Home Bernstein-Walsh type inequalities for derivatives of algebraic polynomials in quasidisks
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Bernstein-Walsh type inequalities for derivatives of algebraic polynomials in quasidisks

  • Fahreddin G. Abdullayev EMAIL logo
Published/Copyright: December 31, 2021
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Open Mathematics
From the journal Open Mathematics Volume 19 Issue 1

Abstract

In this paper, we study Bernstein-Walsh type estimates for the higher-order derivatives of an arbitrary algebraic polynomial on quasidisks.

Keywords: algebraic polynomial; quasiconformal mapping; quasicircle
MSC 2010: 30C10; 30E10; 30C70

1 Introduction

Let C be a complex plane and C ¯ ≔ C ∪ { ∞ } ; G ⊂ C be a bounded Jordan region with boundary L ≔ ∂ G such that 0 ∈ G ; Ω ≔ C ¯ \ G ¯ = ext L ; Δ ≔ Δ ( 0 , 1 ) ≔ { w : ∣ w ∣ > 1 } . Let w = Φ ( z ) be the univalent conformal mapping of Ω onto Δ such that Φ ( ∞ ) = ∞ and lim z → ∞ Φ ( z ) z > 0 ; Ψ ≔ Φ − 1 . For R > 1 , we take L R ≔ { z : ∣ Φ ( z ) ∣ = R } , G R ≔ int L R , and Ω R ≔ ext L R . Let ℘ n denote the class of all algebraic polynomials P n ( z ) of degree at most n ∈ N .

In this work, we consider the following weight function h ( z ) . Let { z j } j = 1 l be the fixed system of distinct points on curve L . For some fixed R 0 , 1 < R 0 < ∞ , and z ∈ G ¯ R 0 , consider generalized Jacobi weight function h ( z ) , which is defined as follows:

(1) h ( z ) ≔ h 0 ( z ) ∏ j = 1 l ∣ z − z j ∣ γ j ,

where γ j > − 2 , for all j = 1 , 2 , … , l , and h 0 is uniformly separated from zero in L R 0 , i.e., there exists a constant c 1 ( G ) > 0 such that h 0 ( z ) ≥ c 1 ( G ) > 0 for all z ∈ G R 0 .

Let 0 < p ≤ ∞ and σ be the two-dimensional Lebesgue measure. For the Jordan region G , we introduce:

(2) ∥ P n ∥ p ≔ ∥ P n ∥ A p ( h , G ) ≔ ∬ G h ( z ) ∣ P n ( z ) ∣ p d σ z 1 / p , 0 < p < ∞ , ∥ P n ∥ ∞ ≔ ∥ P n ∥ A ∞ ( 1 , G ) ≔ max z ∈ G ¯ ∣ P n ( z ) ∣ , p = ∞ ,

and A p ( 1 , G ) ≕ A p ( G ) .

When L is rectifiable, for any p > 0 , let

(3) ∥ P n ∥ ℒ p ( h , G ) ≔ ∫ L h ( z ) ∣ P n ( z ) ∣ p ∣ d z ∣ 1 / p < ∞ , 0 < p < ∞ , ∥ P n ∥ ℒ ∞ ( 1 , G ) ≔ max z ∈ L ∣ P n ( z ) ∣ , p = ∞ ,

and ℒ p ( 1 , L ) ≕ ℒ p ( L ) .

According to well-known Bernstein-Walsh lemma [1], we have:

(4) ∥ P n ∥ C ( G ¯ R ) ≤ R n ∥ P n ∥ C ( G ¯ ) .

Hence, setting R = 1 + 1 n , we see that the C norm of polynomials P n ( z ) in G ¯ R and G ¯ have the same order of growth, that is, the norm ∥ P n ∥ C ( G ¯ ) increases up to multiplication by a constant in G ¯ R .

In [1] also was given some similar estimates for various norms on the right-hand side of (3). Analogous estimation with respect to the quasinorm (3) for p > 0 was obtained in [2] for h ( z ) ≡ 1 (i.e., γ j = 0 for all j = 1 , 2 , … , l ) as follows:

∥ P n ∥ ℒ p ( L R ) ≤ R n + 1 p ∥ P n ∥ ℒ p ( L ) , p > 0 .

Moreover, in [3, Lemma 2.4], this estimate has been generalized for h ( z ) ≠ 1 , defined as in (1) for the γ j > − 1 , for all j = 1 , 2 , … , l , and the following was proved:

(5) ∥ P n ∥ ℒ p ( h , L R ) ≤ R n + 1 + γ ∗ p ∥ P n ∥ ℒ p ( h , L ) , γ ∗ = max { 0 ; γ j : 1 ≤ j ≤ l } .

To give a similar estimation to (5) for the A p ( h , G ) norm, first, we will give the following definition.

Let the function φ map G conformally and univalently onto B ≔ B ( 0 , 1 ) ≔ { w : ∣ w ∣ < 1 } , which is normalized by φ ( 0 ) = 0 and φ ′ ( 0 ) > 0 ; let ψ ≔ φ − 1 .

Definition 1

A bounded Jordan region G is called a κ -quasidisk, 0 ≤ κ < 1 , if any conformal mapping ψ can be extended to a K -quasiconformal, K = 1 + κ 1 − κ , homeomorphism of the plane C ¯ on the C ¯ . In that case, the curve L ≔ ∂ G is called a κ -quasicircle. The region G (curve L ) is called a quasidisk (quasicircle), if it is κ -quasidisk ( κ -quasicircle) with some 0 ≤ κ < 1 .

A simple example of a κ -quasidisk may be a region bounded by two arcs of circle, symmetric with respect to the O X -axis and O Y -axis, each of the arcs crosses the O X -axis at ± ε 0 , where ε 0 > 0 and the angle between the arcs is π ( 1 − κ ) , where 0 ≤ κ < 1 .

A Jordan curve L is called a quasicircle or quasiconformal curve, if it is the image of the unit circle under a quasiconformal mapping of C to C (see [4, p. 105], [5, p. 286]). On the other hand, it was given some geometric criteria of quasiconformality of the curves (see also [6, p. 81], [7, p. 107]). It is well-known that quasicircles can be nonrectifiable (see, e.g., [8], [4, p. 104]).

In [9] (see also [10]), the Bernstein-Walsh type estimates for the norm (2), the regions with quasiconformal boundary, weight function h ( z ) , defined in (1) with γ j > − 2 , and for all p > 0 , are as follows:

∥ P n ∥ A p ( h , G R ) ≤ c 1 R ∗ n + 1 p ∥ P n ∥ A p ( h , G ) ,

where R ∗ ≔ 1 + c 2 ( R − 1 ) , c 2 > 0 and, c 1 ≔ c 1 ( G , p , c 2 ) > 0 constants, independent from n and R .

In [11, Theorem 1.1], analogous estimate was studied for A p ( G ) norm, p > 0 , for arbitrary Jordan region and was obtained: for any P n ∈ ℘ n , R 1 = 1 + 1 n and arbitrary R , R > R 1 , the following estimate

∥ P n ∥ A p ( G R ) ≤ c R n + 2 p ∥ P n ∥ A p ( G R 1 )

is true, where c = 2 e p − 1 1 p 1 + O 1 n , n → ∞ .

Stylianopoulos in [12] replaced the norm ∥ P n ∥ C ( G ¯ ) with norm ∥ P n ∥ A 2 ( G ) on the right-hand side of (4) and found a new version of the Bernstein-Walsh lemma: Assume that L is quasiconformal and rectifiable. Then, there exists a constant c = c ( L ) > 0 depending only on L such that

∣ P n ( z ) ∣ ≤ c n d ( z , L ) ∥ P n ∥ A 2 ( G ) ∣ Φ ( z ) ∣ n + 1 , z ∈ Ω ,

where d ( z , L ) ≔ inf { ∣ ζ − z ∣ : ζ ∈ L } holds for every P n ∈ ℘ n .

In this work, we study for κ -quasidisks G , 0 ≤ κ < 1 , pointwise estimation in unbounded region Ω 1 + ε 0 n − 1 = C ¯ \ G ¯ 1 + ε 0 n − 1 for sufficiently small ε 0 > 0 , for the derivative ∣ P n ( m ) ( z ) ∣ , m = 0 , 1 , 2 , … , in the following type:

(6) ∣ P n ( m ) ( z ) ∣ ≤ η n ( G , h , p , m , d ( z , L ) , ∣ Φ ( z ) ∣ n + 1 ) ∥ P n ∥ p , z ∈ Ω 1 + ε 0 n − 1 ,

where η n ( ⋅ ) → ∞ , as n → ∞ , depending on the properties of the G and h .

Analogous results of (6)-type for m = 0 , different weight function h , unbounded region Ω , and some norms were obtained in [13, pp. 418–428] [14,15,16,17,18,19, 20,21,22] and others.

To obtain estimates for ∣ P n ( m ) ( z ) ∣ on the whole complex plane, it is necessary to use Bernstein-Markov-Nikolsky type estimate for ∣ P n ( m ) ( z ) ∣ , z ∈ G ¯ , of the following type:

(7) ‖ P n ( m ) ‖ ∞ ≤ λ n ( G , h , p ) ‖ P n ‖ p , m = 0 , 1 , 2 , … ,

where λ n ≔ λ n ( G , h , p , m ) > 0 , λ n → ∞ , n → ∞ , is a constant, depending on the geometrical properties of the region G and the weight function h in general.

Many mathematicians have studied inequalities of type (7) since the beginning of the twentieth century [23,24,25]. In recent years, such inequalities for various spaces have been studied (see, i.e., [10], [13, pp. 418–428], [14], [15, Secttion 5.3], [16], [21, pp. 122–133], [26,27, 28,29] (see also the references cited therein). In recent years, analogous estimates to the (7)-type for m = 0 were continued to be studied in [9,10, 19,20,22, 23,24,25, 26,27,28, 29,30,31, 32,33,34] and others for various regions in the complex plane.

Therefore, combining estimates (6) and (7), we obtain in whole complex plane estimate for ∣ P n ( m ) ( z ) ∣ for any m = 1 , 2 , … , :

(8) ∣ P n ( m ) ( z ) ∣ ≤ c 4 ∥ P n ∥ p λ n ( G , h , p ) z ∈ G ¯ 1 + ε 0 n − 1 , η n ( G , h , p , d ( z , L ) ∣ Φ ( z ) ∣ n + 1 ) z ∈ Ω 1 + ε 0 n − 1 ,

where c 4 = c 4 ( G , p ) > 0 is a constant independent of n , h , P n , and λ n ( G , h , p ) → ∞ , η n ( G , h , p , d ( z , L ) ) → ∞ , as n → ∞ , depending on the properties of the G and h .

2 Definitions and main results

Throughout this paper, c , c 0 , c 1 , c 2 , … are positive and ε 0 , ε 1 , ε 2 , … are sufficiently small positive constants (generally, different in different relations), which depends on G in general and, on parameters inessential for the argument, otherwise, the dependence will be explicitly stated. For any k ≥ 0 and m > k , notation i = k , m ¯ means i = k , k + 1 , … , m .

First, we give estimate for ∣ P n ( m ) ( z ) ∣ , z ∈ G , ¯ for m ≥ 0 .

Theorem A

[35, Theorem 1] Let 0 < p ≤ ∞ , G be a κ -quasidisk for some 0 ≤ κ < 1 and h ( z ) be defined by (1). Then, for any P n ∈ ℘ n , n ∈ N , and every m = 0 , 1 , 2 , … , we have:

(9) ∥ P n ( m ) ∥ ∞ ≤ c n γ ∗ + 2 p + m ( 1 + κ ) ∥ P n ∥ p ,

where here and throughout the text,

(10) γ ∗ ≔ max { 0 ; γ j , j = 1 , l ¯ } .

Recall, the estimate (9) is sharp and for m = 0 was given in [32, Theorem 2.1].

For 0 < δ j < δ 0 ≔ 1 4 min { ∣ z i − z j ∣ : i , j = 1 , 2 , … , l , i ≠ j } , let Ω ( z j , δ j ) ≔ Ω ∩ { z : ∣ z − z j ∣ ≤ δ j } ; δ ≔ min 1 ≤ j ≤ l δ j ; U ∞ ( L , δ ) ≔ ⋃ ζ ∈ L U ( ζ , δ ) -infinite open cover of the curve L ; U N ( L , δ ) ≔ ⋃ j = 1 N U j ( L , δ ) ⊂ U ∞ ( L , δ ) -finite open cover of the curve L ; Ω ( δ ) ≔ Ω ( L , δ ) ≔ Ω ∩ U N ( L , δ ) , Ω ^ ≔ Ω \ Ω ( δ ) ; Ω R ( δ ) ≔ Ω ( L R , δ ) ≔ Ω R ∩ U N ( L R , δ ) , Ω ^ R ≔ Ω R \ Ω R ( δ ) .

