On the Growth of Derivatives of Algebraic Polynomials in Regions with a Piecewise Smooth Boundary

Abstract

In this paper, we study the behavior of the m-th $(m\ge 0)$ derivatives of general algebraic polynomials in weighted Bergman spaces defined in regions of the complex plane G bounded by piecewise smooth curves $L=\partial G$ with $\lambda \pi$ $(0<\lambda \le 2)$ exterior angles relative to G. Upper bounds are found for the growth of the m-th derivatives of the polynomials not only inside the unbounded region but also on the closures of this region with both exterior non-zero angles $\lambda \pi$ $(0<\lambda <2)$ and interior zero angles (i.e., exterior angles $2\pi$). The influence of the boundary angles $\lambda \pi$ $(0<\lambda \le 2)$ of the region G and the “growth rate” of the weight function on the behavior of the moduli of polynomials and their derivatives in regions of the complex plane that are “symmetric” with respect to L (bounded and unbounded) is found.

1. Introduction and Preliminaries

Let $\mathbb{C}$ denote the complex plane, $\overline{\mathbb{C}}:=C\cup \left\{\infty \right\};G$ $\subset C$ be a bounded Jordan region (without loss of generality, we will assume that $0\in G)$; $L:=\partial G$; and $\Omega :=\overline{\mathbb{C}}\setminus \overline{G}=extL$. For $w\in \mathbb{C}$ and $\delta >0,$ we set the following: $\Delta (w,\delta ):=\left\{t\in \mathbb{C}:\left|t-w\right|>\delta \right\};\Delta :=\Delta (0,1)$. Let $w=\Phi \left(z\right)$ be the univalent conformal mapping of $\Omega$ onto $\Delta$ such that $\Phi (\infty )=\infty$ and ${lim}_{z\to \infty }{\displaystyle \frac{\Phi \left(z\right)}{z}}>0$; $\Psi :={\Phi }^{-1}$. For $\rho \ge 1,$ let $L_{\rho }$ denote the exterior level curve of L, $G_{\rho }$ denote the inner region of $L_{\rho }$, and ${\Omega }_{\rho }$ denote the outer region of $L_{\rho }$.

Let us denote

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

which is the set of all complex-coefficient algebraic polynomials of degree at most n, where $n\in \mathbb{N}$.

The distance from a point $z\in \mathbb{C}$ to a set $A\subset \mathbb{C}$, denoted by $d(z,A)$, is defined as

$$d(z,A):=inf\left\{\right|z-\xi |:\xi \in A\}.$$

Let ${\left\{z_j\right\}}_{j=1}^l\subset L$ be a fixed sequence of distinct points ordered in the positive direction along L, without a loss of generality. Let $R_0$ be a fixed constant such that $1<R_0<\infty$. Assume that the parameter ${\gamma }_j$ satisfies ${\gamma }_j>-2$ for all $j=1,\dots ,l$. Let $h_0$ be a measurable function satisfying the inequality $h_0\left(z\right)\ge c_0(G,h)>0,z\in G_{R_0},$ for some constant $c_0(G,h)$ depending only on G and h. Under these assumptions, we define the generalized Jacobi weight function as

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

(1)

Let $\sigma$ denote the two-dimensional Lebesgue measure. Given an arbitrary Jordan region G and a real number $p>0$, we present

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

(2)

and, when $L$ is rectifiable,

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

(3)

For any $P_n\in {\wp }_n$, the Bernstein–Walsh Lemma [1] states that the following inequality holds:

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

(4)

Also, in [2], a similar inequality of (4) for the space ${\mathcal{L}}_p\left(L\right)$ is obtained as follows:

$${∥P_n∥}_{{\mathcal{L}}_p\left(L_R\right)}\le R^{n+\frac{1}{p}}{∥P_n∥}_{{\mathcal{L}}_p\left(L\right)},p>0.$$

(5)

In [3] (Th. 1.1), an analogous estimate was studied for the $A_p(1,G)$–norm, $p>0,$ for an arbitrary Jordan region, and it was obtained that, for any $P_n\in {\wp }_n,R_1=1+{\displaystyle \frac{1}{n}}$ and arbitrary $R,R>R_1$, the following estimate,

$${∥P_n∥}_{A_p\left(G_R\right)}\le c{R}^{n+\frac{2}{p}}{∥P_n∥}_{A_p\left(G_{R_1}\right)}.$$

In [4] the case was considered, where the norm ${∥P_n∥}_{C\left(\overline{G}\right)}$ on the right-hand side (4) is replaced by ${∥P_n∥}_{A_2\left(G\right)}$ which leads to a new version of the Bernstein–Walsh lemma: if the curve $L=\partial G$ is quasiconformal and rectifiable, then there exists a constant $c=c\left(L\right)>0$ depending only on $L$ such that

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

(6)

holds for every $P_n\in {\wp }_n.$

On the other hand, using the mean value theorem, for an arbitrary Jordan region G, any polynomial $P_n\in {\wp }_n$, and any $p>0$, we can find that

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

(7)

In particular, estimates of the form (7) for orthonormal polynomials $K_n\left(z\right):=K_n(z,h,G),$ $deg{K}_n=n,$ $n\in \mathbb{N}$, over a region $G$ and a specific weight function h were investigated in [5] (Th.2.1), [6] (Th.5, $h\left(z\right)\equiv 1$), [7] (Th.1, $h\left(z\right)\ne 1$), and others. Similar estimates to (7) were studied, and estimates in the space $A_2(h,G)$ were obtained for any point $z\in G$ in the following form:

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

where $c_6=c_6\left(G\right)>0,$ $\alpha =\alpha (G,h)>0$ and $\beta =\beta (G,h)>0,$ are constants independent of $n,$ z and depending on the properties of the region G and the behavior of the weight function $h.$ By combining (6) and (7), we obtain an estimation on the growth of $\left|P_n\left(z\right)\right|$ in $\mathbb{C}\setminus L$.

Thus, as can be seen from (6) and (7), to study the behavior of the polynomials and their derivatives $\left|P_n^{\left(m\right)}\left(z\right)\right|,$ $m\ge 0,$ in whole $\mathbb{C}$, it is also necessary to know the behavior of these derivatives at the points of $L.$ This means that we need to represent the whole plane as the union of a finite closed region and a region “symmetric to it with respect to the curve L”, for example, $\mathbb{C}=\overline{G}\cup \Omega$, and then study the behavior of $P_n^{\left(m\right)}\left(z\right),$ $m\ge 0$, in each of these regions separately.

“Symmetry with respect to the boundary” can naturally be viewed as a “generalization” of the concepts of symmetry with respect to a line and a circle. Just as there is symmetry with respect to a line or a circle, symmetry can also be defined with respect to a given smooth curve (for example, a quasiconformal reflection; see, for example, [8] (p. 100)). Since every piecewise smooth curve L can be viewed as a quasiconformal (see, for example, [8] (p. 81), Corollary 1), then there is a quasiconformal reflection with respect to it, i.e., a quasiconformal mapping that takes G to $\Omega$ and $\Omega$ to G and leaves the points of L fixed. Such a mapping can be regarded as a “symmetry” with respect to the curve $L.$ It is well known that if L is quasiconformal, then every level curve $L_R$ is also quasiconformal for each $R>1.$ In this context, we study the growth of polynomials in regions ${\overline{G}}_{R_1}$ and ${\Omega }_{R_1}$, which are mutually symmetric with respect to a given curve $L_{R_1}$ for $0<\lambda <2$. In the case of $\lambda =2$, one can also define symmetric regions with respect to a curve with zero angles, but using reflections with respect to a finite number of smooth arcs that make up the curve L results in symmetries.

In [9] (Th. 5, 6), the growth of the modulus of the derivative ($m=1$) of polynomials in unbounded regions with zero and non-zero angles on the boundary was studied, and the results were obtained using a recurrence formula. Further analysis of the results showed that on the right-hand side of the estimates, the value ${\left|\Phi \left(z\right)\right|}^{2(n+1)}$ yields a strong growth of the modulus of the derivative of the polynomial. In the present paper, instead of a recurrence formula, another method was found, and a “weaker” growth of the modulus of the derivative of the polynomial was obtained, and the dependence of the growth on the order of the derivative $m\ge 1$ was indicated.

In the present study, we extend previous investigations on the problem of obtaining pointwise estimates for higher-order derivatives of the modulus $P_n\left(z\right)$ within some unbounded regions characterized by interior zero and exterior non-zero angles. Our goal is to derive estimates in the following form:

$$\left|P_n^{\left(m\right)}\left(z\right)\right|\le {\eta }_n{∥P_n∥}_p,z\in \Omega ,m=1,2,\dots ,n,$$

(8)

where ${\eta }_n:={\eta }_n(G,h,p,m,z), with$ ${\eta }_n(·)\to \infty$ as $n\to \infty$, depends on the properties of G and h.

Similar results of type (8) for some norms and various unbounded regions were obtained by N.A. Lebedev, P.M. Tamrazov, and V.K. Dzyadyk (see, for example, [10] (p. 383) and references therein). A number of studies considered the case $m=0$ with various values of $p>0$ and for various regions. Such studies include the works of [11] for regions with piecewise smooth boundaries having interior angles and for $p>0$, as well as others.

For derivatives of order $m\ge 1$, estimates similar to those given in (8) have been studied in various papers. In particular, ref. [12] studies the cases of $m=1,p>1$ for regions with piecewise smooth boundaries without cusps; ref. [9] studies the cases of $m=1,$ $p>1$ for regions with a piecewise smooth boundary having both zero and non-zero interior angles and others. These results are often based on a recursive formula approach, where the bound for each order of the derivative is derived from the bounds with lower orders, which requires extensive computation. However, in the current work, a direct method is used to obtain estimates of type (8) without relying on the recurrence formula. As a result, estimates for the growth of the modulus of polynomials that are much better than those in the above mentioned works are found in an unbounded region.

Secondly, to finalize the analysis of the growth of $\left|P_n^{\left(m\right)}\left(z\right)\right|$ in the whole complex plane, it is necessary to establish an inequality analogous to the Bernstein–Markov–Nikol’skii inequality in the regions under consideration, as shown below.

$$\parallel P_n^{\left(m\right)}{\parallel }_{\infty } \le {\mu }_n{\parallel P_n\parallel }_p,m=0,1,2,\dots ,n,$$

(9)

where ${\mu }_n:={\mu }_n(G,h,p,m)>0,$ ${\mu }_n\to \infty$, $n\to \infty ,$ is a constant that depends on the properties of G and the weight function h, in general.

The investigation of inequalities of type (9) began with the foundational works in [13,14,15]. Subsequently, similar research has been explored in numerous studies. Numerous studies have addressed these inequalities for $m\ge 0$ within a range of function spaces, including those by [16], [17] (pp. 122–133), [18,19,20,21], and references therein. A more detailed study on this topic has been made by [22] ($m=0)$ for $p>0$ and regions with a piecewise smooth boundary with interior cusps; [11] ($m=0)$ for $p>0$ and regions with piecewise smooth boundaries containing interior angles; [23] ($m\ge 1)$ for $p>1$ and regions with piecewise Dini-smooth boundaries having interior zero angles); and others.

In all the cited works, when assessing the growth of the modulus of a polynomial and its derivatives of order $m\ge 1$, an important role was played by the explicit dependence of the value of ${\mu }_n$ on the right-hand side of the angle formed between the boundary arcs at the points $\left\{z_j\right\}\in L,$ $j=1,2,$ where the weight function has a zero of order ${\gamma }_j\ge 0$ or a pole of order $-2<{\gamma }_j<0.$ This work also indicates the dependence of the value of ${\mu }_n$ on the geometry of a given region and on the weight function.

Thus, by combining the estimates (8) and (9), we find the growth of the m-th derivatives $\left|P_n^{\left(m\right)}\left(z\right)\right|,$ $m=0,1,2,\dots ,$ on the whole complex plane as follows:

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

(10)

where $R=1+c{n}^{-1};$ $c,$ $0<c<1,$ and $c_7=c_7(G,p)>0$ are constants that do not depend on n or $P_n$ but depend on the properties of G and h.

2. Definitions and Notations

Throughout this paper, $c,c_0,c_1,c_2,\dots$ are positive constants, and ${\epsilon }_0,{\epsilon }_1,{\epsilon }_2,\dots$ are sufficiently small positive constants (generally, they are different in different relations). These constants generally depend on G and inessential parameters for the argument. Otherwise, the dependence will be explicitly stated. For any $k\ge 0$ and $m>k,$ the notation $i=\overline{k,m}$ means $i=k,k+1,\dots ,m$.

Definition 1

([24]). The Jordan curve (or arc) L is called K-quasiconformal ($K>1$) if there is a K-quasiconformal mapping f of the region $D\supset L$ such that $f\left(L\right)$ is a circle (or line segment).

