders of growth of the modulus of polynomial.

.