Now, we start to formulate the new results.

2.1 The general estimate (recurrence formula)

First, we present a general estimate for the ∣ P n ( m ) ( z ) ∣ , for which it will be possible to obtain estimates for the derivative for each order m = 1 , 2 , …

Theorem 2

Let p ≥ 2 , G be a κ -quasidisk for some 0 ≤ κ < 1 , and h ( z ) be defined by (1). Then, for any P n ∈ ℘ n , n ∈ N , and every m = 1 , 2 , … , we have:

(11) ∣ P n ( m ) ( z ) ∣ ≤ c 1 ∣ Φ n + 1 ( z ) ∣ ⋅ ∥ P n ∥ p d ( z , L ) A n , p 1 ( z , m ) + ∑ j = 1 m C m j B n , j 1 ( z ) ∣ P n ( m − j ) ( z ) ∣ ,

where c 1 = c 1 ( G , γ , m , p ) > 0 constant independent from n and z ;

A n , p 1 ( z , m ) ≔ n γ + 2 p + m − 1 ( 1 + κ ) , p ≥ 2 , m ≥ 2 , γ > − 2 , n γ + 2 p + m − 1 ( 1 + κ ) , 2 ≤ p < 1 + ( 2 + γ ) ( 1 + κ ) , m = 1 , γ ≥ 0 , ( n ln n ) 1 − 1 p , p = 1 + ( 2 + γ ) ( 1 + κ ) , m = 1 , γ ≥ 0 , n 1 − 1 p , p > 1 + ( 2 + γ ) ( 1 + κ ) , m = 1 , γ ≥ 0 , n 2 p + m − 1 ( 1 + κ ) , p ≥ 2 , m ≥ 1 , − 2 < γ < 0 , B n , j 1 ( z ) ≔ n j ( 1 + κ ) , j = 1 , 2 , … , m ,

if z ∈ Ω ( δ ) , and

A n , p 1 ( z , m ) ≔ n γ + 2 p + m − 1 ( 1 + κ ) , 2 ≤ p < 1 2 + κ + ( γ + 2 ) 1 + κ 2 + κ , γ ≥ 1 1 + κ , ( n ln n ) 1 − 1 p , p = 1 2 + κ + ( γ + 2 ) 1 + κ 2 + κ , γ ≥ 1 1 + κ , n 1 − 1 p , p > 1 2 + κ + ( γ + 2 ) 1 + κ 2 + κ , γ ≥ 1 1 + κ , n 1 − 1 p , p ≥ 2 , 0 ≤ γ < 1 1 + κ , n κ 1 − 2 p + 1 p , p ≥ 2 , − 2 < γ < 0 , B n , j 1 ( z ) ≔ n κ , j = 1 , 2 , … , m ,

if z ∈ Ω ^ ( δ ) .

Theorem 3

Let 1 < p < 2 , G be a κ -quasidisk for some 0 ≤ κ < 1 and h ( z ) be defined by (1). Then, for any P n ∈ ℘ n , n ∈ N , and every m = 1 , 2 , … , we have:

(12) ∣ P n ( m ) ( z ) ∣ ≤ c 2 ∣ Φ n + 1 ( z ) ∣ ∥ P n ∥ p d ( z , L ) A n , p 2 ( z , m ) + ∑ j = 1 m C m j B n , j 1 ( z ) ∣ P n ( m − j ) ( z ) ∣ ,

where c 2 = c 2 ( G , γ , m , p ) > 0 constant independent from n and z ;

A n , p 2 ( z , m ) ≔ n γ ∗ + 2 p + m − 1 ( 1 + κ ) ,

if z ∈ Ω ( δ ) ,

A n , p 2 ( z , m ) ≔ n ( γ + 2 p + m − 1 ) ( 1 + κ ) , 1 < p < 2 , γ ≥ 1 1 + κ , n ( γ + 2 p + m − 1 ) ( 1 + κ ) , p < 1 + γ ( 1 + κ ) , 0 < γ < 1 1 + κ , n κ 2 p − 1 + 1 p ( ln n ) 1 − 1 p , p = 1 + γ ( 1 + κ ) , 0 < γ < 1 1 + κ , n κ 2 p − 1 + 1 p , p > 1 + γ ( 1 + κ ) , 0 < γ < 1 1 + κ , n κ 2 p − 1 + 1 p , 1 < p < 2 , − 2 < γ ≤ 0 ,

if z ∈ Ω ^ ( δ ) , γ ∗ ≔ max { 0 ; γ } , and B n , j 1 ( z ) defined as in Theorem 2.

2.2 Estimate for ∣ P n ( z ) ∣

As can be seen from (12), we also need an estimate for ∣ P n ( z ) ∣ as follows.

Theorem 4

Let p > 1 , G be a κ -quasidisk for some 0 ≤ κ < 1 and h ( z ) be defined by (1). Then, for any P n ∈ ℘ n , n ∈ N , and z ∈ Ω R , we have:

(13) ∣ P n ( z ) ∣ ≤ c 3 ∣ Φ n + 1 ( z ) ∣ d ( z , L ) A n , p 3 ∥ P n ∥ p ,

where c 3 = c 3 ( G , γ , p ) > 0 constant independent from n and z ;

A n , p 3 ≔ n ( γ + 2 p − 1 ) ( 1 + κ ) , 2 ≤ p < 1 2 + κ + ( 2 + γ ) 1 + κ 2 + κ , γ ≥ 1 1 + κ , ( n ln n ) 1 − 1 p , p = 1 2 + κ + ( 2 + γ ) 1 + κ 2 + κ , γ ≥ 1 1 + κ , n 1 − 1 p , p > 1 2 + κ + ( 2 + γ ) 1 + κ 2 + κ , γ ≥ 1 1 + κ , n 1 − 1 p , p ≥ 2 , 0 ≤ γ < 1 1 + κ , n κ 1 − 2 p + 1 p , p ≥ 2 , − 2 < γ < 0 ,

for any p ≥ 2 and

A n , p 3 ≔ n γ + 2 p − 1 ( 1 + κ ) , 1 < p < 2 , γ ≥ 1 1 + κ , n γ + 2 p − 1 ( 1 + κ ) , 1 < p < 1 + γ ( 1 + κ ) , 0 < γ < 1 1 + κ , n κ 2 p − 1 + 1 p ( ln n ) 1 − 1 p , p = 1 + γ ( 1 + κ ) , 0 < γ < 1 1 + κ , n κ 2 p − 1 + 1 p , 1 + γ ( 1 + κ ) < p < 2 , 0 ≤ γ < 1 1 + κ , n κ 2 p − 1 + 1 p , 1 < p < 2 , − 2 < γ < 0 ,

for 1 < p < 2 .

We note that the estimate for ∣ P n ( z ) ∣ was previously received by us in [32, Theorem 3] for p > 0 . But this result is better for p ≥ 2 and coincides with that [32, Theorem 3] for 1 < p < 2 .

2.3 Estimate for ∣ P n ′ ( z ) ∣

Now, using Theorems 2 and 3, we can give an estimate for the ∣ P n ′ ( z ) ∣ for z ∈ Ω R .

Theorem 5

Let p ≥ 2 , G be a κ -quasidisk for some 0 ≤ κ < 1 and h ( z ) be defined by (1). Then, for any P n ∈ ℘ n , n ∈ N , we have:

(14) ∣ P n ′ ( z ) ∣ ≤ c 4 ∣ Φ 2 ( n + 1 ) ( z ) ∣ d ( z , L ) ∥ P n ∥ p A n , p 4 ( z ) ,

where c 4 = c 4 ( G , γ , p ) > 0 constant independent from n and z ;

A n , p 4 ( z ) ≔ n γ + 2 p ( 1 + κ ) , 2 ≤ p < 1 2 + κ + ( 2 + γ ) 1 + κ 2 + κ , γ ≥ 1 1 + κ , n 2 − 1 p + κ ( ln n ) 1 − 1 p , p = 1 2 + κ + ( 2 + γ ) 1 + κ 2 + κ , γ ≥ 1 1 + κ , n 2 − 1 p + κ , p > 1 2 + κ + ( 2 + γ ) 1 + κ 2 + κ , γ ≥ 1 1 + κ , n 2 − 1 p + κ , p ≥ 2 , 0 ≤ γ < 1 1 + κ , n 1 + 1 p + 2 − 2 p κ , p ≥ 2 , − 2 < γ < 0 ,

if z ∈ Ω ( δ ) , and

A n , p 4 ( z ) ≔ n γ + 2 p ( 1 + κ ) , 2 ≤ p < 1 2 + κ + ( 2 + γ ) 1 + κ 2 + κ , γ ≥ 1 1 + κ , n 1 + κ − 1 p ( ln n ) 1 − 1 p , p = 1 2 + κ + ( 2 + γ ) 1 + κ 2 + κ , γ ≥ 1 1 + κ , n 1 + κ − 1 p , p > 1 2 + κ + ( 2 + γ ) 1 + κ 2 + κ , γ ≥ 1 1 + κ , n 1 + κ − 1 p , p ≥ 2 , 0 ≤ γ < 1 1 + κ , n κ 2 − 2 p + 1 p , p ≥ 2 , − 2 < γ < 0 ,

if z ∈ Ω ^ ( δ ) .

Theorem 6

Let 1 < p < 2 , G be a κ -quasidisk for some 0 ≤ κ < 1 and h ( z ) be defined by (1). Then, for any P n ∈ ℘ n , n ∈ N , we have:

(15) ∣ P n ′ ( z ) ∣ ≤ c 5 ∣ Φ 2 ( n + 1 ) ( z ) ∣ d ( z , L ) ∥ P n ∥ p A n , p 5 ( z ) ,

where c 5 = c 5 ( G , γ , p ) > 0 constant independent from n and z ;

A n , p 5 ( z ) ≔ n γ + 2 p ( 1 + κ ) , 1 < p < 2 , γ ≥ 1 1 + κ , n γ + 2 p ( 1 + κ ) , 1 < p < 1 + γ ( 1 + κ ) , 0 < γ < 1 1 + κ , n κ 2 p − 1 + 1 p + 1 + κ , 1 + γ ( 1 + κ ) ≤ p < 2 , 0 ≤ γ < 1 1 + κ , n κ 2 p − 1 + 1 p + 1 + κ , 1 < p < 2 , − 2 < γ < 0 ,

if z ∈ Ω ( δ ) , and

A n , p 5 ( z ) = n γ + 2 p ( 1 + κ ) − 1 , 1 < p < 2 , γ ≥ 1 1 + κ , n γ + 2 p ( 1 + κ ) − 1 , 1 < p < 1 + γ ( 1 + κ ) , 0 < γ < 1 1 + κ , n κ 2 p − 1 + 1 p + κ ( ln n ) 1 − 1 p , p = 1 + γ ( 1 + κ ) , 0 < γ < 1 1 + κ , n κ 2 p − 1 + 1 p + κ , 1 + γ ( 1 + κ ) < p < 2 , 0 ≤ γ < 1 1 + κ , n κ 2 p − 1 + 1 p + κ , 1 < p < 2 , − 2 < γ < 0 ,

if z ∈ Ω ^ ( δ ) .

2.4 Estimate for ∣ P n ″ ( z ) ∣

Considering the estimates obtained in Theorems 5 and 6 for ∣ ∣ P n ′ ( z ) ∣ and Theorem 4 for ∣ P n ( z ) ∣ in Theorems 2 and 3, respectively, we can obtain the following.