Let $F\left(L\right)$ denote the set of all sense-preserving plane homeomorphisms f of the region $D\supset L$ such that $f\left(L\right)$ is a circle (a line segment), and let

$$K_L:=inf\left\{K\left(f\right):f\in F\left(L\right)\right\},$$

where $K\left(f\right)$ is the maximal dilatation of $f.$ Then L is a quasiconformal curve if $K_L<\infty ,$ and L is a K-quasiconformal curve if $K_L\le K.$ A curve L is called a quasiconformal if it is a K-quasiconformal for some $K>1.$

Let $z=z\left(s\right),s\in \left[0,mesL\right]$ denote the natural representation of L.

Definition 2.

We say that a Jordan curve or arc L is smooth if $L$ has a continuous tangent $\theta \left(z\right):=\theta \left(z\right(s\left)\right)$ at every point $z\left(s\right).$ The class of such smooth curves or arcs is denoted by $C_{\theta }$. Then we write $G\in C_{\theta }$ such that $\partial G\in C_{\theta }.$

According to the “three-point” criterion [8] (p. 100), every piecewise smooth curve (without any cusps) or arc is quasiconformal. Moreover, according to [24], we have the following:

Corollary 1.

If $L\in C_{\theta },$ then L is $(1+\epsilon )-$quasiconformal for arbitrary small $\epsilon >0.$

We say that a bounded Jordan curve (or arc) $L$ is locally smooth at the point $z\in L$ if there exists a closed subarc ℓ $\subset L$ containing z such that every open subarc of ℓ containing z is smooth.

Next, we introduce a certain category of regions enclosed by a piecewise smooth boundary curve, where the connecting arcs form interior angles with zero or non-zero exterior angles.

For a Jordan region G with boundary $L=\partial G$, let ${\left\{{\zeta }_j\right\}}_{j=0}^l\subset L$ denote a finite, ordered set of distinct points placed along L, which, without loss of generality, are assumed to follow the positive orientation. We define each arc $L_j$ as the segment of the boundary connecting the consecutive points ${\zeta }_j$ and ${\zeta }_{j+1}$, where the indexing is taken as modulo $l+1$, i.e., ${\zeta }_{l+1}\equiv {\zeta }_0$.

Definition 3.

We say that a Jordan region $G\in P{C}_{\theta }(1;{\lambda }_1,\dots ,{\lambda }_l),0<{\lambda }_j\le 2,j=\overline{1,l}$ if $L=\partial G$ is the union of finitely many smooth arcs ${\left\{L_j\right\}}_{j=0}^l$ connected at points ${\left\{{\zeta }_j\right\}}_{j=0}^l\in L$ such that L has exterior (with respect to $\overline{G}$) angles ${\lambda }_j\pi ,0<{\lambda }_j\le 2,$ at the corner points ${\left\{{\zeta }_j\right\}}_{j=1}^l\in L$ where two arcs $L_{j-1}$ and $L_j,j=\overline{1,l},$ intersect, and L is locally smooth at ${\zeta }_0$.

Without loss of generality, we assume that these points on the curve $L=\partial G$ are located in the positive direction such that $G$ has exterior ${\lambda }_j\pi ,0<{\lambda }_j<2,$ $j=\overline{1,l_1},$ angles at the points ${\left\{{\zeta }_j\right\}}_{j=1}^{l_1}$, $l_1\le l,$ and interior zero angles (i.e., ${\lambda }_j=2$—interior cusps) at the points ${\left\{{\zeta }_j\right\}}_{j=l_1+1}^l$.

It is clear from Definition 3 that each region $G\in P{C}_{\theta }(1;{\lambda }_1,\dots ,{\lambda }_l),0<{\lambda }_j\le 2,j=\overline{1,l},$ may have exterior non-zero ${\lambda }_j\pi ,$ $0<{\lambda }_j<2,$ angles at the points ${\left\{z_j\right\}}_{j=1}^{l_1}\in L,$ and interior zero angles (${\lambda }_j=2$) at the the points ${\left\{z_j\right\}}_{j=l_1+1}^l\in L.$ If $l_1=l=0,$ then the region G does not have such angles; in this case we will write the following: $G\in P{C}_{\theta }\left(1\right)\equiv C_{\theta }$; if $l_1=l\ge 1,$ then G has only ${\lambda }_i\pi ,0<{\lambda }_i<2,i={\overline{1,l}}_1,$ exterior non-zero angles; in this case we will write $G\in P{C}_{\theta }(1;{\lambda }_1,\dots ,{\lambda }_l),0<{\lambda }_j<2,j=\overline{1,l}$; if $l_1=0$ and $l\ge 1,$ then G has only interior zero angles; in this case we will write $G\in P{C}_{\theta }(1;2,\dots ,2).$

Moreover, throughout this paper, we will assume that the ${\left\{z_j\right\}}_{j=1}^l\in L$ points defined in (1) coincide with the ${\left\{{\zeta }_j\right\}}_{j=1}^l\in L$ points specified in Definition 3.

To simplify the exposition and avoid cumbersome calculations, we will take $l_1=1,l=2$ without a loss of generality. After this assumption, we have the $G\in P{C}_{\theta }(1;{\lambda }_1,2),0<{\lambda }_1<2,$ region which has the exterior non-zero angle ${\lambda }_1\pi ,$ $0<{\lambda }_1<2,$ at the point $z_1\in L$ and the interior zero angle at the point $z_2\in L$. Note that the notation “$G\in P{C}_{\theta }(1;{\lambda }_1,{\lambda }_2),0<{\lambda }_1,{\lambda }_2<2$” means that the region G has two exterior non-zero ${\lambda }_j\pi ,$ $0<{\lambda }_j<2,$ angles at the point $z_j\in L,$ $j=1,2.$

For $0<{\delta }_j<{\delta }_0:={\displaystyle \frac{1}{4}}min\left\{\left|z_1-z_2\right|:j=1,2\right\}$, let us assume that $\delta :={\displaystyle \underset{1\le j\le 2}{min}}{\delta }_j,$ $\Omega (z_j,{\delta }_j):=\Omega \cap \left\{z:\left|z-z_j\right|\le {\delta }_j\right\};\Omega \left(\delta \right):={\displaystyle \bigcup _{j=1}^2}\Omega (z_j,\delta ),\widehat{\Omega }:=\Omega \setminus \Omega \left(\delta \right).$

Let $w_j:=\Phi \left(z_j\right),{\phi }_j:=arg{w}_j.$ Without loss of generality, we will assume that ${\phi }_2<2\pi .$ For ${\eta }_j={\displaystyle \underset{t\in \partial \Phi (\Omega (z_j,{\delta }_j))}{min}}\left|t-w_j\right|>0$ and $\eta :=min\left\{{\eta }_j,j=1,2\right\}$ let us set the following:

$$\begin{array}{cc}\hfill {\Delta }_j\left({\eta }_j\right)& :=\left\{t:\left|t-w_j\right|\le {\eta }_j\right\}\subset \Phi (\Omega (z_j,{\delta }_j)),\hfill \\ \hfill \Delta \left(\eta \right)& :={\displaystyle \bigcup _{j=1}^2}{\Delta }_j\left(\eta \right),{\widehat{\Delta }}_j:=\Delta \setminus \Delta \left({\eta }_j\right),\widehat{\Delta }\left(\eta \right):=\Delta \setminus \Delta \left(\eta \right),\hfill \\ \hfill {\Delta }_1^{\prime }\left(\rho \right)& :=\left\{t=R{e}^{i\theta }:R\ge \rho >1,2\pi -{\displaystyle \frac{{\phi }_1+{\phi }_2}{2}}\le \theta <2\pi +{\displaystyle \frac{{\phi }_1+{\phi }_2}{2}}\right\},{\Delta }_1^{\prime }:={\Delta }_1^{\prime }\left(1\right),\hfill \\ \hfill {\Delta }_2^{\prime }\left(\rho \right)& :=\left\{t=R{e}^{i\theta }:R\ge \rho >1,{\displaystyle \frac{{\phi }_1+{\phi }_2}{2}}\le \theta <2\pi -{\displaystyle \frac{{\phi }_1+{\phi }_2}{2}}\right\},{\Delta }_2^{\prime }:={\Delta }_2^{\prime }\left(1\right),\hfill \\ \hfill {\Omega }_R^j& :=\Psi \left({\Delta }_j^{\prime }\left(R\right)\right),L_R^j:=L_R\cap {\overline{\Omega }}_R^j,R\ge 1,\Omega :=\bigcup _{j=1}^2{\Omega }^j.\hfill \end{array}$$

3. Main Results

We begin by setting up the notational framework required for both the formulation of the main results and the subsequent analysis.

$$\begin{array}{ccc}\hfill \tilde{\lambda }& =& \left\{\begin{array}{cc}max\{1,\lambda \}+\epsilon ,& 0<\lambda <2,\\ 2,& \lambda =2;\end{array}\right.\hfill \\ \hfill {\tilde{\lambda }}_j& =& max\{1,{\lambda }_j\}+\epsilon ,0<{\lambda }_j<2,{\lambda }^{*}:=max\left\{{\tilde{\lambda }}_1,{\tilde{\lambda }}_2\right\},\forall \epsilon >0;\hfill \\ \hfill \gamma & :& =\left\{\begin{array}{cc}{\gamma }_1,& \mathrm{if}0<\lambda <2,\\ {\gamma }_2,& \mathrm{if}\lambda =2;\end{array}\right.{\tilde{\gamma }}_i:=max\left\{0;{\gamma }_i\right\},i=1,2;\hfill \\ \hfill \widehat{\gamma }& :& =\left\{\begin{array}{cc}{\tilde{\gamma }}_1,& \mathrm{if}0<\lambda <2,\\ {\tilde{\gamma }}_2,& \mathrm{if}\lambda =2;\end{array}\right.\hfill \\ \hfill {\gamma }^{*}& :& =max\left\{0,{\gamma }_1,{\gamma }_2\right\};\tilde{\gamma }:={\gamma }_j,\mathrm{for}z\in {\Omega }_R^j,j=1,2;{\tilde{\gamma }}^{*}:=max\left\{0;\tilde{\gamma }\right\}.\hfill \end{array}$$

(11)

In all theorems and corollaries throughout this section, the weight function $h\left(z\right)$ is defined by (1) for the case $l=2$. Also we assume that $n\in \mathbb{N}$ and $n\ge m$, where $m\in \mathbb{N}$ is specified individually in each result. Now let us start formulating the new results.

Theorem 1.

Let $p\ge 1$ and $G\in P{C}_{\theta }(1;{\lambda }_1,2)$ for some $0<{\lambda }_1<2$. Suppose the weight function h is defined by (1) with $\gamma >-2$. Then, for every polynomial $P_n\in {\wp }_n$, all $m=0,1,2,\dots ,n$ and $z\in {\Omega }_{R_1}$, the following estimate holds:

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

(12)

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

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

and the parameters $\tilde{\gamma }$ and $\tilde{\lambda }$ are defined in (11).

As Theorem 1 shows, the estimation on the right-hand side for $J_{n,p}^1\left(m\right)$ explicitly expresses the dependence of the growth in the unbounded region’s (${\Omega }_{R_1}$) modulus of the polynomial and its m-th derivatives on the behavior of the boundary angles and the properties of the weight function. Thus, the dependence of the growth of the polynomial’s modulus and its m-th derivatives on the geometry of the given region and weight is formulated.

As a consequence, we can consider cases in which the region has either zero or non-zero angles at both points.

Corollary 2.

Let $p\ge 1$ and $G\in P{C}_{\theta }(1;{\lambda }_1,{\lambda }_2)$ for some parameters $0<{\lambda }_1,{\lambda }_2<2$. Suppose the weight function h is defined by (1) with $\gamma >-2$. Then, for every polynomial $P_n\in {\wp }_n$, all $m=0,1,2,\dots ,n$ and $z\in {\Omega }_R^j$, we have

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

(13)

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

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

The sharpness of the estimations (12) for ${\lambda }_1=1$ and (13) for ${\lambda }_j=1,$ $j=1,2,$ and $m=0$ can be discussed by comparing them with the following:

Remark 1

([25] (Th.17)). For any $n\in \mathbb{N}$ there exists a polynomial $P_n^{*}\in {\wp }_n,$ a region $G_1^{*}\subset \mathbb{C}$, and a compact subset $F^{*}⋐\Omega \setminus {\overline{G}}_1^{*}$ such that

$$\left|P_n^{*}\left(z\right)\right|\ge c_3{\displaystyle \frac{\sqrt{n}}{d(z,L)}}{∥P_n^{*}∥}_{A_2\left(G_1^{*}\right)}{\left|\Phi \left(z\right)\right|}^{n+1},\mathit{for}\mathit{all}z\in F,$$

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

Corollary 3.