Theorem 7

Let p ≥ 2 , G be a κ -quasidisk for some 0 ≤ κ < 1 and h ( z ) be defined by (1). Then, for any P n ∈ ℘ n , n ∈ N , we have:

(16) ∣ P n ″ ( z ) ∣ ≤ c 6 ∣ Φ 3 ( n + 1 ) ( z ) ∣ d ( z , L ) ∥ P n ∥ p A n , p 6 ( z ) ,

where c 6 = c 6 ( G , γ , p ) > 0 constant independent from n and z ;

A n , p 6 ( z ) ≔ n γ + 2 p + 1 ( 1 + κ ) , 2 ≤ p < 1 2 + κ + ( 2 + γ ) 1 + κ 2 + κ , γ ≥ 1 1 + κ , n 1 − 1 p + 2 ( 1 + κ ) ( ln n ) 1 − 1 p , p = 1 2 + κ + ( 2 + γ ) 1 + κ 2 + κ , γ ≥ 1 1 + κ , n 1 − 1 p + 2 ( 1 + κ ) , p > 1 2 + κ + ( 2 + γ ) 1 + κ 2 + κ , γ ≥ 1 1 + κ , n 1 − 1 p + 2 ( 1 + κ ) , 2 ≤ p < 1 + ( 2 + γ ) ( 1 + κ ) , 0 ≤ γ < 1 1 + κ , n 2 + 1 p + 3 − 2 p κ , p ≥ 2 , − 2 < γ < 0 ,

if z ∈ Ω ( δ ) , and

A n , p 6 ( z ) ≔ n γ + 2 p ( 1 + κ ) + κ , 2 ≤ p < 1 2 + κ + ( 2 + γ ) 1 + κ 2 + κ , γ ≥ 1 1 + κ , n 1 + 2 κ − 1 p ( ln n ) 1 − 1 p , p = 1 2 + κ + ( 2 + γ ) 1 + κ 2 + κ , γ ≥ 1 1 + κ , n 1 + 2 κ − 1 p , p > 1 2 + κ + ( 2 + γ ) 1 + κ 2 + κ , γ ≥ 1 1 + κ , n 1 + 2 κ − 1 p , p ≥ 2 , 0 ≤ γ < 1 1 + κ , n κ 2 − 2 p + 1 p + κ , p ≥ 2 , − 2 < γ < 0 ,

if z ∈ Ω ^ ( δ ) .

Theorem 8

Let 1 < p < 2 , G be a κ -quasidisk for some 0 ≤ κ < 1 and h ( z ) be defined by (1). Then, for any P n ∈ ℘ n , n ∈ N , we have:

(17) ∣ P n ″ ( z ) ∣ ≤ c 7 ∣ Φ 3 ( n + 1 ) ( z ) ∣ d ( z , L ) ∥ P n ∥ p A n , p 7 ( z ) ,

where c 7 = c 7 ( G , γ , p ) > 0 constant independent from n and z ;

A n , p 7 ( z ) ≔ n γ + 2 p + 1 ( 1 + κ ) , 1 < p < 2 , γ ≥ 1 1 + κ , n γ + 2 p + 1 ( 1 + κ ) , 1 < p < 1 + γ ( 1 + κ ) , 0 < γ < 1 1 + κ , n κ 2 p + 1 + 1 p + 2 ( ln n ) 1 − 1 p , p = 1 + γ ( 1 + κ ) , 0 < γ < 1 1 + κ , n κ 2 p + 1 + 1 p + 2 , 1 + γ ( 1 + κ ) < p < 2 , 0 ≤ γ < 1 1 + κ , n κ 2 p + 1 + 1 p + 2 , 1 < p < 2 , − 2 < γ < 0 ,

if z ∈ Ω ( δ ) , and

A n , p 7 ( z ) ≔ n γ + 2 p + 1 ( 1 + κ ) , 1 < p < 2 , γ ≥ 1 1 + κ , n γ + 2 p + 1 ( 1 + κ ) , p < 1 + γ ( 1 + κ ) , 0 < γ < 1 1 + κ , n κ 2 p − 1 + 1 p + 2 κ ( ln n ) 1 − 1 p , p = 1 + γ ( 1 + κ ) , 0 < γ < 1 1 + κ , n κ 2 p − 1 + 1 p + 2 κ , p > 1 + γ ( 1 + κ ) , 0 < γ < 1 1 + κ , n κ 2 p − 1 + 1 p + 2 κ , 1 < p < 2 , − 2 < γ ≤ 0 ,

if z ∈ Ω ^ ( δ ) .

2.5 Estimates for ∣ P n ′ ( z ) ∣ and ∣ P n ″ ( z ) ∣ in whole plane

According to (4) (applied to the polynomial Q n − 1 ( z ) ≔ P n ′ ( z ) ), the estimation (9) is true also for the points z ∈ G ¯ R , R = 1 + ε 0 n − 1 , with a different constant. Therefore, combining estimation (9) (for the z ∈ G ¯ R ) with (14), (15), (16), and (17), we will obtain estimation on the growth of ∣ P n ′ ( z ) ∣ and ∣ P n ″ ( z ) ∣ , respectively, in the whole complex plane.

Theorem 9

Let p ≥ 2 , G be a κ -quasidisk for some 0 ≤ κ < 1 and h ( z ) be defined by (1). Then, for any P n ∈ ℘ n , n ∈ N , we have:

∣ P n ′ ( z ) ∣ ≤ c 8 ∥ P n ∥ p n γ + 2 p + 1 ( 1 + κ ) , z ∈ G ¯ R , ∣ Φ 2 ( n + 1 ) ( z ) ∣ d ( z , L ) A n , p 4 ( z ) , z ∈ Ω R ,

where c 8 = c 8 ( G , γ , p ) > 0 constant independent from n and z ; A n , p 4 ( z ) defined as in Theorem 5 for all z ∈ Ω R .

Theorem 10

Let 1 < p < 2 , G be a κ -quasidisk for some 0 ≤ κ < 1 and h ( z ) be defined by (1). Then, for any P n ∈ ℘ n , n ∈ N , we have:

∣ P n ′ ( z ) ∣ ≤ c 9 ∥ P n ∥ p n γ + 2 p + 1 ( 1 + κ ) , z ∈ G ¯ R , ∣ Φ 2 ( n + 1 ) ( z ) ∣ d ( z , L ) A n , p 5 ( z ) , z ∈ Ω R ,

where c 9 = c 9 ( G , γ , p ) > 0 constant independent from n and z ; A n , p 5 ( z ) defined as in Theorem 6 for all z ∈ Ω R .

Theorem 11

Let p ≥ 2 , G be a κ -quasidisk for some 0 ≤ κ < 1 and h ( z ) be defined by (1). Then, for any P n ∈ ℘ n , n ∈ N , we have:

∣ P n ″ ( z ) ∣ ≤ c 10 ∥ P n ∥ p n γ + 2 p + 2 ( 1 + κ ) , z ∈ G ¯ R , ∣ Φ 3 ( n + 1 ) ( z ) ∣ d ( z , L ) A n , p 6 ( z ) , z ∈ Ω R ,

where c 10 = c 10 ( G , γ , p ) > 0 constant independent from n and z ; A n , p 6 ( z ) defined as in Theorem 7 for all z ∈ Ω R .

Theorem 12

Let 1 < p < 2 , G be a κ -quasidisk for some 0 ≤ κ < 1 and h ( z ) be defined by (1). Then, for any P n ∈ ℘ n , n ∈ N , we have:

∣ P n ″ ( z ) ∣ ≤ c 11 ∥ P n ∥ p n γ + 2 p + 2 ( 1 + κ ) , z ∈ G ¯ R , ∣ Φ 3 ( n + 1 ) ( z ) ∣ d ( z , L ) A n , p 7 ( z ) , z ∈ Ω R ,

where c 11 = c 11 ( G , γ , p ) > 0 constant independent from n and z ; A n , p 7 ( z ) defined as in Theorem 8 for all z ∈ Ω R .

Thus, using Theorems 2 and 3 and estimating the ∣ P n ( m ) ( z ) ∣ sequentially for each m ≥ 3 and combining the obtained estimates with Theorem A, we obtain estimates for the ∣ P n ( m ) ( z ) ∣ on the whole complex plane.

3 Some auxiliary results

Throughout this paper, we denote “ a ≼ b ” and “ a ≍ b ” are equivalent to a ≤ b and c 1 a ≤ b ≤ c 2 a for some constants c , c 1 , c 2 , respectively.

Lemma 1

[36] Let G be a quasidisk, z 1 ∈ L , z 2 , z 3 ∈ Ω ∩ { z : ∣ z − z 1 ∣ ≼ d ( z 1 , L r 0 ) } , w j = Φ ( z j ) , j = 1 , 2 , 3 . Then,

  1. The statements ∣ z 1 − z 2 ∣ ≼ ∣ z 1 − z 3 ∣ and ∣ w 1 − w 2 ∣ ≼ ∣ w 1 − w 3 ∣ are equivalent. So they are ∣ z 1 − z 2 ∣ ≍ ∣ z 1 − z 3 ∣ and ∣ w 1 − w 2 ∣ ≍ ∣ w 1 − w 3 ∣ .

  2. If ∣ z 1 − z 2 ∣ ≼ ∣ z 1 − z 3 ∣ , then,

    w 1 − w 3 w 1 − w 2 c 1 ≼ z 1 − z 3 z 1 − z 2 ≼ w 1 − w 3 w 1 − w 2 c 2 ,

    where 0 < r 0 < 1 is a constant, depending on G and κ .

Lemma 2

Let G be a κ -quasidisk for some 0 ≤ κ < 1 . Then,

∣ w 1 − w 2 ∣ 1 − κ ≽ ∣ Ψ ( w 1 ) − Ψ ( w 2 ) ∣ ≽ ∣ w 1 − w 2 ∣ 1 + κ ,

for all w 1 , w 2 ∈ Ω ¯ ′ .

This fact follows from an appropriate result for the mapping f ∈ ∑ ( κ ) [5, p. 287] and estimation for the Ψ ′ [37, Theorem 2.8]:

(18) ∣ Ψ ′ ( τ ) ∣ ≍ d ( Ψ ( τ ) , L ) ∣ τ ∣ − 1 .

Lemma 3

[38] Let L be a κ -quasidisk for some 0 ≤ κ < 1 ; R = 1 + c n . Then, for any fixed ε ∈ ( 0 , 1 ) , there exist a level curve L 1 + ε ( R − 1 ) such that the following holds for any polynomial P n ( z ) ∈ ℘ n , n ∈ N :

∥ P n ∥ ℒ p h ∣ Φ ′ ∣ , L 1 + ε ( R − 1 ) ≼ n 1 p ∥ P n ∥ p , p > 0 .

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

Lemma 4

[19] Let G be a quasidisk and P n ( z ) , deg P n ≤ n , n = 1 , 2 , … , is arbitrary polynomial and weight function h ( z ) satisfied the condition (1). Then, for any R > 1 , p > 0 and n = 1 , 2 , … ,

∥ P n ∥ A p ( h , G R ) ≤ c 3 ( 1 + c ( R − 1 ) ) n + 1 p ∥ P n ∥ A p ( h , G ) ,

where c , and c 3 are independent of n and G .

This fact shows that the order of norms ∥ P n ∥ A p ( h , G 1 + c / n ) and ∥ P n ∥ A p ( h , G ) for arbitrary polynomials P n ( z ) have the same growth order.

4 Proof of theorems

Proof of Theorems 2 and 3

The proofs of Theorems 2 and 3 are as follows. Let G ∈ Q ( κ ) for some 0 ≤ κ < 1 and let R = 1 + 1 n , R 1 ≔ 1 + R − 1 2 . For z ∈ Ω , let us set:

H n ( z ) ≔ P n ( z ) Φ n + 1 ( z ) .

Let us represent the m th derivative of H n , p ( z ) as follows:

H n ( m ) ( z ) ≔ P n ( z ) Φ n + 1 ( z ) ( m ) = ∑ j = 0 m C m j 1 Φ n + 1 ( z ) ( j ) P n ( m − j ) ( z ) = P n ( m ) ( z ) Φ n + 1 ( z ) + ∑ j = 1 m C m j 1 Φ n + 1 ( z ) ( j ) P n ( m − j ) ( z ) ,

where C m j = m j . Therefore,

P n ( m ) ( z ) = Φ n + 1 ( z ) P n ( z ) Φ n + 1 ( z ) ( m ) − ∑ j = 1 m C m j 1 Φ n + 1 ( z ) ( j ) P n ( m − j ) ( z ) , z ∈ Ω .

Then,

(19) ∣ P n ( m ) ( z ) ∣ ≤ ∣ Φ n + 1 ( z ) ∣ P n ( z ) Φ n + 1 ( z ) ( m ) + ∑ j = 1 m C m j 1 Φ n + 1 ( z ) ( j ) ∣ P n ( m − j ) ( z ) ∣ .

As can be seen from (19), the following three statements on the right side need to be evaluated for z ∈ Ω to obtain the evaluation for ∣ P n ( m ) ( z ) ∣ :

(A) P n ( z ) Φ n + 1 ( z ) ( m ) ; (B) 1 Φ n + 1 ( z ) ( j ) ; (C) ∣ P n ( m − j ) ( z ) ∣ .

Now let us start the evaluations in order.

(A) Since the function H n ( z ) ≔ P n ( z ) Φ n + 1 ( z ) , H n ( ∞ ) = 0 , is analytic in Ω , continuous on Ω ¯ , then Cauchy integral representation for the region Ω R 1 gives

H n ( m ) ( z ) = − 1 2 π i ∫ L R 1 H n ( ζ ) d ζ ( ζ − z ) m + 1 , z ∈ Ω R , m ≥ 1 .

Then,

(20) P n ( z ) Φ n + 1 ( z ) ( m ) ≤ 1 2 π ∫ L R 1 P n ( ζ ) Φ n + 1 ( ζ ) ∣ d ζ ∣ ∣ ζ − z ∣ m + 1 ≤ 1 2 π d ( z , L R 1 ) ∫ L R 1 ∣ P n ( ζ ) ∣ ∣ d ζ ∣ ∣ ζ − z ∣ m .

We denote

(21) A n ( z ) ≔ ∫ L R 1 ∣ P n ( ζ ) ∣ ∣ d ζ ∣ ∣ ζ − z ∣ m ,

and estimate these integrals separately.

For this, we give some notations.

Let w j ≔ Φ ( z j ) , φ j ≔ arg w j . Without loss of generality, we will assume that φ l < 2 π . For η j = min t ∈ ∂ Φ ( Ω ( z j , δ j ) ) ∣ t − w j ∣ > 0 and η ≔ min { η j , j = 1 , l ¯ } , let us set:

Δ j ( η j ) ≔ { t : ∣ t − w j ∣ ≤ η j } ⊂ Φ ( Ω ( z j , δ j ) ) , Δ ( η ) ≔ ⋃ j = 1 l Δ j ( η ) , Δ ^ j = Δ \ Δ ( η j ) ; Δ ^ ( η ) ≔ Δ \ Δ ( η ) ; Δ 1 ′ ≔ Δ 1 ′ ( 1 ) , Δ 1 ′ ( ρ ) ≔ t = R ⋅ e i θ : R ≥ ρ > 1 , φ 0 + φ 1 2 ≤ θ < φ 1 + φ 2 2 , Δ j ′ ≔ Δ j ′ ( 1 ) , Δ j ′ ( ρ ) ≔ t = R e i θ : R ≥ ρ > 1 , φ j − 1 + φ j 2 ≤ θ < φ j + φ 0 2 , j = 2 , 3 … , l ,

where φ 0 = 2 π − φ l ; Ω j ≔ Ψ ( Δ j ′ ) , L R 1 j ≔ L R 1 ∩ Ω j ; Ω = ⋃ j = 1 l Ω j .

For simplicity, only one “critical” point z 1 on the boundary is taken for h ( z ) , i.e., l = 1 . To estimate A n ( z ) , first by replacing the variable τ = Φ ( ζ ) and multiplying the numerator and denominator of the integrant by ∣ Ψ ( τ ) − Ψ ( w 1 ) ∣ γ p ∣ Ψ ′ ( τ ) ∣ 2 p and applying the Hölder inequality, we obtain:

A n ( z ) = ∫ L R 1 ∣ P n ( ζ ) ∣ ∣ d ζ ∣ ∣ ζ − z ∣ m = ∑ i = 1 2 ∫ F R 1 i ∣ Ψ ( τ ) − Ψ ( w 1 ) ∣ γ j p P n ( Ψ ( τ ) ) ( Ψ ′ ( τ ) ) 2 p ∣ Ψ ′ ( τ ) ∣ 1 − 2 p ∣ Ψ ( τ ) − Ψ ( w 1 ) ∣ γ p ∣ Ψ ( τ ) − Ψ ( w ) ∣ m ∣ d τ ∣ ≤ ∑ i = 1 2 ∫ F R 1 i ∣ Ψ ( τ ) − Ψ ( w 1 ) ∣ γ ∣ P n ( Ψ ( τ ) ) ∣ p ∣ Ψ ′ ( τ ) ∣ 2 ∣ d τ ∣ 1 p × ∫ F R 1 i ∣ Ψ ′ ( τ ) ∣ 1 − 2 p ∣ Ψ ( τ ) − Ψ ( w 1 ) ∣ γ p ∣ Ψ ( τ ) − Ψ ( w ) ∣ m q ∣ d τ ∣ 1 q ≕ ∑ i = 1 2 A n i ( z ) ,

where F R 1 1 ≔ Φ ( L R 1 1 ) = Δ 1 ′ ∩ { τ : ∣ τ ∣ = R 1 } , F R 1 2 ≔ Φ ( L R 1 ) \ F R 1 1 and

A n i ( z ) ≔ ∫ F R 1 i ∣ f n , p ( τ ) ∣ p ∣ d τ ∣ 1 p ∫ F R 1 i ∣ Ψ ′ ( τ ) ∣ 2 − q ∣ Ψ ( τ ) − Ψ ( w 1 ) ∣ γ ( q − 1 ) ∣ Ψ ( τ ) − Ψ ( w ) ∣ q m ∣ d τ ∣ 1 q ≕ J n , 1 i ⋅ J n , 2 i ( z ) , f n , p ( τ ) ≔ ( Ψ ( τ ) − Ψ ( w 1 ) ) γ p P n ( Ψ ( τ ) ) ( Ψ ′ ( τ ) ) 2 p , ∣ τ ∣ = R 1 .

By applying Lemma 3, we obtain:

J n , 1 i ≼ n 1 p ∥ P n ∥ p , i = 1 , 2 .

For the estimation of the integral

(22) ( J n , 2 i ( z ) ) q = ∫ F R 1 i ∣ Ψ ′ ( τ ) ∣ 2 − q ∣ Ψ ( τ ) − Ψ ( w 1 ) ∣ γ ( q − 1 ) ∣ Ψ ( τ ) − Ψ ( w ) ∣ q m ∣ d τ ∣

for i = 1 , 2 , we set:

(23) E R 1 11 ≔ { τ : τ ∈ F R 1 1 , ∣ τ − w 1 ∣ < c 1 ( R 1 − 1 ) } ,

E R 1 12 ≔ { τ : τ ∈ F R 1 1 , c 1 ( R 1 − 1 ) ≤ ∣ τ − w 1 ∣ < η } , E R 1 13 ≔ { τ : τ ∈ Φ ( L R 1 ) , ∣ τ − w 1 ∣ ≥ η } ,

where 0 < c 1 < η is chosen so that { τ : ∣ τ − w 1 ∣ < c 1 ( R 1 − 1 ) } ∩ Δ ≠ ∅ and Φ ( L R 1 ) = ⋃ k = 1 3 E R 1 1 k . Considering these notations, from (22), we have:

J n , 2 1 ( z ) + J n , 2 2 ( z ) ≕ J 2 ( z ) = J 2 ( E R 1 11 ) + J 2 ( E R 1 12 ) + J 2 ( E R 1 13 ) ≕ J 2 1 ( z ) + J 2 2 ( z ) + J 2 3 ( z )

and, consequently,

(24) A n ( z ) = A n 1 ( z ) + A n 2 ( z ) ≼ n 1 p ∥ P n ∥ p ⋅ ( J 2 1 ( z ) + J 2 2 ( z ) + J 2 3 ( z ) ) ≕ A n , 1 ( z ) + A n , 2 ( z ) + A n , 3 ( z ) ,

where

A n , k ( z ) ≔ n 1 p ∥ P n ∥ p ⋅ J 2 k ( z ) , k = 1 , 2 , 3 , ( J 2 k ( z ) ) q ≔ ∫ E R 1 1 k ∣ Ψ ′ ( τ ) ∣ 2 − q ∣ d τ ∣ ∣ Ψ ( τ ) − Ψ ( w 1 ) ∣ γ ( q − 1 ) ∣ Ψ ( τ ) − Ψ ( w ) ∣ q m , k = 1 , 2 , 3 .

For any k = 1 , 2 , we denote

(25) E R 1 , 1 1 k ≔ { τ ∈ E R 1 1 k : ∣ Ψ ( τ ) − Ψ ( w 1 ) ∣ ≥ ∣ Ψ ( τ ) − Ψ ( w ) ∣ } , E R 1 , 2 1 k ≔ E R 1 1 k \ E R 1 , 1 1 k , ( I ( E R 1 , 1 1 k ) ) q ≔ ∫ E R 1 , 1 1 k ∣ Ψ ′ ( τ ) ∣ 2 − q ∣ d τ ∣ ∣ Ψ ( τ ) − Ψ ( w ) ∣ γ ( q − 1 ) + q m , if γ ≥ 0 , ∫ E R 1 , 1 1 k ∣ Ψ ( τ ) − Ψ ( w 1 ) ∣ ( − γ ) ( q − 1 ) ∣ Ψ ′ ( τ ) ∣ 2 − q ∣ d τ ∣ ∣ Ψ ( τ ) − Ψ ( w ) ∣ q m , if γ < 0 , ( I ( E R 1 , 2 1 k ) ) q ≔ ∫ E R 1 , 2 1 k ∣ Ψ ′ ( τ ) ∣ 2 − q ∣ d τ ∣ ∣ Ψ ( τ ) − Ψ ( w 1 ) ∣ γ ( q − 1 ) + q m , k = 1 , 2 ,

and estimate the last integrals.

Given the possible values q ( q > 2 and q < 2 ) and γ ( − 2 < γ < 0 and γ ≥ 0 ), we will consider the cases separately.

Case 1. Let 1 < q ≤ 2 ( p ≥ 2 ) . Then,

( I ( E R 1 , 1 1 k ) ) q = ∫ E R 1 , 1 1 k ∣ Ψ ′ ( τ ) ∣ 2 − q ∣ d τ ∣ ∣ Ψ ( τ ) − Ψ ( w 1 ) ∣ γ ( q − 1 ) ∣ Ψ ( τ ) − Ψ ( w ) ∣ q m .

1.1. Let γ ≥ 0 . If z ∈ Ω ( δ ) , applying Lemma 2 to (18), we get:

(26) ( I ( E R 1 , 1 11 ) ) q ≤ ∫ E R 1 , 1 11 ∣ Ψ ′ ( τ ) ∣ 2 − q ∣ d τ ∣ ∣ Ψ ( τ ) − Ψ ( w ) ∣ γ ( q − 1 ) + q m ≍ ∫ E R 1 , 1 11 d ( Ψ ( τ ) , L ) ∣ τ ∣ − 1 2 − q ∣ d τ ∣ ∣ Ψ ( τ ) − Ψ ( w ) ∣ γ ( q − 1 ) + q m ≼ n ( 2 − q ) ∫ E R 1 , 1 11 ∣ d τ ∣ ∣ τ − w ∣ [ γ ( q − 1 ) + q m − ( 2 − q ) ] ( 1 + κ ) ≼ n ( 2 − q ) + [ γ ( q − 1 ) + q m − ( 2 − q ) ] ( 1 + κ ) mes E R 1 , 1 11 ≼ n ( 2 − q ) + [ γ ( q − 1 ) + q m − ( 2 − q ) ] ( 1 + κ ) − 1 ,

I ( E R 1 , 1 11 ) ≼ n γ p + m ( 1 + κ ) − 1 + 1 − 2 p κ − 1 p ,

( I ( E R 1 , 2 11 ) ) q ≼ ∫ E R 1 , 2 11 d ( Ψ ( τ ) , L ) ∣ τ ∣ − 1 2 − q ∣ d τ ∣ ∣ Ψ ( τ ) − Ψ ( w 1 ) ∣ γ ( q − 1 ) + q m

≼ n ( 2 − q ) ∫ E R 1 , 2 11 ∣ d τ ∣ ∣ τ − w 1 ∣ [ γ ( q − 1 ) + q m − ( 2 − q ) ] ( 1 + κ ) ≼ n ( 2 − q ) + [ γ ( q − 1 ) + q m − ( 2 − q ) ] ( 1 + κ ) mes E R 1 , 2 11 ≼ n ( 2 − q ) + [ γ ( q − 1 ) + q m − ( 2 − q ) ] ( 1 + κ ) − 1 , I ( E R 1 , 2 11 ) ≼ n γ p + m ( 1 + κ ) − 1 + 1 − 2 p κ − 1 p ,