Let $p\ge 1$ and $G\in P{C}_{\theta }(1;2,2)$. Then, for every polynomial $P_n\in {\wp }_n$, all $m=0,1,2,\dots n$, and $z\in {\Omega }_R^j$, we have

$$\left|P_n^{\left(m\right)}\left(z\right)\right|\le c_4{\displaystyle \frac{\left|{\Phi }^{n-m+1}\left(z\right)\right|}{d(z,L)}}{J}_{n,p}^3\left(m\right){∥P_n∥}_p,$$

(14)

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

$$J_{n,p}^3\left(m\right):=\left\{\begin{array}{cc}{n}^{2\left({\displaystyle \frac{{\tilde{\gamma }}^{*}+2}{p}}+m-1\right)},& 1\le p<{\displaystyle \frac{2({\tilde{\gamma }}^{*}+2)+1}{\tilde{\lambda }+1}},\\ n^{2m+1-{\displaystyle \frac{1}{p}}}{\left(\ln n\right)}^{1-{\displaystyle \frac{1}{p}}},& p={\displaystyle \frac{2({\tilde{\gamma }}^{*}+2)+1}{\tilde{\lambda }+1}},\\ n^{2m+1-{\displaystyle \frac{1}{p}}},& p>{\displaystyle \frac{2({\tilde{\gamma }}^{*}+2)+1}{\tilde{\lambda }+1}}.\end{array}\right.$$

Remark 2.

Note that in [9] (Th. 5, 6), results similar to those in Theorem 1 and Corollaries 2 and 3 were obtained for $p>1,$ $m=1$, as well as those using a recurrence formula. The results obtained in Theorem 1 and Corollaries 2 and 3, using a different method without a recurrence formula, extend the corresponding results in [9] (Th. 5, 6) to the cases $p\ge 1$ and $m\ge 1$. Furthermore, in [9] (Th. 5, 6), the right-hand side contains the quantity ${\left|\Phi \left(z\right)\right|}^{2(n+1)}$, whereas in the results obtained here, the right-hand side contains ${\left|\Phi \left(z\right)\right|}^{n-m+1},m=1,2,\dots ,n$, $n=1,2,\dots ,$ which significantly improves the corresponding estimate in [9].

Now, we can state the uniform estimate of $\left|P_n^{\left(m\right)}\left(z\right)\right|,$ $m\ge 0,$ for $z\in \overline{G}.$

Theorem 2.

Let $p>0$ and $G\in P{C}_{\theta }(1;\lambda ,2)$ for some parameter $0<\lambda <2$. Then, for every polynomial $P_n\in {\wp }_n$, all $m=0,1,2,\dots ,n$, and arbitrarily small $\epsilon >0$, the following estimate holds:

$${∥P_n^{\left(m\right)}∥}_{\infty }\le c_5{\mu }_{n,m}{∥P_n∥}_p,$$

(15)

where $c_5=c_5(G,\gamma ,\lambda ,p,m,\epsilon )>0$ is a constant independent of z and $n;$ the numbers $\tilde{\gamma }$ and $\widehat{\lambda }$ are defined as in (11) and

$${\mu }_{n,m}:=\left\{\begin{array}{cc}{n}^{\left({\displaystyle \frac{2+\widehat{\gamma }}{p}}+m\right)\tilde{\lambda }},& \mathit{if}(2+\widehat{\gamma })·\tilde{\lambda }>1,\\ n^{m\tilde{\lambda }+{\displaystyle \frac{1}{p}}}{\left(\ln n\right)}^{{\displaystyle \frac{1}{p}}},& \mathit{if}(2+\widehat{\gamma })·\tilde{\lambda }=1,\\ n^{m\tilde{\lambda }+{\displaystyle \frac{1}{p}}},& \mathit{if}(2+\widehat{\gamma })·\tilde{\lambda }<1,\end{array}\right.$$

and the parameters $\tilde{\gamma }$, $\widehat{\gamma }$ and $\tilde{\lambda }$ are defined in (11).

Similar to the above, Theorem 2 also demonstrates the dependence of the growth of the modulus of a polynomial and its m-th derivatives on the closure of a bounded region $\overline{G}$ on the behavior of the boundary angles and the properties of the weight function. Thus, the dependence of the growth of the modulus of a polynomial and its m-th derivatives on the geometry of a given region and the weight is also explicitly stated.

We can separately consider the cases where the L curve has the same type of angle at both points: exterior non-zero or interior zero angles. In this case, from Theorem 2, we obtain the following:

Corollary 4.

Under the conditions of Theorem 2, the relation (15) is satisfied for $G\in P{C}_{\theta }(1;{\lambda }_1,{\lambda }_2)$ for ${\mu }_{n,m}={\tilde{\mu }}_{n,m,1},$ where

$${\tilde{\mu }}_{n,m,1}=\left\{\begin{array}{cc}{n}^{\left({\displaystyle \frac{2+{\gamma }^{*}}{p}}+m\right){\lambda }^{*}},& \mathit{if}(2+{\gamma }^{*})·{\lambda }^{*}>1,\\ n^{m{\lambda }^{*}+{\displaystyle \frac{1}{p}}}{\left(\ln n\right)}^{{\displaystyle \frac{1}{p}}},& \mathit{if}(2+{\gamma }^{*})·{\lambda }^{*}=1,\\ n^{m{\lambda }^{*}+{\displaystyle \frac{1}{p}}},& \mathit{if}(2+{\gamma }^{*})·{\lambda }^{*}<1,\end{array}\right.$$

(16)

and ${\gamma }^{*}$ and ${\lambda }^{*}$ are defined as in (11).

Corollary 5.

Under the conditions of Theorem 2, the relation (15) is satisfied for $G\in P{C}_{\theta }(1;2,2)$ if we take

$${\tilde{\mu }}_{n,m,2}=n^{2\left({\displaystyle \frac{2+{\gamma }^{*}}{p}}+m\right)},$$

(17)

and ${\gamma }^{*}$ is defined as in (11).

The following fact demonstrates the accuracy of the estimates (15)–(17) in some special cases.

Remark 3

([25] (Th.17)).

1. 

For any $n\in \mathbb{N}$ there exists a polynomial $Q_n\in {\wp }_n$ such that the inequality

$${∥Q_n∥}_{\infty }\ge c_{6}n{∥Q_n∥}_{A_2\left(B\right)}$$

is true for the unit disk B and the weight function $h\left(z\right)\equiv 1$.

2. 

For any $n\in \mathbb{N}$ there exists a polynomial $R_n\in {\wp }_n$ such that the inequality

$${∥R_n^{\prime }∥}_{\infty }\ge c_7{n}^2{∥R_n∥}_{A_2\left(B\right)}$$

is true for the unit disk B and the weight function $h\left(z\right)\equiv 1$.

According to (4) (applied to the polynomial $Q_{n-m}\left(z\right):=P_n^{\left(m\right)}\left(z\right)$), the estimate (15) and their corollaries ((16) and (17)) are also valid for $z\in {\overline{G}}_R$ with a different constant. Therefore, by combining the corresponding estimates (15)–(17) (for the $z\in {\overline{G}}_R$) with the estimates (12) and (13), we obtain an estimate for the growth of $|P_n^{\left(m\right)}\left(z\right)|$ in the whole complex plane.

Now, let us present the corresponding statements. When Theorem 1 is combined with Theorem 2, we find the following:

Theorem 3.

Let $p\ge 1$ and $G\in P{C}_{\theta }(1;\lambda ,2)$ for some parameter $0<\lambda <2$. Then, for every polynomial $P_n\in {\wp }_n$, all $m=0,1,2,\dots ,n$, and arbitrarily small $\epsilon >0$, the following estimate holds:

$${∥P_n^{\left(m\right)}∥}_{\infty }\le c_8{∥P_n∥}_p\left\{\begin{array}{cc}{\displaystyle \frac{\left|{\Phi }^{n-m+1}\left(z\right)\right|}{d(z,L)}}{J}_{n,p}^1\left(m\right),& z\in {\Omega }_{1+{\displaystyle \frac{1}{n}}},\\ {\mu }_{n,m},& z\in {\overline{G}}_{1+{\displaystyle \frac{1}{n}}},\end{array}\right.$$

where $c_8=c_8(G,\gamma ,\lambda ,p,m,\epsilon )>0$ is a constant independent of z and $n;$ $J_{n,p}^1\left(m\right)$ and ${\mu }_{n,m}$ are defined as in (12) and (15), respectively.

By combining Corollaries 2 and 3 with Corollaries 4 and 5, respectively, we obtain the following:

Corollary 6.

Let $p\ge 1$ and $G\in P{C}_{\theta }(1;{\lambda }_1,{\lambda }_2)$ for some parameters $0<{\lambda }_1,{\lambda }_2<2$. Then, for every polynomial $P_n\in {\wp }_n$, all $m=1,2,\dots ,n$, and arbitrarily small $\epsilon >0$, the following estimate holds:

$${∥P_n^{\left(m\right)}∥}_{\infty }\le c_9{∥P_n∥}_p\left\{\begin{array}{cc}{\displaystyle \frac{\left|{\Phi }^{n-m+1}\left(z\right)\right|}{d(z,L)}}{J}_{n,p}^2\left(m\right),& z\in {\Omega }_{1+{\displaystyle \frac{1}{n}}},\\ {\tilde{\mu }}_{n,m,1},& z\in {\overline{G}}_{1+{\displaystyle \frac{1}{n}}},\end{array}\right.$$

where $c_9=c_9(G,\gamma ,\lambda ,p,m,\epsilon )>0$ is a constant independent of z and $n;$ $J_{n,p}^2\left(m\right)$ and ${\tilde{\mu }}_{n,m,1}$ are defined as in (13) and (16), respectively.

Corollary 7.

Let $p\ge 1$ and $G\in P{C}_{\theta }(1;2,2).$ Then, for every polynomial $P_n\in {\wp }_n$, all $m=0,1,2,\dots n$, and $z\in {\Omega }_R^j$, we have

$${∥P_n^{\left(m\right)}∥}_{\infty }\le c_{10}{∥P_n∥}_p\left\{\begin{array}{cc}{\displaystyle \frac{\left|{\Phi }^{n-m+1}\left(z\right)\right|}{d(z,L)}}{J}_{n,p}^3\left(m\right),& z\in {\Omega }_{1+{\displaystyle \frac{1}{n}}},\\ {\tilde{\mu }}_{n,m,2},& z\in {\overline{G}}_{1+{\displaystyle \frac{1}{n}}},\end{array}\right.$$

where $c_{10}=c_{10}(G,\gamma ,\lambda ,p,m,\epsilon )>0$ is a constant independent of z and $n;$ $J_{n,p}^3\left(m\right)$ and ${\tilde{\mu }}_{n,m,2}$ are defined as in (14) and (17), respectively.

Thus, in Theorem 3 and Corollaries 6 and 7, we provided precise estimates for the growth of the modulus of the polynomial itself and its m-th derivatives for a bounded and unbounded region with a piecewise smooth curve. The influence of the boundary angle and the degree of the “zero” or “pole” of the weight function on this growth is also indicated. This result extends and significantly refines the corresponding result in [9]. The method used allows us to study similar problems in various regions and spaces without resorting to a recurring formula.

4. Some Auxiliary Results

Lemma 1

([12]). Let $L=\partial G$ be a K-quasiconformal curve, $z_1\in L,$ $z_2,z_3\in \Omega \cap \{z:\left|z-z_1\right|⪯d(z_1,L_{R_0})\};$ $w_j=\Phi \left(z_j\right),$ $j=1,2,3.$ 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 $\left|z_1-z_2\right|\asymp \left|z_1-z_3\right|$ and $\left|w_1-w_2\right|\asymp \left|w_1-w_3\right|$ are also equivalent.

(b) 

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

$${\left|{\displaystyle \frac{w_1-w_3}{w_1-w_2}}\right|}^{K^{-2}}⪯\left|{\displaystyle \frac{z_1-z_3}{z_1-z_2}}\right|⪯{\left|{\displaystyle \frac{w_1-w_3}{w_1-w_2}}\right|}^{K^2},$$

where $R_0>1$ is a constant, depending on $G.$

Corollary 8.

Under the assumptions of Lemma 1, if $z_3\in L_{R_0}$, then

$${\left|w_1-w_2\right|}^{K^2}⪯\left|z_1-z_2\right|⪯{\left|w_1-w_2\right|}^{K^{-2}}.$$

Corollary 9.