(27) ( I ( E R 1 , 1 12 ) ) q ≼ ∫ E R 1 , 1 12 ∣ Ψ ′ ( τ ) ∣ 2 − q ∣ d τ ∣ ∣ Ψ ( τ ) − Ψ ( w ) ∣ γ ( q − 1 ) + q m ≼ ∫ E R 1 , 1 12 d ( Ψ ( τ ) , L ) ∣ τ ∣ − 1 2 − q ∣ d τ ∣ ∣ Ψ ( τ ) − Ψ ( w ) ∣ γ ( q − 1 ) + q m ≼ n ( 2 − q ) ∫ E R 1 , 1 11 ∣ d τ ∣ ∣ τ − w ∣ [ γ ( q − 1 ) + q m − ( 2 − q ) ] ( 1 + κ ) ≼ n ( 2 − q ) n [ γ ( q − 1 ) + q m − ( 2 − q ) ] ( 1 + κ ) − 1 , [ γ ( q − 1 ) + q m − ( 2 − q ) ] ( 1 + κ ) > 1 , ln n , [ γ ( q − 1 ) + q m − ( 2 − q ) ] ( 1 + κ ) = 1 , 1 , [ γ ( q − 1 ) + q m − ( 2 − q ) ] ( 1 + κ ) < 1 , I ( E R 1 , 1 12 ) ≼ n γ p + m ( 1 + κ ) − 1 + 1 − 2 p κ − 1 p , γ > − 2 , m ≥ 2 , n γ p + m ( 1 + κ ) − 1 + 1 − 2 p κ − 1 p , p < 1 + ( 2 + γ ) ( 1 + κ ) , m = 1 , n 1 − 2 p ( ln n ) ( 1 − 1 p ) , p = 1 + ( 2 + γ ) ( 1 + κ ) , m = 1 , n 1 − 2 p , p > 1 + ( 2 + γ ) ( 1 + κ ) , m = 1 ,

(28) ( I ( E R 1 , 2 12 ) ) q ≤ ∫ E R 1 , 2 12 ∣ Ψ ′ ( τ ) ∣ 2 − q ∣ d τ ∣ ∣ Ψ ( τ ) − Ψ ( w 1 ) ∣ γ ( q − 1 ) + q m ≼ n ( 2 − q ) ∫ E R 1 , 2 12 ∣ d τ ∣ ∣ τ − w 1 ∣ [ γ ( q − 1 ) + q m − ( 2 − q ) ] ( 1 + κ )

≼ n ( 2 − q ) n [ γ ( q − 1 ) + q m − ( 2 − q ) ] ( 1 + κ ) − 1 , [ γ ( q − 1 ) + q m − ( 2 − q ) ] ( 1 + κ ) > 1 , ln n , [ γ ( q − 1 ) + q m − ( 2 − q ) ] ( 1 + κ ) = 1 , 1 , [ γ ( q − 1 ) + q m − ( 2 − q ) ] ( 1 + κ ) < 1 ,

I ( E R 1 , 2 12 ) ≼ n γ p + m ( 1 + κ ) − 1 + 1 − 2 p κ − 1 p , γ > − 2 , m ≥ 2 , n γ p + m ( 1 + κ ) − 1 + 1 − 2 p κ − 1 p , p < 1 + ( 2 + γ ) ( 1 + κ ) , m = 1 , n 1 − 2 p ( ln n ) 1 − 1 p , p = 1 + ( 2 + γ ) ( 1 + κ ) , m = 1 , n 1 − 2 p , p > 1 + ( 2 + γ ) ( 1 + κ ) , m = 1 ,

I ( E R 1 , 1 12 ) + I ( E R 1 , 2 12 ) ≼ n γ + 2 p + m ( 1 + κ ) − 1 + κ + 1 p , γ > − 2 , m ≥ 2 , n γ p + m ( 1 + κ ) − 1 + 1 − 2 p κ − 1 p , p < 1 + ( 2 + γ ) ( 1 + κ ) , m = 1 , n 1 − 2 p ( ln n ) 1 − 1 p , p = 1 + ( 2 + γ ) ( 1 + κ ) , m = 1 , n 1 − 2 p , p > 1 + ( 2 + γ ) ( 1 + κ ) , m = 1 .

For τ ∈ E R 1 13 , we see that η < ∣ τ − w 1 ∣ < 2 π R ˙ 1 , ∣ τ − w ∣ ≥ η − c 1 . Therefore, ∣ Ψ ( τ ) − Ψ ( w 1 ) ∣ ≽ 1 , from Lemma 1 and for ∣ τ − w 1 ∣ ≥ η , ∣ Ψ ( τ ) − Ψ ( w ) ∣ ≽ ∣ τ − w ∣ 1 + κ , from Lemma 2. Then, for w ∈ Δ ( w 1 , η ) , applying (18), we obtain:

( J 2 3 ( z ) ) q ≼ ∫ E R 1 13 ∣ Ψ ′ ( τ ) ∣ 2 − q ∣ d τ ∣ ∣ Ψ ( τ ) − Ψ ( w ) ∣ q m ≼ ∫ E R 1 13 d ( Ψ ( τ ) , L ) ∣ τ ∣ − 1 2 − q ∣ d τ ∣ ∣ Ψ ( τ ) − Ψ ( w ) ∣ q m ≼ n ( 2 − q ) ∫ E R 1 , 1 11 ∣ d τ ∣ ∣ τ − w ∣ [ q m − ( 2 − q ) ] ( 1 + κ ) ≼ n ( 2 − q ) n [ q m − ( 2 − q ) ] ( 1 + κ ) − 1 , [ q m − ( 2 − q ) ] ( 1 + κ ) > 1 , m ≥ 2 , n [ q m − ( 2 − q ) ] ( 1 + κ ) − 1 , [ q m − ( 2 − q ) ] ( 1 + κ ) > 1 , m = 1 , ln n , [ q m − ( 2 − q ) ] ( 1 + κ ) = 1 , m = 1 , 1 , [ q m − ( 2 − q ) ] ( 1 + κ ) < 1 , m = 1 ,

(29) J 2 3 ( z ) ≼ n 2 p + m ( 1 + κ ) − 1 + κ + 1 p , p ≥ 2 , m ≥ 2 , n 2 p + m ( 1 + κ ) − 1 + κ + 1 p , p < 1 + 2 ( 1 + κ ) , m = 1 , n 1 − 2 p ( ln n ) 1 − 1 p , p = 1 + 2 ( 1 + κ ) , m = 1 , n 1 − 2 p , p > 1 + 2 ( 1 + κ ) , m = 1 , z ∈ Ω ( δ ) , J 2 3 ( z ) ≼ n κ 1 − 2 p , z ∈ Ω ^ ( δ ) .

Combining (26)–(29), for p ≥ 2 , γ ≥ 0 and z ∈ Ω ( δ ) , we obtain:

(30) ∑ k = 1 3 J 2 k ( z ) ≼ n γ p + m ( 1 + κ ) − 1 + 1 − 2 p κ − 1 p , γ > − 2 , m ≥ 2 , n γ p + m ( 1 + κ ) − 1 + 1 − 2 p κ − 1 p , 2 ≤ p < 1 + ( 2 + γ ) ( 1 + κ ) , m = 1 , n 1 − 2 p ( ln n ) ( 1 − 1 p ) , p = 1 + ( 2 + γ ) ( 1 + κ ) , m = 1 , n 1 − 2 p , p > 1 + ( 2 + γ ) ( 1 + κ ) , m = 1 .

If z ∈ Ω ^ ( δ ) , then

(31) ( J 2 1 ( z ) ) q ≼ ∫ E R 1 11 ∣ Ψ ′ ( τ ) ∣ 2 − q ∣ d τ ∣ ∣ Ψ ( τ ) − Ψ ( w 1 ) ∣ γ ( q − 1 ) ≼ ∫ E R 1 11 d ( Ψ ( τ ) , L ) ∣ τ ∣ − 1 2 − q ∣ d τ ∣ ∣ τ − w 1 ∣ γ ( q − 1 ) ( 1 + κ ) ≼ n ( 2 − q ) ∫ E R 1 11 ∣ d τ ∣ ∣ τ − w 1 ∣ [ γ ( q − 1 ) − ( 2 − q ) ] ( 1 + κ ) ≼ n ( 2 − q ) + [ γ ( q − 1 ) − ( 2 − q ) ] ( 1 + κ ) mes E R 1 11 ≼ n ( 2 − q ) + [ γ ( q − 1 ) − ( 2 − q ) ] ( 1 + κ ) − 1 , J 2 1 ( z ) ≼ n γ p ( 1 + κ ) − 1 + 1 − 2 p κ − 1 p .

( J 2 2 ( z ) ) q ≼ ∫ E R 1 12 ∣ Ψ ′ ( τ ) ∣ 2 − q ∣ d τ ∣ ∣ Ψ ( τ ) − Ψ ( w 1 ) ∣ γ ( q − 1 ) ≼ ∫ E R 1 12 d ( Ψ ( τ ) , L ) ∣ τ ∣ − 1 2 − q ∣ d τ ∣ ∣ τ − w 1 ∣ γ ( q − 1 ) ( 1 + κ ) ≼ n ( 2 − q ) ∫ E R 1 11 ∣ d τ ∣ ∣ τ − w 1 ∣ [ γ ( q − 1 ) − ( 2 − q ) ] ( 1 + κ ) ≼ n ( 2 − q ) n [ γ ( q − 1 ) − ( 2 − q ) ] ( 1 + κ ) − 1 , [ γ ( q − 1 ) − ( 2 − q ) ] ( 1 + κ ) > 1 , ln n , [ γ ( q − 1 ) − ( 2 − q ) ] ( 1 + κ ) = 1 , 1 , [ γ ( q − 1 ) − ( 2 − q ) ] ( 1 + κ ) < 1 , J 2 2 ( z ) ≼ n γ p ( 1 + κ ) − 1 + 1 − 2 p κ − 1 p , 2 ≤ p < 1 2 + κ + ( γ + 2 ) 1 + κ 2 + κ , γ > 1 1 + κ , n 1 − 2 p ( ln n ) ( 1 − 1 p ) , p = 1 2 + κ + ( γ + 2 ) 1 + κ 2 + κ , γ > 1 1 + κ , n 1 − 2 p , p > 1 2 + κ + ( γ + 2 ) 1 + κ 2 + κ , γ > 1 1 + κ , n 1 − 2 p , p ≥ 2 , − 2 < γ ≤ 1 1 + κ ,

( J 2 3 ( z ) ) q ≼ ∫ E R 1 13 ∣ Ψ ′ ( τ ) ∣ 2 − q ∣ d τ ∣ ∣ Ψ ( τ ) − Ψ ( w 1 ) ∣ γ ( q − 1 ) ≼ ∫ E R 1 13 d ( Ψ ( τ ) , L ) ∣ τ ∣ − 1 2 − q ∣ d τ ∣ ≼ n κ ( 2 − q ) , J 2 3 ( z ) ≼ n κ 1 − 2 p .

From (24)–(31), we obtain:

(32) A n ( z ) ≼ ∥ P n ∥ p n γ + 2 p + m − 1 ( 1 + κ ) , γ > − 2 , p ≥ 2 , m ≥ 2 , n γ + 2 p + m − 1 ( 1 + κ ) , 2 ≤ p < 1 + ( 2 + γ ) ( 1 + κ ) , m = 1 , ( n ln n ) ( 1 − 1 p ) , p = 1 + ( 2 + γ ) ( 1 + κ ) , m = 1 , n ( 1 − 1 p ) , p > 1 + ( 2 + γ ) ( 1 + κ ) , m = 1 ,

if z ∈ Ω ( δ ) and

(33) A n ( z ) ≼ ∥ P n ∥ p n γ + 2 p + m − 1 ( 1 + κ ) , 2 ≤ p < 1 2 + κ + ( γ + 2 ) 1 + κ 2 + κ , γ > 1 1 + κ , ( n ln n ) 1 − 1 p , p = 1 2 + κ + ( γ + 2 ) 1 + κ 2 + κ , γ > 1 1 + κ , n 1 − 1 p , p > 1 2 + κ + ( γ + 2 ) 1 + κ 2 + κ , γ > 1 1 + κ , n 1 − 1 p , p ≥ 2 , − 2 < γ ≤ 1 1 + κ ,

if z ∈ Ω ^ ( δ ) .

1.2. If γ < 0 , for w ∈ Δ ( w 1 , η ) ∩ Ω R ( δ ) , such that ∣ Ψ ( τ ) − Ψ ( w 1 ) ∣ ≼ ∣ Ψ ( τ ) − Ψ ( w ) ∣ , according to Lemma 1, analogously we have:

(34) ( I ( E R 1 , 1 11 ) ) q ≼ ∫ E R 1 , 1 11 d ( Ψ ( τ ) , L ) ∣ τ ∣ − 1 2 − q ∣ Ψ ( τ ) − Ψ ( w 1 ) ∣ ( − γ ) ( q − 1 ) ∣ d τ ∣ ∣ Ψ ( τ ) − Ψ ( w ) ∣ q m ≼ n 2 − q ∫ E R 1 , 1 11 ∣ τ − w 1 ∣ ( − γ ) ( q − 1 ) ( 1 − κ ) ∣ τ − w ∣ [ q m − ( 2 − q ) ] ( 1 + κ ) ∣ d τ ∣ ≼ n 2 − q + [ q m − ( 2 − q ) ] ( 1 + κ ) mes E R 1 11 ≼ n 2 − q + [ q m − ( 2 − q ) ] ( 1 + κ ) − 1 , I ( E R 1 , 1 11 ) ≼ n m ( 1 + κ ) − 1 + 1 − 2 p κ − 1 p ,

(35) ( I ( E R 1 , 2 11 ) ) q ≍ ∫ E R 1 , 2 11 d ( Ψ ( τ ) , L ) ∣ τ ∣ − 1 2 − q ∣ d τ ∣ ∣ Ψ ( τ ) − Ψ ( w 1 ) ∣ γ ( q − 1 ) + q m ≼ n ( 2 − q ) ∫ E R 1 , 2 11 ∣ d τ ∣ ∣ τ − w 1 ∣ [ γ ( q − 1 ) + q m − ( 2 − q ) ] ( 1 + κ ) ≼ n ( 2 − q ) + [ γ ( q − 1 ) + q m − ( 2 − q ) ] ( 1 + κ ) mes E R 1 , 2 11 ≼ n ( 2 − q ) + [ γ ( q − 1 ) + q m − ( 2 − q ) ] ( 1 + κ ) − 1 , I ( E R 1 , 2 11 ) ≼ n γ p + m ( 1 + κ ) − 1 + 1 − 2 p κ − 1 p .

For τ ∈ E R 1 12 , we see that ∣ τ − w 1 ∣ < η and from Lemma 1, ∣ Ψ ( τ ) − Ψ ( w 1 ) ∣ ≼ 1 . Then, for w ∈ Δ ( w 1 , η ) ∩ Ω R ( δ ) , such that ∣ Ψ ( τ ) − Ψ ( w 1 ) ∣ ≼ ∣ Ψ ( τ ) − Ψ ( w ) ∣ , applying Lemma 2, we obtain:

(36) ( I ( E R 1 , 1 12 ) ) q = ∫ E R 1 , 1 12 Ψ ( τ ) − Ψ ( w 1 ) Ψ ( τ ) − Ψ ( w ) ( − γ ) ( q − 1 ) d ( Ψ ( τ ) , L ) ∣ τ ∣ − 1 2 − q ∣ d τ ∣ ∣ Ψ ( τ ) − Ψ ( w ) ∣ q m + γ ( q − 1 ) ≼ n 2 − q ∫ E R 1 , 1 12 τ − w 1 τ − w ( − γ ) ( q − 1 ) 1 c 1 ∣ d τ ∣ ∣ Ψ ( τ ) − Ψ ( w ) ∣ q m + γ ( q − 1 ) − ( q − 2 ) ≼ n 2 − q ∫ E R 1 , 1 12 ∣ d τ ∣ ∣ τ − w ∣ [ q m + γ ( q − 1 ) − ( 2 − q ) ] ( 1 + κ ) ≼ n ( 2 − q ) n [ q m + γ ( q − 1 ) − ( 2 − q ) ] ( 1 + κ ) − 1 , q m + γ ( q − 1 ) − ( 2 − q ) ( 1 + κ ) > 1 , ln n , q m + γ ( q − 1 ) − ( 2 − q ) ( 1 + κ ) = 1 , 1 , q m + γ ( q − 1 ) − ( 2 − q ) ( 1 + κ ) < 1 ,

I ( E R 1 , 1 12 ) ≼ n γ p + m ( 1 + κ ) − 1 + 1 − 2 p κ − 1 p , γ > − 2 , m ≥ 2 , γ > − 2 , n γ p + m ( 1 + κ ) − 1 + 1 − 2 p κ − 1 p , 2 ≤ p < 1 + ( 2 + γ ) ( 1 + κ ) , m = 1 , γ > − 2 + 1 1 + κ , n 1 − 2 p ( ln n ) ( 1 − 1 p ) , p = 1 + ( 2 + γ ) ( 1 + κ ) , m = 1 , γ > − 2 + 1 1 + κ , n 1 − 2 p , p > 1 + ( 2 + γ ) ( 1 + κ ) , m = 1 , γ > − 2 + 1 1 + κ , n 1 − 2 p , p ≥ 2 , m = 1 , − 2 < γ ≤ − 2 + 1 1 + κ ,

(37) ( I ( E R 1 , 2 12 ) ) q ≍ ∫ E R 1 , 2 12 d ( Ψ ( τ ) , L ) ∣ τ ∣ − 1 2 − q ∣ d τ ∣ ∣ Ψ ( τ ) − Ψ ( w ) ∣ q m + γ ( q − 1 ) ≼ n 2 − q ∫ E R 1 , 2 12 ∣ d τ ∣ ∣ τ − w ∣ [ q m + γ ( q − 1 ) − ( 2 − q ) ] ( 1 + κ ) ≼ n ( 2 − q ) n [ q m + γ ( q − 1 ) − ( 2 − q ) ] ( 1 + κ ) − 1 , q m + γ ( q − 1 ) − ( 2 − q ) ( 1 + κ ) > 1 , ln n , q m + γ ( q − 1 ) − ( 2 − q ) ( 1 + κ ) = 1 , 1 , q m + γ ( q − 1 ) − ( 2 − q ) ( 1 + κ ) < 1 ,

I ( E R 1 , 2 12 ) ≼ n γ p + m ( 1 + κ ) − 1 + 1 − 2 p κ − 1 p , γ > − 2 , m ≥ 2 , γ > − 2 , n γ p + m ( 1 + κ ) − 1 + 1 − 2 p κ − 1 p , p < 1 + ( 2 + γ ) ( 1 + κ ) , m = 1 , γ > − 2 + 1 1 + κ , n 1 − 2 p ( ln n ) 1 − 1 p , p = 1 + ( 2 + γ ) ( 1 + κ ) , m = 1 , γ > − 2 + 1 1 + κ , n 1 − 2 p , p > 1 + ( 2 + γ ) ( 1 + κ ) , m = 1 , γ > − 2 + 1 1 + κ , n 1 − 2 p , p ≥ 2 , m = 1 , − 2 < γ ≤ − 2 + 1 1 + κ .

For τ ∈ E R 1 13 and each w ∈ Δ ( w 1 , η ) ∩ Ω R ( δ ) , we see that η < ∣ τ − w 1 ∣ < 2 π R ˙ 1 . Therefore, from Lemma 1 and applying (18), we obtain:

(38) ( I ( E R 1 13 ) ) q ≔ ∫ E R 1 , 1 13 ∣ Ψ ( τ ) − Ψ ( w 1 ) ∣ ( − γ ) ( q − 1 ) ∣ Ψ ′ ( τ ) ∣ 2 − q ∣ d τ ∣ ∣ Ψ ( τ ) − Ψ ( w ) ∣ q m ≍ ∫ E R 1 13 d ( Ψ ( τ ) , L ) ∣ τ ∣ − 1 2 − q ∣ d τ ∣ ∣ Ψ ( τ ) − Ψ ( w ) ∣ q m ≼ n 2 − q ∫ E R 1 13 ∣ d τ ∣ ∣ τ − w ∣ [ q m − ( 2 − q ) ] ( 1 + κ ) ≼ n 1 − q + [ q m − ( 2 − q ) ] ( 1 + κ ) , I ( E R 1 13 ) ≼ n m ( 1 + κ ) − 1 + 1 − 2 p κ − 1 p .

Therefore, combining (34)–(38) in case of γ < 0 for z ∈ Ω ( δ ) , we have:

(39) ∑ k = 1 3 J 2 k ( z ) ≼ n m ( 1 + κ ) − 1 + 1 − 2 p κ − 1 p + n γ p + m ( 1 + κ ) − 1 + 1 − 2 p κ − 1 p , p ≥ 2 , m ≥ 2 , γ > − 2 , n γ p + m ( 1 + κ ) − 1 + 1 − 2 p κ − 1 p , p < 1 + ( 2 + γ ) ( 1 + κ ) , m = 1 , γ > − 2 + 1 1 + κ , n 1 − 2 p ( ln n ) 1 − 1 p , p = 1 + ( 2 + γ ) ( 1 + κ ) , m = 1 , γ > − 2 + 1 1 + κ , n 1 − 2 p , p > 1 + ( 2 + γ ) ( 1 + κ ) , m = 1 , γ > − 2 + 1 1 + κ , n 1 − 2 p , p ≥ 2 , m = 1 , − 2 < γ ≤ − 2 + 1 1 + κ , ≼ n m ( 1 + κ ) − 1 + 1 − 2 p κ − 1 p .

If z ∈ Ω ^ ( δ ) , then ∣ w − w 1 ∣ ≥ η , from Lemma 2 and (18), we obtain:

(40) ( J 2 1 ( z ) ) q = ∫ E R 1 11 ∣ Ψ ′ ( τ ) ∣ 2 − q ∣ d τ ∣ ∣ Ψ ( τ ) − Ψ ( w 1 ) ∣ γ ( q − 1 ) ∣ Ψ ( τ ) − Ψ ( w ) ∣ q m ≼ ∫ E R 1 11 d ( Ψ ( τ ) , L ) ∣ τ ∣ − 1 2 − q ∣ Ψ ( τ ) − Ψ ( w 1 ) ∣ ( − γ ) ( q − 1 ) ∣ d τ ∣ ≼ n κ ( 2 − q ) ∫ E R 1 11 1 n ( − γ ) ( q − 1 ) ( 1 − κ ) ∣ d τ ∣ ≼ n κ ( 2 − q ) mes E R 1 11 ≼ n κ ( 2 − q ) − 1 ≼ 1 , J 2 1 ( z ) ≼ 1 ,

( J 2 2 ( z ) ) q ≼ ∫ E R 1 12 d ( Ψ ( τ ) , L ) ∣ τ ∣ − 1 2 − q ∣ Ψ ( τ ) − Ψ ( w 1 ) ∣ ( − γ ) ( q − 1 ) ∣ d τ ∣ ≼ ∫ E R 1 12 d ( Ψ ( τ ) , L ) ∣ τ ∣ − 1 2 − q ∣ d τ ∣ ≼ n κ ( 2 − q ) , J 2 2 ( z ) ≼ n κ 1 − 2 p , ( J 2 3 ( z ) ) q ≼ ∫ E R 1 13 ∣ Ψ ′ ( τ ) ∣ 2 − q ∣ d τ ∣ ∣ Ψ ( τ ) − Ψ ( w 1 ) ∣ γ ( q − 1 ) ≼ ∫ E R 1 13 d ( Ψ ( τ ) , L ) ∣ τ ∣ − 1 2 − q ∣ d τ ∣ ≼ n κ ( 2 − q ) , J 2 3 ( z ) ≼ n κ 1 − 2 p .

Combining the last three estimates, in case of γ < 0 for z ∈ Ω ^ ( δ ) , we have:

(41) ∑ k = 1 3 J 2 k ( z ) ≼ n κ 1 − 2 p .

Then, for γ < 0 , from (39)–(41), we obtain:

∑ k = 1 3 J 2 k ( z ) ≼ n m ( 1 + κ ) − 1 + 1 − 2 p κ − 1 p , z ∈ Ω ( δ ) , n κ 1 − 2 p , z ∈ Ω ^ ( δ ) ,

and, consequently, in this case from (24), we have:

(42) A n ( z ) ≼ n 1 p ∥ P n ∥ p n m ( 1 + κ ) − 1 + 1 − 2 p κ − 1 p , z ∈ Ω ( δ ) , n κ 1 − 2 p , z ∈ Ω ^ ( δ ) .