If $L\in C_{\theta },$ then

$${\left|w_1-w_2\right|}^{1+\epsilon }⪯\left|z_1-z_2\right|⪯{\left|w_1-w_2\right|}^{1-\epsilon },$$

for all $\epsilon >0.$

We will also use the following estimate for $\left|{\Psi }^{\prime }\right|$ (see, for example, [26] (Th.2.8)):

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

(18)

The following lemma is a consequence of the results given in [6,27].

Lemma 2.

Let $G\in P{C}_{\theta }(1;\lambda ,\dots ,{\lambda }_l),0<{\lambda }_j<2,j=1,2,\dots ,l$. Then

(i) 

For any $w\in {\Delta }_j\left(\eta \right)$, $|w-w_j{|}^{{\lambda }_j+\epsilon }⪯|\Psi \left(w\right)-\Psi \left(w_j\right)| ⪯ |w-w_j{|}^{{\lambda }_j-\epsilon }$, $|w-w_j{|}^{{\lambda }_j-1+\epsilon } ⪯ |{\Psi }^{\prime }\left(w\right)| ⪯ |w-w_j{|}^{{\lambda }_j-1-\epsilon }$;

(ii) 

For any $w\in \widehat{\Delta }\left(\eta \right)$, ${\left(\right|w|-1)}^{1+\epsilon }{⪯d(\Psi \left(w\right),L)|⪯(\left|w\right|-1)}^{1-\epsilon }$, ${\left(\left|w\right|-1\right)}^{\epsilon }⪯\left|{\Psi }^{\prime }\left(w\right)\right|⪯{\left(\left|w\right|-1\right)}^{-\epsilon }.$

Let ${\left\{z_j\right\}}_{j=1}^l$ be a fixed system of distinct points on curve L ordered in the positive direction and the weight function $h\left(z\right)$ be defined as in (1).

Lemma 3

([12] (Lemma 3.5)). Let $L=\partial G$ be a K-quasiconformal curve, $withR=1+{\displaystyle \frac{c}{n}}.$ Then, for any fixed $\epsilon \in (0,1),$ there exist a level curve $L_{1+\epsilon (R-1)}$ such that the following holds for anypolynomial $P_n\left(z\right)\in {\wp }_n$, $n\in \mathbb{N}:$

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

(19)

Lemma 4

([12] (Lemma 3.6)). Let L $=\partial G$ be a K-quasiconformal curve; let $h\left(z\right)$ be defined as in (1). Then, for arbitrary $P_n\left(z\right)\in {\wp }_n,$ any $R>1$, and $n=1,2,\dots ,$ we have

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

(20)

where $\tilde{R}=1+c(R-1)$ and c is independent of n and $R.$

5. Proofs

Proof of Theorem 1.

The proof of Theorem 1 will be carried out by separating the cases $p\ge 2$ and $1<p<2$. Let $G\in C_{\theta }(1;{\lambda }_1,2)$ with $0<{\lambda }_1<2$, $R=1+{\displaystyle \frac{{\epsilon }_0}{n}},R_1:=1+{\displaystyle \frac{R-1}{3}}$, and $R_2:=1+{\displaystyle \frac{2(R-1)}{3}}$. For $z\in \Omega$ and $0\le m<n$, let us define

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

Noting that $H_{n,m}\left(z\right)$ is analytic in the region $\Omega$ and continuous on its closure $\overline{\Omega }$ while also taking into account that $H_{n,m}\left(\infty \right)=0$, we can apply the Cauchy integral representation for unbounded regions to obtain

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

Taking the modulus of both parts and considering that $\left|{\Phi }^{n-m+1}\left(\zeta \right)\right|=R_1^{n-m+1}>1,$ for $\zeta \in L_{R_1}$, we obtain

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

Therefore,

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

(21)

Let us express the Cauchy m-th derivative formula for $P_n\left(\zeta \right)$ within the integral

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

and taking $\zeta \in L_{R_1}$ we find

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

(22)

It is denote by

$$\underset{L_{R_1}}{\int }{\displaystyle \frac{\left|d\zeta \right|}{{\left|t-\zeta \right|}^{m+1}}}=:J_1\left(L_{R_1}\right);\underset{L_{R_2}}{\int }\left|P_n\left(t\right)\right|\left|dt\right|=:J_2\left(L_{R_2}\right),$$

(23)

and we estimate these integrals separately.

(A)

Estimation of $J_1\left(L_{R_1}\right).$ By replacing the variables $\tau =\Phi \left(\zeta \right),\zeta \in L_{R_1},w=\Phi \left(t\right),t\in L_{R_2}$ and taking into account (18), we have

$$J_1\left(L_{R_1}\right)=\underset{\left|\tau \right|=R_1}{\int }{\displaystyle \frac{\left|{\Psi }^{\prime }\left(\tau \right)\right|\left|d\tau \right|}{{\left|\Psi \left(\tau \right)-\Psi \left(w\right)\right|}^{m+1}}}.$$

(24)

To estimate the integral, for $\rho \ge 1$, we introduce the following notations:

$$\begin{array}{cc}\hfill E_{\rho ,j}^1& :=\{\tau :\tau \in \Phi \left(L_{\rho }^j\right),\left|\tau -w_j\right|<c_j(\rho -1)\},E_{\rho ,j}^{11}:=\left\{\tau \in E_{\rho ,j}^1:|\tau -w_j| \ge |\tau -w|\right\},\hfill \\ \hfill E_{\rho ,j}^{12}& :=E_{\rho ,j}^1\setminus E_{\rho ,j}^{11};\hfill \\ \hfill E_{\rho ,j}^2& :=\{\tau :\tau \in \Phi \left(L_{\rho }^j\right),c_j(\rho -1)\le \left|\tau -w_j\right|<\eta \},E_{\rho ,j}^{21}:=\left\{\tau \in E_{\rho ,j}^2:|\tau -w_j| \ge |\tau -w|\right\},\hfill \\ \hfill E_{\rho ,j}^{22}& :=E_{\rho ,j}^2\setminus E_{\rho ,j}^{21};\hfill \\ \hfill E_{\rho ,j}^3& :=\{\tau :\tau \in \Phi \left(L_{\rho }\right)\}\setminus \left(E_{\rho ,j}^1\cup E_{\rho ,j}^2\right),j=1,2,\hfill \end{array}$$

and for $\rho =R_1$, the estimates $J_1\left(L_{R_1}\right)$ can be determined separately for the specified sets.

$$J_1\left(L_{R_1}\right)=\sum _{i,j=1}^2{J}_1\left(E_{R_1,j}^i\right)+J_1\left(E_{R_1,j}^3\right).$$

(25)

By applying Lemmas 1, 2, and (18), we obtain the following estimates:

(I)

 

$$J_1\left(E_{R_1,1}^1\right)=\underset{E_{R_1,1}^1}{\int }{\displaystyle \frac{\left|{\Psi }^{\prime }\left(\tau \right)\right|\left|d\tau \right|}{{\left|\Psi \left(\tau \right)-\Psi \left(w\right)\right|}^{m+1}}}\asymp \underset{E_{R_1,1}^1}{\int }{\displaystyle \frac{{\left|\tau -w_1\right|}^{{\lambda }_1-1-\epsilon }}{{\left|\tau -w\right|}^{(m+1)({\lambda }_1+\epsilon )}}}\left|d\tau \right|.$$

(a)

For $1\le {\lambda }_1<2,$

$$\begin{array}{cc}\hfill J_1\left(E_{R_1,1}^1\right)& \asymp \underset{E_{R_1,1}^1}{\int }{\displaystyle \frac{{\left|\tau -w_1\right|}^{{\lambda }_1-1-\epsilon }}{{\left|\tau -w\right|}^{(m+1)({\lambda }_1+\epsilon )}}}\left|d\tau \right|⪯{\left({\displaystyle \frac{1}{n}}\right)}^{{\lambda }_1-1-\epsilon }\underset{E_{R_1,1}^1}{\int }{\displaystyle \frac{\left|d\tau \right|}{{\left|\tau -w\right|}^{(m+1)({\lambda }_1+\epsilon )}}}\hfill \\ \hfill & ⪯{\left({\displaystyle \frac{1}{n}}\right)}^{{\lambda }_1-1-\epsilon }{n}^{(m+1)({\lambda }_1+\epsilon )-1}=n^{m{\lambda }_1+\epsilon }.\hfill \end{array}$$

(b)

For $0<{\lambda }_1<1,$

$$\begin{array}{cc}\hfill J_1\left(E_{R_1,1}^1\right)& \asymp \underset{E_{R_1,1}^1}{\int }{\displaystyle \frac{{\left|\tau -w_1\right|}^{{\lambda }_1-1-\epsilon }}{{\left|\tau -w\right|}^{(m+1)({\lambda }_1+\epsilon )}}}\left|d\tau \right|=\underset{E_{R_1,1}^1}{\int }{\displaystyle \frac{\left|d\tau \right|}{{\left|\tau -w\right|}^{1-{\lambda }_1+(m+1)({\lambda }_1+\epsilon )}}}\hfill \\ \hfill & ⪯n^{1-{\lambda }_1+(m+1)({\lambda }_1+\epsilon )}mes\left(E_{R_1,1}^1\right)⪯n^{1-{\lambda }_1+(m+1)({\lambda }_1+\epsilon )-1}=n^{m{\lambda }_1+\epsilon }.\hfill \end{array}$$

(II)

 

$$\begin{array}{cc}\hfill J_1\left(E_{R_1,1}^2\right)& =\underset{E_{R_1,1}^2}{\int }{\displaystyle \frac{\left|{\Psi }^{\prime }\left(\tau \right)\right|\left|d\tau \right|}{{\left|\Psi \left(\tau \right)-\Psi \left(w\right)\right|}^{m+1}}}\asymp \underset{E_{R_1,1}^2}{\int }{\displaystyle \frac{{\left|\tau -w_1\right|}^{{\lambda }_1-1-\epsilon }}{{\left|\tau -w\right|}^{(m+1)({\lambda }_1+\epsilon )}}}\left|d\tau \right|\hfill \\ \hfill & =\underset{E_{R_1,1}^{21}}{\int }{\displaystyle \frac{\left|{\Psi }^{\prime }\left(\tau \right)\right|\left|d\tau \right|}{{\left|\Psi \left(\tau \right)-\Psi \left(w\right)\right|}^{m+1}}}\left|d\tau \right|+\underset{E_{R_1,1}^{22}}{\int }{\displaystyle \frac{{\left|\tau -w_1\right|}^{{\lambda }_1-1-\epsilon }}{{\left|\tau -w\right|}^{(m+1)({\lambda }_1+\epsilon )}}}\left|d\tau \right|.\hfill \end{array}$$

(a)

For $1\le {\lambda }_1<2,$

$$\begin{array}{cc}\hfill J_1\left(E_{R_1,1}^{21}\right)& =\underset{E_{R_1,1}^{21}}{\int }{\displaystyle \frac{\left|{\Psi }^{\prime }\left(\tau \right)\right|\left|d\tau \right|}{{\left|\Psi \left(\tau \right)-\Psi \left(w\right)\right|}^{m+1}}}\left|d\tau \right|=\underset{\Psi \left(E_{R_1,1}^{21}\right)}{\int }{\displaystyle \frac{\left|d\zeta \right|}{{\left|\zeta -t\right|}^{m+1}}}\hfill \\ \hfill & ⪯\underset{d(\zeta ,L_{R_2})}{\overset{c}{\int }}{\displaystyle \frac{\left|d\zeta \right|}{{\left|\zeta -t\right|}^{m+1}}}⪯{\displaystyle \frac{1}{d^m(\zeta ,L_{R_2})}}⪯n^{m{\lambda }_1+\epsilon };\hfill \end{array}$$

$$J_1\left(E_{R_1,1}^{22}\right)=\underset{E_{R_1,1}^{22}}{\int }{\displaystyle \frac{{\left|\tau -w_1\right|}^{{\lambda }_1-1-\epsilon }}{{\left|\tau -w\right|}^{(m+1)({\lambda }_1+\epsilon )}}}\left|d\tau \right|⪯\underset{E_{R_1,1}^{22}}{\int }{\displaystyle \frac{\left|d\tau \right|}{{\left|\tau -w\right|}^{(m+1){\lambda }_1-{\lambda }_1+1+\epsilon }}}⪯n^{m{\lambda }_1+\epsilon }.$$

(b)