Therefore, combining (30) and (42), for any γ > − 2 , p ≥ 2 , we obtain:

(43) A n ( z ) ≼ ∥ P n ∥ p n γ + 2 p + m − 1 ( 1 + κ ) , p ≥ 2 , m ≥ 2 , γ > − 2 , n γ + 2 p + m − 1 ( 1 + κ ) , 2 ≤ p < 1 + ( 2 + γ ) ( 1 + κ ) , m = 1 , γ ≥ 0 , ( n ln n ) 1 − 1 p , p = 1 + ( 2 + γ ) ( 1 + κ ) , m = 1 , γ ≥ 0 , n 1 − 1 p , p > 1 + ( 2 + γ ) ( 1 + κ ) , m = 1 , γ ≥ 0 , n 2 p + m − 1 ( 1 + κ ) , p ≥ 2 , m ≥ 1 , − 2 < γ < 0 ,

if z ∈ Ω ( δ ) , and

(44) A n ( z ) ≼ ∥ P n ∥ p n γ + 2 p + m − 1 ( 1 + κ ) , 2 ≤ p < 1 2 + κ + ( γ + 2 ) 1 + κ 2 + κ , γ > 1 1 + κ , ( n ln n ) 1 − 1 p , p = 1 2 + κ + ( γ + 2 ) 1 + κ 2 + κ , γ > 1 1 + κ , n 1 − 1 p , p > 1 2 + κ + ( γ + 2 ) 1 + κ 2 + κ , γ > 1 1 + κ , n 1 − 1 p , p ≥ 2 , 0 ≤ γ ≤ 1 1 + κ , n κ 1 − 2 p + 1 p , p ≥ 2 , − 2 < γ < 0 ,

if z ∈ Ω ^ ( δ ) .

Case 2. Let q > 2 ( p < 2 ) . Then, 2 − q < 0 , and so

(45) ( I ( E R 1 , 1 1 k ) ) q ≔ ∫ E R 1 , 1 1 k ∣ d τ ∣ ∣ Ψ ′ ( τ ) ∣ q − 2 ∣ Ψ ( τ ) − Ψ ( w ) ∣ γ ( q − 1 ) + q m , if γ ≥ 0 , ∫ E R 1 , 1 1 k ∣ Ψ ( τ ) − Ψ ( w 1 ) ∣ ( − γ ) ( q − 1 ) ∣ d τ ∣ ∣ Ψ ′ ( τ ) ∣ q − 2 ∣ Ψ ( τ ) − Ψ ( w ) ∣ q m , if γ < 0 , ( I ( E R 1 , 2 1 k ) ) q ≔ ∫ E R 1 , 2 1 k ∣ d τ ∣ ∣ Ψ ′ ( τ ) ∣ q − 2 ∣ Ψ ( τ ) − Ψ ( w 1 ) ∣ γ ( q − 1 ) + q , k = 1 , 2 , ( J 2 3 ( z ) ) q ≔ ∫ E R 1 13 ∣ Ψ ′ ( τ ) ∣ 2 − q ∣ d τ ∣ ∣ Ψ ( τ ) − Ψ ( w 1 ) ∣ γ ( q − 1 ) ∣ Ψ ( τ ) − Ψ ( w ) ∣ q m .

2.1. If γ ≥ 0 and z ∈ Ω ( δ ) , applying Lemmas 1 and 2 to (45), we obtain:

(46) ( I ( E R 1 , 1 11 ) ) q ≼ ∫ E R 1 , 1 11 ∣ τ ∣ − 1 d ( Ψ ( τ ) , L ) q − 2 ∣ d τ ∣ ∣ Ψ ( τ ) − Ψ ( w ) ∣ γ ( q − 1 ) + q m ≼ n κ ( q − 2 ) ∫ E R 1 , 1 11 ∣ d τ ∣ ∣ τ − w ∣ [ γ ( q − 1 ) + q m ] ( 1 + κ ) ≼ n [ γ ( q − 1 ) + q m ] ( 1 + κ ) + κ ( q − 2 ) mes E R 1 , 1 11 ≼ n [ γ ( q − 1 ) + q m ] ( 1 + κ ) + κ ( q − 2 ) − 1 , I ( E R 1 , 1 11 ) ≼ n ( γ p + m ) ( 1 + κ ) − 1 − 2 p − 1 κ − 1 p ,

(47) ( I ( E R 1 , 2 11 ) ) q ≼ ∫ E R 1 , 2 11 ∣ τ ∣ − 1 d ( Ψ ( τ ) , L ) q − 2 ∣ d τ ∣ ∣ Ψ ( τ ) − Ψ ( w 1 ) ∣ γ ( q − 1 ) + q m ≼ n κ ( q − 2 ) ∫ E R 1 , 2 11 ∣ d τ ∣ ∣ τ − w 1 ∣ γ ( q − 1 ) + q m ≼ n [ γ ( q − 1 ) + q m ] ( 1 + κ ) + κ ( q − 2 ) mes E R 1 , 2 11 ≼ n [ γ ( q − 1 ) + q m ] ( 1 + κ ) + κ ( q − 2 ) − 1 , I ( E R 1 , 2 11 ) ≼ n ( γ p + m ) ( 1 + κ ) − 1 − 2 p − 1 κ − 1 p ,

( I ( E R 1 , 1 12 ) ) q ≼ n κ ( q − 2 ) ∫ E R 1 , 1 12 ∣ d τ ∣ ∣ τ − w ∣ [ γ ( q − 1 ) + q m ] ( 1 + κ ) ≼ n κ ( q − 2 ) + [ γ ( q − 1 ) + q m ] ( 1 + κ ) − 1 , I ( E R 1 , 1 12 ) ≼ n ( γ p + m ) ( 1 + κ ) − 1 − 2 p − 1 κ − 1 p ,

(48) ( I ( E R 1 , 2 12 ) ) q ≼ ∫ E R 1 , 2 12 ∣ d τ ∣ ∣ Ψ ′ ( τ ) ∣ q − 2 ∣ Ψ ( τ ) − Ψ ( w 1 ) ∣ γ ( q − 1 ) + q m ≼ n κ ( q − 2 ) ∫ E R 1 , 2 12 ∣ d τ ∣ ∣ τ − w 1 ∣ [ γ ( q − 1 ) + q m ] ( 1 + κ ) ≼ n κ ( q − 2 ) + [ γ ( q − 1 ) + q m ] ( 1 + κ ) − 1 , I ( E R 1 , 2 12 ) ≼ n ( γ p + m ) ( 1 + κ ) − 1 − 2 p − 1 κ − 1 p .

For τ ∈ E R 1 13 , we see that η < ∣ τ − w 1 ∣ < 2 π R ˙ 1 . Therefore, from Lemma 1, we have ∣ Ψ ( τ ) − Ψ ( w 1 ) ∣ ≽ 1 . For w ∈ Δ ( w 1 , η ) ∣ Ψ ( τ ) − Ψ ( w ) ∣ ≽ ∣ τ − w ∣ 1 + κ . Then, applying Lemma 2, we obtain:

(49) ( J 2 3 ( z ) ) q ≼ n κ ( q − 2 ) ∫ E R 1 13 ∣ d τ ∣ ∣ τ − w ∣ q m ( 1 + κ ) ≼ n κ ( q − 2 ) + q m ( 1 + κ ) − 1 , J 2 3 ( z ) ≼ n m ( 1 + κ ) − 1 − 2 p − 1 κ − 1 p .

If z ∈ Ω ^ ( δ ) , then ∣ w − w 1 ∣ ≥ η , from (18), we have:

( J 2 1 ( z ) ) q ≼ ∫ E R 1 11 ∣ τ ∣ − 1 d ( Ψ ( τ ) , L ) q − 2 ∣ d τ ∣ ∣ Ψ ( τ ) − Ψ ( w 1 ) ∣ γ ( q − 1 ) ≼ n κ ( q − 2 ) ∫ E R 1 11 ∣ d τ ∣ ∣ τ − w 1 ∣ γ ( q − 1 ) ( 1 + κ ) ≼ n κ ( q − 2 ) + γ ( q − 1 ) ( 1 + κ ) mes E R 1 11 ≼ n κ ( q − 2 ) + γ ( q − 1 ) ( 1 + κ ) − 1 , J 2 1 ( z ) ≼ n γ p ( 1 + κ ) − 1 − 2 p − 1 κ − 1 p ,

( J 2 2 ( z ) ) q ≼ ∫ E R 1 12 ∣ τ ∣ − 1 d ( Ψ ( τ ) , L ) q − 2 ∣ d τ ∣ ∣ Ψ ( τ ) − Ψ ( w 1 ) ∣ γ ( q − 1 ) ≼ n κ ( q − 2 ) ∫ E R 1 12 ∣ d τ ∣ ∣ τ − w 1 ∣ γ ( q − 1 ) ( 1 + κ ) ≼ n κ ( q − 2 ) n γ ( q − 1 ) ( 1 + κ ) − 1 , γ ( q − 1 ) ( 1 + κ ) > 1 , ln n , γ ( q − 1 ) ( 1 + κ ) = 1 , 1 , γ ( q − 1 ) ( 1 + κ ) < 1 , J 2 2 ( z ) ≼ n γ p ( 1 + κ ) − 1 − 2 p − 1 κ − 1 p , γ > p − 1 1 + κ , n κ 2 p − 1 ( ln n ) 1 − 1 p , γ = p − 1 1 + κ , n κ 2 p − 1 , γ < p − 1 1 + κ ,

( J 2 3 ( z ) ) q = ∫ E R 1 13 ∣ d τ ∣ ∣ Ψ ′ ( τ ) ∣ q − 2 ∣ Ψ ( τ ) − Ψ ( w 1 ) ∣ γ ( q − 1 ) ∣ Ψ ( τ ) − Ψ ( w ) ∣ q m ≼ ∫ E R 1 13 ∣ d τ ∣ ∣ Ψ ′ ( τ ) ∣ q − 2 ≼ ∫ E R 1 13 ∣ τ ∣ − 1 d ( Ψ ( τ ) , L ) q − 2 ∣ d τ ∣ ≼ n κ ( q − 2 ) , J 2 3 ( z ) ≼ n κ 2 p − 1 .

From (46)–(49) and (24), for γ ≥ 0 , 1 < p < 2 , m ≥ 1 , we have:

(50) A n ( z ) ≼ ∥ P n ∥ p n γ + 2 p + m − 1 ( 1 + κ ) ,

if z ∈ Ω ( δ ) and

(51) A n ( z ) ≼ ∥ P n ∥ p n ( γ + 2 p + m − 1 ) ( 1 + κ ) , γ > p − 1 1 + κ , n κ 2 p − 1 + 1 p ( ln n ) 1 − 1 p , γ = p − 1 1 + κ , n κ 2 p − 1 + 1 p , γ < p − 1 1 + κ ,

if z ∈ Ω ^ ( δ ) .

2.2. Let γ < 0 . For z ∈ Ω ( δ ) , according to Lemma 1, we have:

(52) ( I ( E R 1 , 1 11 ) ) q = ∫ E R 1 , 1 11 Ψ ( τ ) − Ψ ( w 1 ) Ψ ( τ ) − Ψ ( w ) ( − γ ) ( q − 1 ) ∣ d τ ∣ ∣ Ψ ′ ( τ ) ∣ q − 2 ∣ Ψ ( τ ) − Ψ ( w ) ∣ q m + γ ( q − 1 ) ≼ n κ ( q − 2 ) ∫ E R 1 , 1 11 ∣ τ − w 1 ∣ ∣ τ − w ∣ ( − γ ) ( q − 1 ) 1 c 1 ∣ d τ ∣ ∣ τ − w ∣ [ q m + γ ( q − 1 ) ] ( 1 + κ ) ≼ n κ ( 2 − q ) + [ γ ( q − 1 ) + q m ] ( 1 + κ ) mes E R 1 , 1 11 ≼ n κ ( 2 − q ) + [ γ ( q − 1 ) + q m ] ( 1 + κ ) − 1 , I ( E R 1 , 1 11 ) ≼ n γ p + m ( 1 + κ ) − 1 − 2 p − 1 κ − 1 p ,

( I ( E R 1 , 2 11 ) ) q ≼ n κ ( q − 2 ) ∫ E R 1 , 2 11 ∣ d τ ∣ ∣ Ψ ( τ ) − Ψ ( w ) ∣ γ ( q − 1 ) + q m ≼ n κ ( q − 2 ) ∫ E R 1 11 ∣ d τ ∣ ∣ τ − w 1 ∣ [ q m + γ ( q − 1 ) ] ( 1 + κ ) ≼ n κ ( q − 2 ) + [ q m + γ ( q − 1 ) ] ( 1 + κ ) mes E R 1 , 2 11 ≼ n κ ( q − 2 ) + [ q m + γ ( q − 1 ) ] ( 1 + κ ) − 1 , I ( E R 1 , 2 11 ) ≼ n γ p + m ( 1 + κ ) − 1 − 2 p − 1 κ − 1 p ,