For $0<{\lambda }_1<1,$

$$\begin{array}{cc}\hfill J_1\left(E_{R_1,1}^2\right)& =\underset{E_{R_1,1}^2}{\int }{\displaystyle \frac{\left|{\Psi }^{\prime }\left(\tau \right)\right|\left|d\tau \right|}{{\left|\Psi \left(\tau \right)-\Psi \left(w\right)\right|}^{m+1}}}\asymp \underset{E_{R_1,1}^2}{\int }{\displaystyle \frac{{\left|\tau -w_1\right|}^{{\lambda }_1-1-\epsilon }}{{\left|\tau -w\right|}^{(m+1)({\lambda }_1+\epsilon )}}}\left|d\tau \right|\hfill \\ \hfill & =\underset{E_{R_1,1}^{21}}{\int }{\displaystyle \frac{\left|d\tau \right|}{{\left|\tau -w\right|}^{1-{\lambda }_1+(m+1)({\lambda }_1+\epsilon )}}}+\underset{E_{R_1,1}^{22}}{\int }{\displaystyle \frac{\left|d\tau \right|}{{\left|\tau -w_1\right|}^{1-{\lambda }_1+(m+1)({\lambda }_1+\epsilon )}}}\hfill \\ \hfill & ⪯n^{m{\lambda }_1+\epsilon }+n^{m{\lambda }_1+\epsilon }⪯n^{m{\lambda }_1+\epsilon }.\hfill \end{array}$$

By applying (18) and Corollary 9, we obtain the following:

$$J_1\left(E_{R_1,2}^k\right)\asymp \underset{E_{R_1,2}^1}{\int }{\displaystyle \frac{d(\Psi (\tau ),L)}{{\left|\Psi \left(\tau \right)-\Psi \left(w\right)\right|}^{m+1}(\left|\tau \right|-1)}}\left|d\tau \right|⪯n\underset{E_{R_1,2}^k}{\int }{\displaystyle \frac{\left|d\tau \right|}{{\left|\tau -w\right|}^{2m}}}⪯n^{2m},k=1,2;$$

$$J_1\left(E_{R_1,j}^3\right)\asymp \underset{E_{R_1,j}^3}{\int }{\displaystyle \frac{d(\Psi (\tau ),L)}{{\left|\Psi \left(\tau \right)-\Psi \left(w\right)\right|}^{m+1}(\left|\tau \right|-1)}}\left|d\tau \right|⪯n\underset{E_{R_1,2}^1}{\int }{\displaystyle \frac{\left|d\tau \right|}{{\left|\tau -w\right|}^{m(1+\epsilon )}}}⪯n^{2(1+\epsilon )},j=1,2,$$

and by combining the estimates for all $0<{\lambda }_1\le 2,$ we obtain

$$J_1\left(L_{R_1}\right)⪯n^{m\tilde{\lambda }}.$$

(26)

(B)

Estimation of $J_2\left(L_{R_2}\right)$.

Let us first assume that $p>1.$ By replacing the variable $t=\Psi \left(w\right)$, multiplying the integrand numerator and denominator by $h^{\frac{1}{p}}\left(t\right)$, and using the Hölder inequality, we have

$$\begin{array}{cc}\hfill J_2\left(L_{R_2}\right)& =\underset{L_{R_2}}{\int }\left|P_n\left(t\right)\right|\left|dt\right|=\underset{\left|w\right|=R_2}{\int }{\displaystyle \frac{h^{\frac{1}{p}}(\Psi \left(w\right))\left|P_n(\Psi \left(w\right))\right|{\left|{\Psi }^{\prime }\left(w\right)\right|}^{1-\frac{2}{p}}{\left|{\Psi }^{\prime }\left(w\right)\right|}^{\frac{2}{p}}}{h^{\frac{1}{p}}(\Psi \left(w\right))}}\left|dw\right|\hfill \\ \hfill & \le \underset{\left|w\right|=R_2}{\int }{\left(h(\Psi \left(w\right)){\left|P_n(\Psi \left(w\right))\right|}^p{\left|{\Psi }^{\prime }\left(w\right)\right|}^2\left|dw\right|\right)}^{{\displaystyle \frac{1}{p}}}\times {\left(\underset{\left|w\right|=R_2}{\int }{\displaystyle \frac{{\left|{\Psi }^{\prime }\left(w\right)\right|}^{q(1-\frac{2}{p})}}{h^{\frac{q}{p}}(\Psi \left(w\right))}}\right)}^{{\displaystyle \frac{1}{q}}}\hfill \\ \hfill & =:J_{21}\left(L_{R_2}\right)\times J_{22}\left(L_{R_2}\right),{\displaystyle \frac{1}{p}}+{\displaystyle \frac{1}{q}}=1.\hfill \end{array}$$

(27)

According to Lemma 3, we have

$$J_{21}\left(L_{R_2}\right)⪯n^{\frac{1}{p}}{∥P_n∥}_p.$$

(28)

Now, we estimate the following integral:

$${\left[J_{22}\left(L_{R_2}\right)\right]}^q:=\underset{L_{R_2}}{\int }{\displaystyle \frac{{\left|{\Psi }^{\prime }\left(w\right)\right|}^{2-q}}{{\displaystyle \prod _{j=1}^2}{\left|\Psi \left(w\right)-\Psi \left(w_j\right)\right|}^{{\gamma }_j(q-1)}}}\left|dw\right|.$$

(29)

To estimate this integral, we set the following as

$$\begin{array}{cc}\hfill E_{R_2,j}^1:=& \{w:w\in \Phi \left(L_{R_2}^j\right),\left|w-w_j\right|<c_j(R_2-1)\};\hfill \\ \hfill E_{R_2,j}^2:=& \{w:w\in \Phi \left(L_{R_2}^j\right),c_j(R_2-1)\le \left|w-w_j\right|<\eta \};\hfill \\ \hfill E_{R_2,j}^3:=& \{w:w\in \Phi \left(L_{R_2}^j\right),\eta \le \left|w-w_j\right|<diam\overline{G}\},\hfill \end{array}$$

where $0<c_j<\eta$ is chosen so that $\{w:\left|w-w_j\right|<c_j(R_2-1)\}\cap \Delta \ne \varnothing$ and $\Phi \left(L_{R_2}\right)=\Phi \left({\displaystyle \bigcup _{j=1}^2}{L}_2^j\right)={\displaystyle \bigcup _{j=1}^2}\Phi \left(L_{R_2}^j\right)={\displaystyle \bigcup _{j=1}^2}{\displaystyle \bigcup _{i=1}^3}{E}_{R_2,i}^j$. Taking these notations into account, from (29), we obtain

$$\begin{array}{cc}\hfill {\left[J_{22}\left(L_{R_2}\right)\right]}^q& ⪯\sum _{j=1}^2\underset{\Phi \left(L_{R_2}^j\right)}{\int }{\displaystyle \frac{{\left|{\Psi }^{\prime }\left(w\right)\right|}^{2-q}}{{\displaystyle \prod _{j=1}^2}{\left|\Psi \left(w\right)-\Psi \left(w_j\right)\right|}^{{\gamma }_j(q-1)}}}\left|dw\right|\hfill \\ \hfill & \asymp \sum _{j=1}^2\underset{\Phi \left(L_{R_2}^j\right)}{\int }{\displaystyle \frac{{\left|{\Psi }^{\prime }\left(w\right)\right|}^{2-q}}{{\left|\Psi \left(w\right)-\Psi \left(w_j\right)\right|}^{{\gamma }_j(q-1)}}}\left|dw\right|=:\sum _{j=1}^2{B}_{n,2}\left(w_j\right),\hfill \end{array}$$

(30)

since the points $z_1$ and $z_2$ are isolated. Therefore, we need to estimate $B_{n,2}\left(w_j\right)$ for $j=1,2$.

$$B_{n,2}\left(w_j\right)=\sum _{i=1}^3\underset{E_{R_2,j}^i}{\int }{\displaystyle \frac{{\left|{\Psi }^{\prime }\left(w\right)\right|}^{2-q}}{{\left|\Psi \left(w\right)-\Psi \left(w_j\right)\right|}^{{\gamma }_j(q-1)}}}\left|dw\right|=\sum _{i=1}^3{B}_{n,2}^i\left(w_j\right).$$

(31)

Case 1.

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

(I)

For ${\gamma }_1,{\gamma }_2\ge 0$, we have

$$\begin{array}{cc}\hfill B_{n,2}^1\left(w_1\right)& =\underset{E_{R_2,1}^1}{\int }{\displaystyle \frac{{\left|{\Psi }^{\prime }\left(w\right)\right|}^{2-q}}{{\left|\Psi \left(w\right)-\Psi \left(w_1\right)\right|}^{{\gamma }_1(q-1)}}}\left|dw\right|⪯\underset{E_{R_2,1}^1}{\int }{\displaystyle \frac{{\left|w-w_1\right|}^{({\lambda }_1-1-\epsilon )(2-q)}}{{\left|w-w_1\right|}^{{\gamma }_1(q-1)({\lambda }_1+\epsilon )}}}\left|dw\right|\hfill \\ \hfill & ⪯n^{{\gamma }_1(q-1)({\lambda }_1+\epsilon )-({\lambda }_1-1-\epsilon )(2-q)}\underset{E_{R_2,1}^1}{\int }\left|dw\right|⪯n^{\left[{\gamma }_1(q-1)-(2-q)\right]{\lambda }_1+(2-q)+\epsilon }mes\left(E_{R_2,1}^1\right)\hfill \\ \hfill & ⪯n^{\left[{\gamma }_1(q-1)-(2-q)\right]{\lambda }_1+(1-q)+\epsilon };\hfill \end{array}$$

$$\begin{array}{cc}\hfill B_{n,2}^2\left(w_1\right)& =\underset{E_{R_2,1}^2}{\int }{\displaystyle \frac{{\left|{\Psi }^{\prime }\left(w\right)\right|}^{2-q}}{{\left|\Psi \left(w\right)-\Psi \left(w_1\right)\right|}^{{\gamma }_1(q-1)}}}\left|dw\right|\asymp \underset{E_{R_2,1}^2}{\int }{\left({\displaystyle \frac{d(\Psi (w),L)}{\left|w\right|-1}}\right)}^{2-q}{\displaystyle \frac{\left|dw\right|}{{\left|\Psi \left(w\right)-\Psi \left(w_1\right)\right|}^{{\gamma }_1(q-1)}}}\hfill \\ \hfill & ⪯n^{2-q}\underset{E_{R_2,1}^2}{\int }{\displaystyle \frac{\left|dw\right|}{{\left|w-w_1\right|}^{\left[{\gamma }_1(q-1)-(2-q)\right]({\lambda }_1+\epsilon )}}}\hfill \\ \hfill & ⪯n^{2-q}\left\{\begin{array}{cc}{n}^{\left[{\gamma }_1(q-1)-(2-q)\right]{\lambda }_1-1+\epsilon },& \left[{\gamma }_1(q-1)-(2-q)\right]{\lambda }_1+\epsilon >1,\\ \ln n,& \left[{\gamma }_1(q-1)-(2-q)\right]{\lambda }_1+\epsilon =1,\\ 1,& \left[{\gamma }_1(q-1)-(2-q)\right]{\lambda }_1+\epsilon <1;\end{array}\right.\hfill \end{array}$$

$$\begin{array}{cc}\hfill B_{n,2}^1\left(w_2\right)& =\underset{E_{R_2,2}^1}{\int }{\displaystyle \frac{{\left|{\Psi }^{\prime }\left(w\right)\right|}^{2-q}}{{\left|\Psi \left(w\right)-\Psi \left(w_2\right)\right|}^{{\gamma }_2(q-1)}}}\left|dw\right|\asymp \underset{E_{R_2,2}^1}{\int }{\left({\displaystyle \frac{d(\Psi (w),L)}{\left|w\right|-1}}\right)}^{2-q}{\displaystyle \frac{\left|dw\right|}{{\left|\Psi \left(w\right)-\Psi \left(w_2\right)\right|}^{{\gamma }_2(q-1)}}}\hfill \\ \hfill & ⪯n^{2-q}\underset{E_{R_2,2}^1}{\int }{\displaystyle \frac{\left|dw\right|}{{\left|\Psi \left(w\right)-\Psi \left(w_2\right)\right|}^{{\gamma }_2(q-1)-(2-q)}}}⪯n^{2-q}\underset{E_{R_2,2}^1}{\int }{\displaystyle \frac{\left|dw\right|}{{\left|w-w_2\right|}^{2\left[{\gamma }_2(q-1)-(2-q)\right]}}}\hfill \\ \hfill & ⪯n^{2-q+2\left[{\gamma }_2(q-1)-(2-q)\right]}mes\left(E_{R_2,2}^1\right)⪯n^{2{\gamma }_2(q-1)-(2-q)-1};\hfill \end{array}$$

(32)

$$\begin{array}{cc}\hfill B_{n,2}^2\left(w_2\right)& ⪯n^{2-q}\underset{E_{R_2,2}^2}{\int }{\displaystyle \frac{\left|dw\right|}{{\left|\Psi \left(w\right)-\Psi \left(w_2\right)\right|}^{({\gamma }_2+1)(q-1)-1}}}⪯n^{2-q}\underset{E_{R_2,2}^2}{\int }{\displaystyle \frac{\left|dw\right|}{{\left|w-w_2\right|}^{2\left[{\gamma }_2(q-1)-(2-q)\right]}}}\hfill \\ \hfill & ⪯\left\{\begin{array}{cc}{n}^{2{\gamma }_2(q-1)-(2-q)-1},& 2\left[{\gamma }_2(q-1)-(2-q)\right]>1,\\ n^{2-q}\ln n,& 2\left[{\gamma }_2(q-1)-(2-q)\right]=1,\\ n^{2-q},& 2\left[{\gamma }_2(q-1)-(2-q)\right]<1.\end{array}\right.\hfill \end{array}$$

(33)

(II)

Let us now use ${\gamma }_1,{\gamma }_2<0$. Since $({\lambda }_1-1)(2-q)+(-{\gamma }_1)(q-1){\lambda }_1+1={\lambda }_1(2-q)+(-{\gamma }_1)(q-1){\lambda }_1+1-(2-q)$ and ${\lambda }_1(2-q)+(-{\gamma }_1)(q-1){\lambda }_1>0$ for all $0<{\lambda }_1<2,$ $1<q\le 2$ and ${\gamma }_1<0,$ we obtain the following:

$$\begin{array}{cc}\hfill B_{n,2}^1\left(w_1\right)& ⪯\underset{E_{R_2,1}^1}{\int }{\displaystyle \frac{{\left|w-w_1\right|}^{({\lambda }_1-1-\epsilon )(2-q)}}{{\left|w-w_1\right|}^{{\gamma }_1(q-1)({\lambda }_1+\epsilon )}}}\left|dw\right|⪯{\left({\displaystyle \frac{1}{n}}\right)}^{({\lambda }_1-1)(2-q)+(-{\gamma }_1)(q-1){\lambda }_1+\epsilon }mes\left(E_{R_2,1}^1\right)\hfill \\ \hfill & ⪯n^{2-q};\hfill \end{array}$$

(34)

$$\begin{array}{cc}\hfill B_{n,2}^2\left(w_1\right)& ⪯\underset{E_{R_2,1}^2}{\int }{\displaystyle \frac{{\left|w-w_1\right|}^{({\lambda }_1-1-\epsilon )(2-q)}}{{\left|w-w_1\right|}^{{\gamma }_1(q-1)({\lambda }_1+\epsilon )}}}\left|dw\right|=\underset{E_{R_2,1}^2}{\int }{\left|w-w_1\right|}^{({\lambda }_1(2-q)+(-{\gamma }_1)(q-1){\lambda }_1-(2-q)+\epsilon }\left|dw\right|\hfill \\ \hfill & ⪯n^{2-q}\underset{E_{R_2,1}^2}{\int }\left|dw\right|⪯n^{2-q};\hfill \end{array}$$

(35)

$$\begin{array}{cc}\hfill B_{n,2}^1\left(w_2\right)+B_{n,2}^2\left(w_2\right)& =n^{2-q}\underset{E_{R_2,2}^1\cup E_{R_2,2}^2}{\int }{\left|\Psi \left(w\right)-\Psi \left(w_2\right)\right|}^{-({\gamma }_2+1)(q-1)+1}\left|dw\right|\hfill \\ \hfill & ⪯n^{2-q}mes(E_{R_2,2}^1\cup E_{R_2,2}^2)⪯n^{2-q}({\displaystyle \frac{1}{n}}+1)⪯n^{2-q}.\hfill \end{array}$$

(36)

For $w\in E_{R_2,1}^3$, $\left|\Psi \left(w\right)-\Psi \left(w_j\right)\right|⪰1$; then for $j=1,2$, we obtain the following:

$$B_{n,2}^3\left(w_j\right)⪯\underset{E_{R_2,j}^3}{\int }{\left({\displaystyle \frac{d(\Psi (w),L)}{\left|w\right|-1}}\right)}^{2-q}\left|dw\right|⪯\underset{E_{R_2,j}^3}{\int }{\left(\left|w\right|-1\right)}^{-\epsilon (2-q)}\left|dw\right|⪯n^{\epsilon }.$$

(37)

By combining estimates (27)–(37), in this case, we have that

$$J_2\left(L_{R_2}\right)⪯{∥P_n∥}_p\left\{\begin{array}{cc}{n}^{\left(\frac{\tilde{\gamma }+2}{p}-1\right)\tilde{\lambda }},& p<\frac{\tilde{\lambda }(\tilde{\gamma }+2)+1}{\tilde{\lambda }+1},\\ {\left(n\ln n\right)}^{1-\frac{1}{p}},& p=\frac{\tilde{\lambda }(\tilde{\gamma }+2)+1}{\tilde{\lambda }+1},\\ n^{1-\frac{1}{p}},& p>\frac{\tilde{\lambda }(\tilde{\gamma }+2)+1}{\tilde{\lambda }+1},\end{array}\right.$$

where $\tilde{\gamma }:=max\left\{0;\tilde{\gamma }\right\}.$ Therefore,

$$J_2\left(L_{R_2}\right)⪯{∥P_n∥}_p\left\{\begin{array}{ccc}{n}^{\left(\frac{\tilde{\gamma }+2}{p}-1\right)\tilde{\lambda }},& 2\le p<\frac{\tilde{\lambda }(\tilde{\gamma }+2)+1}{\tilde{\lambda }+1},& \tilde{\gamma }>1+\frac{1}{\tilde{\lambda }},\\ {\left(n\ln n\right)}^{1-\frac{1}{p}},& p=\frac{\tilde{\lambda }(\tilde{\gamma }+2)+1}{\tilde{\lambda }+1},& \tilde{\gamma }>1+\frac{1}{\tilde{\lambda }},\\ n^{1-\frac{1}{p}},& p>\frac{\tilde{\lambda }(\tilde{\gamma }+2)+1}{\tilde{\lambda }+1};& \tilde{\gamma }>1+\frac{1}{\tilde{\lambda }},\\ n^{1-\frac{1}{p}},& p\ge 2& 0\le \tilde{\gamma }\le 1+\frac{1}{\tilde{\lambda }},\end{array}\right.$$

(38)

for ${\gamma }_1,{\gamma }_2\ge 0$, and

$$J_2\left(L_{R_2}\right)⪯n^{1-\frac{1}{p}}{∥P_n∥}_p,$$

for $-2<{\gamma }_1,{\gamma }_2<0$. Therefore, by combining (22), (23), and (26)–(38) for case $p\ge 2,$ ${\gamma }_1>-2,$ ${\gamma }_2>-2,$ and for all $z\in {\Omega }_{R_1},$ we obtain

$$\left|P_n^{\left(m\right)}\left(z\right)\right|⪯\frac{\left|{\Phi }^{n-m+1}\left(z\right)\right|}{d(z,L_{R_1})}{∥P_n∥}_p\left\{\begin{array}{ccc}{n}^{\left(\frac{\tilde{\gamma }+2}{p}+m-1\right)\tilde{\lambda }},& 2\le p<\frac{\tilde{\lambda }(\tilde{\gamma }+2)+1}{\tilde{\lambda }+1},& \tilde{\gamma }>1+\frac{1}{\tilde{\lambda }},\\ n^{m\tilde{\lambda }+1-\frac{1}{p}}{\left(\ln n\right)}^{1-\frac{1}{p}},& p=\frac{\tilde{\lambda }(\tilde{\gamma }+2)+1}{\tilde{\lambda }+1},& \tilde{\gamma }>1+\frac{1}{\tilde{\lambda }},\\ n^{m\tilde{\lambda }+1-\frac{1}{p}},& p>\frac{\tilde{\lambda }(\tilde{\gamma }+2)+1}{\tilde{\lambda }+1},& \tilde{\gamma }>1+\frac{1}{\tilde{\lambda }},\\ n^{m\tilde{\lambda }+1-\frac{1}{p}},& p\ge 2,& -2<\tilde{\gamma }\le 1+\frac{1}{\tilde{\lambda }}.\end{array}\right.$$

(39)

Case 2.

Now, let $q>2(1<p<2)$.

(I)

For ${\gamma }_1,{\gamma }_2\ge 0$, according to Lemmas 1 and 2, we consistently obtain the following:

$$\begin{array}{cc}\hfill B_{n,2}^1\left(w_1\right)& ⪯\underset{E_{R_2,1}^1}{\int }{\displaystyle \frac{\left|dw\right|}{{\left|{\Psi }^{\prime }\left(w\right)\right|}^{q-2}{\left|\Psi \left(w\right)-\Psi \left(w_1\right)\right|}^{{\gamma }_1(q-1)}}}\hfill \\ \hfill & ⪯\underset{E_{R_2,1}^1}{\int }{\displaystyle \frac{\left|dw\right|}{{\left|w-w_1\right|}^{({\lambda }_1-1+\epsilon )(q-2)}{\left|w-w_1\right|}^{{\gamma }_1(q-1)({\lambda }_1+\epsilon )}}}\hfill \\ \hfill & ⪯n^{{\gamma }_1(q-1){\lambda }_1+({\lambda }_1-1+\epsilon )(q-2)}mes\left(E_{R_2,1}^1\right)⪯n^{{\gamma }_1(q-1){\lambda }_1+({\lambda }_1-1+\epsilon )(q-2)-1};\hfill \end{array}$$

(40)

$$\begin{array}{cc}\hfill B_{n,2}^2\left(w_1\right)& =\underset{E_{R_2,1}^2}{\int }{\displaystyle \frac{\left|dw\right|}{{\left|{\Psi }^{\prime }\left(w\right)\right|}^{q-2}{\left|\Psi \left(w\right)-\Psi \left(w_1\right)\right|}^{{\gamma }_1(q-1)}}}\hfill \\ \hfill & \asymp \underset{E_{R_2,1}^2}{\int }{\left({\displaystyle \frac{\left|w\right|-1}{d(\Psi (w),L)}}\right)}^{q-2}{\displaystyle \frac{\left|dw\right|}{{\left|\Psi \left(w\right)-\Psi \left(w_1\right)\right|}^{{\gamma }_1(q-1)}}}\hfill \\ \hfill & ⪯n^{2-q}\underset{E_{R_2,1}^2}{\int }{\displaystyle \frac{\left|dw\right|}{{\left|w-w_1\right|}^{\left[{\gamma }_1(q-1)+(q-2)\right]{\lambda }_1+\epsilon }}}\hfill \\ \hfill & ⪯\left\{\begin{array}{cc}{n}^{\left[{\gamma }_1(q-1)+(q-2)\right]{\lambda }_1-1+(2-q)+\epsilon },& \left[{\gamma }_1(q-1)+(q-2)\right]{\lambda }_1+\epsilon >1,\\ n^{2-q}\ln n,& \left[{\gamma }_1(q-1)+(q-2)\right]{\lambda }_1+\epsilon =1,\\ n^{2-q},& \left[{\gamma }_1(q-1)+(q-2)\right]{\lambda }_1+\epsilon <1;\end{array}\right.\hfill \end{array}$$

(41)

$$\begin{array}{cc}\hfill B_{n,2}^1\left(w_2\right)& ⪯\underset{E_{R_2,2}^1}{\int }{\displaystyle \frac{\left|dw\right|}{{\left|{\Psi }^{\prime }\left(w\right)\right|}^{q-2}{\left|\Psi \left(w\right)-\Psi \left(w_2\right)\right|}^{{\gamma }_2(q-1)}}}\hfill \\ \hfill & \asymp \underset{E_{R_2,2}^1}{\int }{\left({\displaystyle \frac{\left|w\right|-1}{d(\Psi (w),L)}}\right)}^{q-2}{\displaystyle \frac{\left|dw\right|}{{\left|\Psi \left(w\right)-\Psi \left(w_2\right)\right|}^{{\gamma }_2(q-1)}}}\hfill \\ \hfill & ⪯{\left({\displaystyle \frac{1}{n}}\right)}^{q-2}·n^{2\left[{\gamma }_2(q-1)+(q-2)\right]}\underset{E_{R_2,2}^1}{\int }\left|dw\right|\hfill \\ \hfill & ⪯n^{2\left[{\gamma }_2(q-1)+(q-2)\right]-(q-2)}mes\left(E_{R_2,2}^1\right)⪯n^{2\left[{\gamma }_2(q-1)+(q-2)\right]+1-q};\hfill \end{array}$$

(42)

$$\begin{array}{cc}\hfill B_{n,2}^2\left(w_2\right)& ⪯\underset{E_{R_2,2}^2}{\int }{\displaystyle \frac{\left|dw\right|}{{\left|{\Psi }^{\prime }\left(w\right)\right|}^{q-2}{\left|\Psi \left(w\right)-\Psi \left(w_2\right)\right|}^{{\gamma }_2(q-1)}}}\hfill \\ \hfill & \asymp \underset{E_{R_2,2}^2}{\int }{\left({\displaystyle \frac{\left|w\right|-1}{d(\Psi (w),L)}}\right)}^{q-2}{\displaystyle \frac{\left|dw\right|}{{\left|\Psi \left(w\right)-\Psi \left(w_2\right)\right|}^{{\gamma }_2(q-1)}}}\hfill \\ \hfill & ⪯n^{2-q}\underset{E_{R_2,2}^2}{\int }{\displaystyle \frac{\left|dw\right|}{{\left|w-w_2\right|}^{2\left[{\gamma }_2(q-1)+(q-2)\right]}}}\hfill \\ \hfill & ⪯\left\{\begin{array}{cc}{n}^{2\left[{\gamma }_2(q-1)+(q-2)\right]+1-q},& 2\left[{\gamma }_2(q-1)+(q-2)\right]>1,\\ n^{2-q}\ln n,& 2\left[{\gamma }_2(q-1)+(q-2)\right]=1,\\ n^{2-q},& 2\left[{\gamma }_2(q-1)+(q-2)\right]<1.\end{array}\right.\hfill \end{array}$$

(43)

(II)

For $-2<{\gamma }_1,{\gamma }_2<0$, we obtain the following:

$$\begin{array}{cc}\hfill B_{n,2}^1\left(w_1\right)& ⪯\underset{E_{R_2,1}^1}{\int }{\displaystyle \frac{{\left|\Psi \left(w\right)-\Psi \left(w_1\right)\right|}^{(-{\gamma }_1)(q-1)}}{{\left|{\Psi }^{\prime }\left(w\right)\right|}^{q-2}}}\left|dw\right|\hfill \\ \hfill & ⪯n^{2-q}\underset{E_{R_2,1}^1}{\int }{\displaystyle \frac{\left|dw\right|}{{\left|w-w_1\right|}^{(q-2){\lambda }_1+\epsilon }}}⪯n^{2-q+(q-2){\lambda }_1+\epsilon }mes\left(E_{R_2,1}^1\right)⪯n^{1-q+(q-2){\lambda }_1+\epsilon };\hfill \end{array}$$

(44)

$$\begin{array}{cc}\hfill B_{n,2}^2\left(w_1\right)& ⪯\underset{E_{R_2,1}^2}{\int }{\displaystyle \frac{{\left|\Psi \left(w\right)-\Psi \left(w_1\right)\right|}^{(-{\gamma }_1)(q-1)}}{{\left|{\Psi }^{\prime }\left(w\right)\right|}^{q-2}}}\left|dw\right|\hfill \\ \hfill & ⪯n^{2-q}\underset{E_{R_2,1}^2}{\int }{\displaystyle \frac{\left|dw\right|}{{\left|w-w_1\right|}^{(q-2){\lambda }_1+\epsilon }}}⪯\left\{\begin{array}{cc}{n}^{(q-2){\lambda }_1-(q-1)+\epsilon },& (q-2){\lambda }_1+\epsilon >1,\\ n^{2-q}\ln n,& (q-2){\lambda }_1+\epsilon =1,\\ n^{2-q},& (q-2){\lambda }_1+\epsilon <1;\end{array}\right.\hfill \end{array}$$

(45)

$$\begin{array}{cc}\hfill B_{n,2}^1\left(w_2\right)& ⪯\underset{E_{R_2,2}^1}{\int }{\displaystyle \frac{{\left|\Psi \left(w\right)-\Psi \left(w_2\right)\right|}^{(-{\gamma }_2)(q-1)}}{{\left|{\Psi }^{\prime }\left(w\right)\right|}^{q-2}}}\left|dw\right|⪯n^{2-q}\underset{E_{R_2,2}^1}{\int }{\displaystyle \frac{\left|dw\right|}{{\left|w-w_2\right|}^{2(q-2)}}}\hfill \\ \hfill & ⪯n^{2-q+2(q-2)}mes\left(E_{R_2,2}^1\right)⪯n^{(q-2)-1};\hfill \end{array}$$

(46)

$$\begin{array}{cc}\hfill B_{n,2}^2\left(w_2\right)& ⪯n^{2-q}\underset{E_{R_2,2}^2}{\int }{\displaystyle \frac{\left|dw\right|}{{\left|w-w_2\right|}^{2(q-2)}}}⪯\left\{\begin{array}{cc}{n}^{(q-2)-1},& 2(q-2)>1,\\ n^{2-q}\ln n,& 2(q-2)=1,\\ n^{2-q},& 2(q-2)<1.\end{array}\right.\hfill \end{array}$$

(47)

For $w\in E_{R_2,1}^3$, $\left|\Psi \left(w\right)-\Psi \left(w_j\right)\right|⪰1$, and then, for $j=1,2$, we have:

$$B_{n,2}^3\left(w_j\right)⪯\underset{E_{R_2,j}^3}{\int }{\displaystyle \frac{\left|dw\right|}{{\left|{\Psi }^{\prime }\left(w\right)\right|}^{q-2}}}⪯\underset{E_{R_2,j}^3}{\int }{\displaystyle \frac{\left|dw\right|}{{\left(\left|w\right|-1\right)}^{\epsilon }}}⪯n^{\epsilon }.$$

(48)

By combining (40)–(48), we obtain

$$J_2\left(L_{R_2}\right)⪯{∥P_n∥}_p\left\{\begin{array}{ccc}{n}^{\left({\displaystyle \frac{\tilde{\gamma }+2}{p}}-1\right)\tilde{\lambda }},& 1<p<{\displaystyle \frac{\tilde{\lambda }(\tilde{\gamma }+2)+1}{\tilde{\lambda }+1}},& \tilde{\gamma }<{\displaystyle \frac{1}{\tilde{\lambda }}},\\ {\left(n\ln n\right)}^{1-{\displaystyle \frac{1}{p}}},& p={\displaystyle \frac{\tilde{\lambda }(\tilde{\gamma }+2)+1}{\tilde{\lambda }+1}},& \tilde{\gamma }<{\displaystyle \frac{1}{\tilde{\lambda }}},\\ n^{1-{\displaystyle \frac{1}{p}}},& 2>p>{\displaystyle \frac{\tilde{\lambda }(\tilde{\gamma }+2)+1}{\tilde{\lambda }+1}},& \tilde{\gamma }<{\displaystyle \frac{1}{\tilde{\lambda }}},\\ n^{\left({\displaystyle \frac{\tilde{\gamma }+2}{p}}-1\right)\tilde{\lambda }},& 1<p<2& \tilde{\gamma }\ge {\displaystyle \frac{1}{\tilde{\lambda }}},\end{array}\right.$$

for ${\gamma }_1,{\gamma }_2\ge 0$, and

$$J_2\left(L_{R_2}\right)⪯{∥P_n∥}_p\left\{\begin{array}{cc}{n}^{\left({\displaystyle \frac{2}{p}}-1\right)\tilde{\lambda }},& 1<p<{\displaystyle \frac{2\tilde{\lambda }+1}{\tilde{\lambda }+1}},\\ {\left(n\ln n\right)}^{1-{\displaystyle \frac{1}{p}}},& p={\displaystyle \frac{2\tilde{\lambda }+1}{\tilde{\lambda }+1}},\\ n^{1-{\displaystyle \frac{1}{p}}},& 2>p>{\displaystyle \frac{2\tilde{\lambda }+1}{\tilde{\lambda }+1}},\end{array}\right.$$

for $-2<{\gamma }_1,{\gamma }_2<0$, where $\tilde{\gamma }:={\gamma }_j,$ for $z\in {\Omega }_R^j,$ $j=1,2.$

By substituting the last obtained estimates into (22) and taking into account (26), for the case $1<p<2,{\gamma }_1>-2,{\gamma }_2>-2,$ we obtain the following:

$$\left|P_n^{\left(m\right)}\left(z\right)\right|⪯{\displaystyle \frac{\left|{\Phi }^{n-m+1}\left(z\right)\right|}{d(z,L_{R_1})}}{∥P_n∥}_p\left\{\begin{array}{ccc}{n}^{\left({\displaystyle \frac{\tilde{\gamma }+2}{p}}+m-1\right)\tilde{\lambda }},& 1<p<{\displaystyle \frac{\tilde{\lambda }(\tilde{\gamma }+2)+1}{\tilde{\lambda }+1}},& \tilde{\gamma }<{\displaystyle \frac{1}{\tilde{\lambda }}},\\ n^{m\tilde{\lambda }+1-{\displaystyle \frac{1}{p}}}{\left(\ln n\right)}^{1-{\displaystyle \frac{1}{p}}},& p={\displaystyle \frac{\tilde{\lambda }(\tilde{\gamma }+2)+1}{\tilde{\lambda }+1}},& \tilde{\gamma }<{\displaystyle \frac{1}{\tilde{\lambda }}},\\ n^{m\tilde{\lambda }+1-{\displaystyle \frac{1}{p}}},& 2>p>{\displaystyle \frac{\tilde{\lambda }(\tilde{\gamma }+2)+1}{\tilde{\lambda }+1}},& \tilde{\gamma }<{\displaystyle \frac{1}{\tilde{\lambda }}},\\ n^{\left({\displaystyle \frac{\tilde{\gamma }+2}{p}}+m-1\right)\tilde{\lambda }},& 1<p<2& \tilde{\gamma }\ge {\displaystyle \frac{1}{\tilde{\lambda }}},\end{array}\right.$$

for ${\gamma }_1,{\gamma }_2\ge 0$, and

$$\left|P_n^{\left(m\right)}\left(z\right)\right|⪯{\displaystyle \frac{\left|{\Phi }^{n-m+1}\left(z\right)\right|}{d(z,L_{R_1})}}{∥P_n∥}_p\left\{\begin{array}{cc}{n}^{\left({\displaystyle \frac{2}{p}}+m-1\right)\tilde{\lambda }},& 1<p<{\displaystyle \frac{2\tilde{\lambda }+1}{\tilde{\lambda }+1}},\\ n^{m\tilde{\lambda }+1-{\displaystyle \frac{1}{p}}}{\left(\ln n\right)}^{1-{\displaystyle \frac{1}{p}}},& p={\displaystyle \frac{2\tilde{\lambda }+1}{\tilde{\lambda }+1}},\\ n^{m\tilde{\lambda }+1-{\displaystyle \frac{1}{p}}},& 2>p>{\displaystyle \frac{2\tilde{\lambda }+1}{\tilde{\lambda }+1}},\end{array}\right.$$

for $-2<{\gamma }_1,{\gamma }_2<0$. Next, we combine the last two inequalities corresponding to the cases ${\gamma }_j\ge 0$ and $-2<{\gamma }_j<0$ for each $j=1,2.$

$$\left|P_n^{\left(m\right)}\left(z\right)\right|⪯{\displaystyle \frac{\left|{\Phi }^{n-m+1}\left(z\right)\right|}{d(z,L_{R_1})}}{∥P_n∥}_p\left\{\begin{array}{ccc}{n}^{\left({\displaystyle \frac{{\tilde{\gamma }}^{*}+2}{p}}+m-1\right)\tilde{\lambda }},& 1<p<{\displaystyle \frac{\tilde{\lambda }({\tilde{\gamma }}^{*}+2)+1}{\tilde{\lambda }+1}},& -2<\tilde{\gamma }<{\displaystyle \frac{1}{\tilde{\lambda }}},\\ n^{m\tilde{\lambda }+1-{\displaystyle \frac{1}{p}}}{\left(\ln n\right)}^{1-{\displaystyle \frac{1}{p}}},& p={\displaystyle \frac{\tilde{\lambda }({\tilde{\gamma }}^{*}+2)+1}{\tilde{\lambda }+1}},& -2<\tilde{\gamma }<{\displaystyle \frac{1}{\tilde{\lambda }}},\\ n^{m\tilde{\lambda }+1-{\displaystyle \frac{1}{p}}},& 2>p>{\displaystyle \frac{\tilde{\lambda }({\tilde{\gamma }}^{*}+2)+1}{\tilde{\lambda }+1}},& -2<\tilde{\gamma }<{\displaystyle \frac{1}{\tilde{\lambda }}},\\ n^{\left({\displaystyle \frac{\tilde{\gamma }+2}{p}}+m-1\right)\tilde{\lambda }},& 1<p<2,& \tilde{\gamma }\ge {\displaystyle \frac{1}{\tilde{\lambda }}}.\end{array}\right.$$

Now, let us consider the case $p=1.$ By multiplying the numerator and denominator of the integrand of the inner integral by h in (22), we obtain

$$\begin{array}{cc}\hfill \left|P_n^{\left(m\right)}\left(z\right)\right|& ⪯{\displaystyle \frac{\left|{\Phi }^{n-m+1}\left(z\right)\right|}{d(z,L_{R_1})}}·\underset{t\in L_{R_2}}{sup}\left({\displaystyle \frac{1}{h\left(t\right)}}\underset{L_{R_1}}{\int }{\displaystyle \frac{\left|d\zeta \right|}{{\left|t-\zeta \right|}^{m+1}}}\right)\left(\underset{L_{R_2}}{\int }h\left(t\right)\left|P_n\left(t\right)\right|\left|dt\right|\right)\hfill \\ \hfill & ⪯{\displaystyle \frac{\left|{\Phi }^{n-m+1}\left(z\right)\right|}{d(z,L_{R_1})}}{∥P_n∥}_{{\mathcal{L}}_1(h,L_{R_2})}·\underset{t\in L_{R_2}}{sup}\left({\displaystyle \frac{1}{{\displaystyle \prod _{j=1}^2}{\left|t-z_j\right|}^{{\gamma }_j}}}\underset{L_{R_1}}{\int }{\displaystyle \frac{\left|d\zeta \right|}{{\left|t-\zeta \right|}^{m+1}}}\right).\hfill \end{array}$$

(49)

After replacing the variable $t=\Psi \left(w\right)$ and using Lemmas 1 and 2 and according to (18), (24), and (26), we obtain

$${\displaystyle \frac{1}{{\left|t-z_1\right|}^{{\gamma }_1}}}={\displaystyle \frac{1}{{\left|\Psi \left(w\right)-\Psi \left(w_1\right)\right|}^{{\gamma }_1}}}={\displaystyle \frac{1}{{\left|w-w_1\right|}^{{\gamma }_1{\tilde{\lambda }}_1}}}⪯\left\{\begin{array}{cc}{n}^{{\gamma }_1{\tilde{\lambda }}_1},& \mathrm{for}{\gamma }_1\ge 0,\\ 1,& \mathrm{for}{\gamma }_1<0;\end{array}\right.$$

(50)

$${\displaystyle \frac{1}{{\left|t-z_2\right|}^{{\gamma }_2}}}={\displaystyle \frac{1}{{\left|\Psi \left(w\right)-\Psi \left(w_2\right)\right|}^{{\gamma }_2}}}={\displaystyle \frac{1}{{\left|w-w_2\right|}^{2{\gamma }_2}}}⪯\left\{\begin{array}{cc}{n}^{2{\gamma }_2},& \mathrm{for}{\gamma }_2\ge 0,\\ 1,& \mathrm{for}{\gamma }_2<0;\end{array}\right.$$

(51)

$$\underset{L_{R_1}}{\int }{\displaystyle \frac{\left|d\zeta \right|}{{\left|t-\zeta \right|}^{m+1}}}=\underset{\left|\tau \right|=R_1}{\int }{\displaystyle \frac{\left|{\Psi }^{\prime }\left(\tau \right)\right|\left|d\tau \right|}{{\left|\Psi \left(\tau \right)-\Psi \left(w\right)\right|}^{m+1}}}=J_1\left(L_{R_1}\right)⪯n^{m\tilde{\lambda }}.$$

(52)

According to Lemmas 1, 3, and (18), we find that

$${∥P_n∥}_{{\mathcal{L}}_1(h,L_{R_2})}=\underset{L_{R_2}}{\int }h\left(t\right)\left|P_n\left(t\right)\right|\left|dt\right|\le \underset{t\in L_{R_2}}{sup}\left|{\Phi }^{\prime }\left(t\right)\right|\underset{L_{R_2}}{\int }{\displaystyle \frac{h\left(t\right)}{\left|{\Phi }^{\prime }\left(t\right)\right|}}\left|P_n\left(t\right)\right|\left|dt\right|⪯n^{\tilde{\lambda }}{∥P_n∥}_1.$$

(53)

Therefore, by combining Estimates (49)–(53), we find that

$$\left|P_n^{\left(m\right)}\left(z\right)\right|⪯{\displaystyle \frac{\left|{\Phi }^{n-m+1}\left(z\right)\right|}{d(z,L_{R_1})}}{n}^{({\tilde{\gamma }}^{*}+m+1)\tilde{\lambda }}{∥P_n∥}_1.$$

Therefore, for any $p=1$, the proof of Theorem 1 is completed.

In order to finalize the proof, we verify that $d(z,L_{R_1})⪰d(z,L)$ for all $z\in {\Omega }_R$. To this end, we now introduce the notations that will be used in the remainder of the proof. For $0<{\delta }_j<{\delta }_0:={\displaystyle \frac{1}{4}}min\left\{\left|z_i-z_j\right|:i,j=1,2,\dots ,l,i\ne j\right\}$; 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 $L=\partial G$ we set $U_{\infty }(L,\delta ):={\displaystyle \bigcup _{\zeta \in L}}U(\zeta ,\delta )$—infinite open cover of the curve L; $U(L,\delta ):={\displaystyle \bigcup _{j=1}^N}{U}_j(L,\delta )\subset U_{\infty }(L,\delta )$—finite open cover of the curve $L;$ $\Omega \left(\delta \right):=\Omega (L,\delta ):=\Omega \cap U_N(L,\delta ).$ Now, for $z\notin \Omega (L_{R_1},d(L_{R_1},L_R)),$ we have $d(z,L_{R_1})⪰\delta ⪰d(z,L).$ Next, let $z\in \Omega (L_{R_1},d(L_{R_1},L_R)).$ It is denote by ${\xi }_1\in L_{R_1}$ such that $d(z,L_{R_1})=\left|z-{\xi }_1\right|,$ and point ${\xi }_2\in L$ such that $d(z,L)=\left|z-{\xi }_2\right|.$ For $w=\Phi \left(z\right),t_1=\Phi \left({\xi }_1\right),$ $t_2=\Phi \left({\xi }_2\right),$ we have the following: $\left|w-t_1\right|\ge \left|\left|w-t_2\right|-\left|t_2-t_1\right|\right|\ge \left|\left|w-t_2\right|-{\displaystyle \frac{1}{2}}\left|w-t_2\right|\right|\ge {\displaystyle \frac{1}{2}}\left|w-t_2\right|.$ Then, according to Lemma 1, we obtain the following: $d(z,L_{R_1})⪰d(z,L).$ □

Proof of Theorem 2.

First of all, we present the theorem that we will use, along with its corollaries. Then we prove the estimate for $\left|P_n^{\left(m\right)}\left(z\right)\right|,$ $z\in \overline{G}$ for each $m\ge 0$.

Theorem 4

([22]). Let $p>0;G\in P{C}_{\theta }(1;\lambda ,2),$ for some $0<\lambda <2;$ $h\left(z\right)$ be defined as in (1). Then, for any $P_n\in {\wp }_n,n\in \mathbb{N},{\gamma }_j>-2,$ $j=1,2,$ and arbitrary small $\epsilon >0,$ we have

$${∥P_n∥}_{\infty }\le c_{10}{\mu }_{n,0}{∥P_n∥}_p,$$

(54)

where $c_{10}=c_{10}(G,{\gamma }_1,{\gamma }_2,\lambda ,p,\epsilon )>0$ is the constant independent of z and $n,$ and the numbers ${\mu }_{n,0},\widehat{\gamma }$ $,\tilde{\lambda }$ are defined as in (15) for $m=0$ and in (11), respectively.

Corollary 10.

Under the conditions of Theorem 4, for $G\in P{C}_{\theta }(1;{\lambda }_1,{\lambda }_2),$ the relation (54) is satisfied for

$${\tilde{\mu }}_{n,m,1}=\left\{\begin{array}{cc}{n}^{{\displaystyle \frac{2+{\gamma }^{*}}{p}}{\lambda }^{*}},& \mathit{if}(2+{\gamma }^{*})·{\lambda }^{*}>1,\\ {\left(n\ln n\right)}^{{\displaystyle \frac{1}{p}}},& \mathit{if}(2+{\gamma }^{*})·{\lambda }^{*}=1,\\ n^{{\displaystyle \frac{1}{p}}},& \mathit{if}(2+{\gamma }^{*})·{\lambda }^{*}<1,\end{array}\right.$$

where ${\gamma }^{*}$ and ${\lambda }^{*}$ are defined as in (11).

Corollary 11.

Under the conditions of Theorem 4, for $G\in P{C}_{\theta }(1;2,2),$ the relation (1) is satisfied for

$${\tilde{\mu }}_{n,1}=n^{{\displaystyle \frac{2(2+{\gamma }^{*})}{p}}},$$

where ${\gamma }^{*}$ is defined as in (11).

Now we proceed to the proof of the estimate for $\left|P_n^{\left(m\right)}\left(z\right)\right|,$ $z\in \overline{G}$, for each $m\ge 0$, and for the region $G\in P{C}_{\theta }(1;\lambda ,2)$. Let $z\in L$ be an arbitrary fixed point, wgere $B(z,d(z,L_{R_1})):=\left\{\xi :\left|\xi -z\right|<d(z,L_{R_1})\right\}$. According to the Cauchy integral formula for derivatives, we have that

$$P_n^{\left(m\right)}\left(z\right)={\displaystyle \frac{m!}{2\pi i}}\underset{\partial B(z,d(z,L_{R_1}))}{\int }{\displaystyle \frac{P_n\left(t\right)}{{(t-z)}^{m+1}}}dt,m=0,1,2,\dots$$

Then, by applying the well-known Bernstein–Walsh inequality (4), we obtain the following:

$$\begin{array}{cc}\hfill \left|P_n^{\left(m\right)}\left(z\right)\right|& \le {\displaystyle \frac{m!}{2\pi }}\underset{z\in \partial B(z,d(z,L_{R_1}))}{max}\left|P_n\left(t\right)\right|·\underset{\partial B(z,d(z,L_{R_1}))}{\int }{\displaystyle \frac{\left|dt\right|}{{\left|t-z\right|}^{m+1}}}\hfill \\ \hfill & \le {\displaystyle \frac{m!}{2\pi }}\underset{t\in {\overline{G}}_{R_1}}{max}\left|P_n\left(t\right)\right|·{\displaystyle \frac{2\pi d(z,L_R)}{d^{m+1}(z,L_R)}}⪯{\displaystyle \frac{{∥P_n∥}_{C\left(\overline{G}\right)}}{d^m(z,L_R)}}.\hfill \end{array}$$

If $p=\infty ,$ we obtain

$$\left|P_n^{\left(m\right)}\left(z\right)\right|⪯{\displaystyle \frac{1}{d^m(z,L_R)}}{∥P_n∥}_{\infty },\forall z\in L.$$

If $0<p<\infty$, by applying Theorem 4 and using the Lemmas 1 and 2, for all $\forall z\in L,$ we obtain

$$\left|P_n^{\left(m\right)}\left(z\right)\right|⪯{\mu }_{n,0}{∥P_n∥}_p·n^{m\tilde{\lambda }}⪯{\mu }_{n,m}{\parallel P_n\parallel }_p.$$

(55)

Since $z\in L$ is arbitrary, we complete the proof of Theorem 2.

The proofs of Corollaries 4 and 5 are obtained in a completely similar way, that is, using Corollaries 10 and 11 in (55) in place of Theorem 4. □

6. Conclusions

In this paper, we studied the growth properties of the m-th ($m\ge 1$) derivatives of algebraic polynomials in weighted Bergman spaces in bounded and unbounded regions. The object we studied are regions with piecewise smooth curves containing both ${\lambda }_j\pi$ $(0<{\lambda }_1<2,$ ${\lambda }_2=2)$ (zero and non-zero) angles on the boundary. We clearly demonstrated the influence of the angle ${\lambda }_j\pi$ on the boundary of the region and the order of the “zero” or “pole” of the weight function at the boundary points on the growth of the polynomial (Theorem 1). Note that [9] studied a similar question but only for $m=1$ derivatives and used a recursive method. Here, we applied a different approach, solving the problem for all $m\ge 1$ and obtaining a much stronger result than in [9].

Further, for all $m\ge 0$, we proved an estimate for the m-th derivative (Theorem 2). Here, we also found the influence of the ${\lambda }_j\pi$ angles on the growth of the polynomials and demonstrated the accuracy of the estimate obtained.

By combining the obtained estimates (Theorems 1 and 2), we found estimates for the growth of polynomials over the whole complex plane, explicitly indicating the influence of the ${\lambda }_j\pi$ angles and the “zeros” and “poles” of the weight function.

It is of interest to extend these results to broader classes of regions with specific characteristics, as well as to obtain analogs of these results in other weighted spaces and with other weight functions.