(53) ( I ( E R 1 , 1 12 ) ) q ≼ ∫ E R 1 , 1 12 ∣ Ψ ( τ ) − Ψ ( w 1 ) ∣ ( − γ ) ( q − 1 ) ∣ d τ ∣ ∣ Ψ ′ ( τ ) ∣ q − 2 ∣ Ψ ( τ ) − Ψ ( w ) ∣ q m ≼ ∫ E R 1 , 1 12 ∣ d τ ∣ ∣ Ψ ′ ( τ ) ∣ q − 2 ∣ Ψ ( τ ) − Ψ ( w ) ∣ q m ≼ n κ ( q − 2 ) ∫ E R 1 , 1 12 ∣ d τ ∣ ∣ τ − w ∣ q m ( 1 + κ ) ≼ n κ ( q − 2 ) + q m ( 1 + κ ) − 1 , I ( E R 1 , 1 12 ) ≼ n m ( 1 + κ ) − 1 − 2 p − 1 κ − 1 p ,

( I ( E R 1 , 2 12 ) ) q ≼ ∫ E R 1 , 2 12 ∣ d τ ∣ ∣ Ψ ′ ( τ ) ∣ q − 2 ∣ Ψ ( τ ) − Ψ ( w ) ∣ q m ≼ n κ ( q − 2 ) ∫ E R 1 , 2 12 ∣ d τ ∣ ∣ τ − w ∣ q m ( 1 + κ ) ≼ n κ ( q − 2 ) + q m ( 1 + κ ) − 1 , I ( E R 1 , 2 12 ) ≼ n m ( 1 + κ ) − 1 − 2 p − 1 κ − 1 p ,

(54) ( J 2 3 ( z ) ) q ≼ ∫ E R 1 13 ∣ Ψ ( τ ) − Ψ ( w 1 ) ∣ ( − γ ) ( q − 1 ) ∣ d τ ∣ ∣ Ψ ′ ( τ ) ∣ q − 2 ∣ Ψ ( τ ) − Ψ ( w ) ∣ q m ≼ n κ ( q − 2 ) ∫ E R 1 13 ∣ d τ ∣ ∣ τ − w ∣ q m ( 1 + κ ) ≼ n κ ( q − 2 ) + q m ( 1 + κ ) − 1 , J 2 3 ( z ) ≼ n m ( 1 + κ ) − 1 − 2 p − 1 κ − 1 p .

For z ∈ Ω ^ ( δ ) , analogously we obtain:

(55) J 2 i ( z ) ≼ n 2 p − 1 κ , i = 1 , 2 , 3 .

So, for γ < 0 , from (24), we have:

(56) A n ( z ) ≼ ∥ P n ∥ p n 2 p + m ( 1 + κ ) , if z ∈ Ω ( δ ) , n 2 p − 1 κ + 1 p , if z ∈ Ω ^ ( δ ) .

Therefore, for any γ ≥ − 2 , 1 < p < 2 , m ≥ 1 , from (50), (51), and (56), we obtain:

(57) A n ( z ) ≼ ∥ P n ∥ p n ( γ + 2 p + m − 1 ) ( 1 + κ ) , γ ≥ 0 , n ( 2 p + m − 1 ) ( 1 + κ ) , γ < 0 ,

if z ∈ Ω ( δ ) , and

(58) A n ( z ) ≼ ∥ P n ∥ p n ( γ + 2 p + m − 1 ) ( 1 + κ ) , γ > p − 1 1 + κ , n κ 2 p − 1 + 1 p ( ln n ) 1 − 1 p , γ = p − 1 1 + κ , n κ 2 p − 1 + 1 p , − 2 < γ < p − 1 1 + κ ,

if z ∈ Ω ^ ( δ ) .

(B) Now, we begin to estimate the 1 Φ n + 1 ( z ) ( j ) .

Since Φ ( ∞ ) = ∞ , then Cauchy integral representation for the region Ω R gives:

1 Φ n + 1 ( z ) ( j ) = − 1 2 π i ∫ L R 1 1 Φ n + 1 ( ζ ) d ζ ( ζ − z ) j + 1 , z ∈ Ω R .

If we go from both sides to the module, we obtain:

1 Φ n + 1 ( z ) ( j ) ≤ 1 2 π ∫ L R 1 1 ∣ Φ n + 1 ( ζ ) ∣ ∣ d ζ ∣ ∣ ζ − z ∣ j + 1 ≤ 1 2 π ∫ L R 1 ∣ d ζ ∣ ∣ ζ − z ∣ j + 1 .

Replacing the variable τ = Φ ( ζ ) and according to (18), we obtain:

(59) 1 Φ n + 1 ( z ) ( j ) ≼ ∫ ∣ τ ∣ = R 1 d ( Ψ ( τ ) , L ) ∣ τ ∣ − 1 ∣ d τ ∣ ∣ Ψ ( τ ) − Ψ ( w ) ∣ j + 1 ≼ n ∫ ∣ τ ∣ = R 1 ∣ d τ ∣ ∣ τ − w ∣ j ( 1 + κ ) ≼ n j ( 1 + κ ) , if z ∈ Ω ( δ ) , n κ , if z ∈ Ω ^ ( δ ) , j = 1 , m ¯ .

Combining estimates (19)–(24), (43), (44), (57), (58), and (59), we obtain:

(60) ∣ P n ( m ) ( z ) ∣ ≤ ∣ Φ n + 1 ( z ) ∣ A n ( z ) d ( z , L ) + ∑ j = 1 m C m j ∣ P n ( m − j ) ( z ) ∣ n j ( 1 + κ ) , if z ∈ Ω ( δ ) , n κ , if z ∈ Ω ^ ( δ ) ,

where for any γ > − 2 , p ≥ 2 , m ≥ 1

A n ( z ) ≼ ∥ P n ∥ p n γ p + m ( 1 + κ ) − 1 + 1 − 2 p κ − 1 p , p ≥ 2 , m ≥ 2 , γ > − 2 , n γ p + m ( 1 + κ ) − 1 + 1 − 2 p κ − 1 p , 2 ≤ p < 1 + ( 2 + γ ) ( 1 + κ ) , m = 1 , γ ≥ 0 , n 1 − 2 p ( ln n ) 1 − 1 p , p = 1 + ( 2 + γ ) ( 1 + κ ) , m = 1 , γ ≥ 0 , n 1 − 2 p , p > 1 + ( 2 + γ ) ( 1 + κ ) , m = 1 , γ ≥ 0 , n m ( 1 + κ ) − 1 + 1 − 2 p κ − 1 p , p ≥ 2 , m ≥ 1 , γ < 0 ,

if z ∈ Ω ( δ ) and

A n ( z ) ≼ ∥ P n ∥ p n γ p ( 1 + κ ) − 1 + 1 − 2 p κ − 1 p , 2 ≤ p < 1 2 + κ + ( γ + 2 ) 1 + κ 2 + κ , γ > 1 1 + κ , n 1 − 2 p ( ln n ) ( 1 − 1 p ) , p = 1 2 + κ + ( γ + 2 ) 1 + κ 2 + κ , γ > 1 1 + κ , n 1 − 2 p , p > 1 2 + κ + ( γ + 2 ) 1 + κ 2 + κ , γ > 1 1 + κ , n 1 − 2 p , p ≥ 2 , 0 ≤ γ ≤ 1 1 + κ , n κ 1 − 2 p , p ≥ 2 , γ < 0 ,

if z ∈ Ω ^ ( δ ) ; for any γ ≥ − 2 , 1 < p < 2 , m ≥ 1 ,

A n ( z ) ≼ ∥ P n ∥ p n ( γ ∗ p + m ) ( 1 + κ ) − 1 − 2 p − 1 κ − 1 p , γ > − 2 , if z ∈ Ω ( δ ) , n γ p ( 1 + κ ) − 1 − 2 p − 1 κ − 1 p , γ > p − 1 1 + κ , if z ∈ Ω ^ ( δ ) , n κ 2 p − 1 ( ln n ) ( 1 − 1 p ) , γ = p − 1 1 + κ , if z ∈ Ω ^ ( δ ) , n κ 2 p − 1 , − 2 < γ < p − 1 1 + κ , if z ∈ Ω ^ ( δ ) ,

and γ ∗ ≔ max { 0 ; γ } .

(C) ∣ P n ( m − j ) ( z ) ∣ . Estimate ∣ P n ( m − j ) ( z ) ∣ for m = 2 , j = 1 , 2 , and we carry out separately as follows. Therefore, the proof of Theorems 2 and 3 is completed.□

Proof of Theorem 4

Now let us start the evaluations of ∣ P n ( z ) ∣ . For this, we will make the necessary evaluations by writing the above proof for m = 0 . Since the function H n ( z ) ≔ P n ( z ) Φ n + 1 ( z ) , H n ( ∞ ) = 0 , is analytic in Ω , continuous on Ω ¯ , then Cauchy integral representation for the region Ω R 1 gives

H n ( z ) = − 1 2 π i ∫ L R 1 H n ( ζ ) d ζ ζ − z , z ∈ Ω R .

Then,

(61) P n ( z ) Φ n + 1 ( z ) ≤ 1 2 π ∫ L R 1 P n ( ζ ) Φ n + 1 ( ζ ) ∣ d ζ ∣ ∣ ζ − z ∣ ≤ 1 2 π d ( z , L R 1 ) ∫ L R 1 ∣ P n ( ζ ) ∣ ∣ d ζ ∣ ,

and so,

∣ P n ( z ) ∣ ≼ ∣ Φ n + 1 ( z ) ∣ d ( z , L R 1 ) ∫ L R 1 ∣ P n ( ζ ) ∣ ∣ d ζ ∣ .

We denote

(62) A n ≔ ∫ L R 1 ∣ P n ( ζ ) ∣ ∣ d ζ ∣ ,

and estimate this integral.

To estimate A n ( z ) , first replacing the variable τ = Φ ( ζ ) and multiplying the numerator and denominator of the integrant by ∣ Ψ ( τ ) − Ψ ( w 1 ) ∣ γ p ∣ Ψ ′ ( τ ) ∣ 2 p and applying the Hölder inequality, we obtain:

A n = ∫ L R 1 ∣ P n ( ζ ) ∣ ∣ d ζ ∣ = ∑ i = 1 2 ∫ F R 1 i ∣ Ψ ( τ ) − Ψ ( w 1 ) ∣ γ j p P n ( Ψ ( τ ) ) ( Ψ ′ ( τ ) ) 2 p ∣ Ψ ′ ( τ ) ∣ 1 − 2 p ∣ Ψ ( τ ) − Ψ ( w 1 ) ∣ γ p ∣ d τ ∣ ≤ ∑ i = 1 2 ∫ F R 1 i ∣ Ψ ( τ ) − Ψ ( w 1 ) ∣ γ ∣ P n ( Ψ ( τ ) ) ∣ p ∣ Ψ ′ ( τ ) ∣ 2 ∣ d τ ∣ 1 p ∫ F R 1 i ∣ Ψ ′ ( τ ) ∣ 1 − 2 p ∣ Ψ ( τ ) − Ψ ( w 1 ) ∣ γ p q ∣ d τ ∣ 1 q ≕ ∑ i = 1 2 A n i ,

where F R 1 1 ≔ Φ ( L R 1 1 ) = Δ 1 ′ ∩ { τ : ∣ τ ∣ = R 1 } , F R 1 2 ≔ Φ ( L R 1 ) \ F R 1 1 and

A n i ≔ ∫ F R 1 i ∣ f n , p ( τ ) ∣ p ∣ d τ ∣ 1 p ∫ F R 1 i ∣ Ψ ′ ( τ ) ∣ 2 − q ∣ Ψ ( τ ) − Ψ ( w 1 ) ∣ γ ( q − 1 ) ∣ d τ ∣ 1 q ≕ J n , 1 i ⋅ J n , 2 i , f n , p ( τ ) ≔ ( Ψ ( τ ) − Ψ ( w 1 ) ) γ p P n ( Ψ ( τ ) ) ( Ψ ′ ( τ ) ) 2 p , ∣ τ ∣ = R 1 .

Applying Lemma 3, we We denote:

J n , 1 i ≼ n 1 p ∥ P n ∥ p , i = 1 , 2 .

Therefore, we need to evaluate the following integrals:

( J n , 2 i ) q = ∫ F R 1 i ∣ Ψ ′ ( τ ) ∣ 2 − q ∣ Ψ ( τ ) − Ψ ( w 1 ) ∣ γ ( q − 1 ) ∣ d τ ∣ , i = 1 , 2 .

For the estimation of the integral J n , 2 i , for i = 1 , 2 , we use notations (23) and (24) and, consequently, we need to evaluate the following statement:

(63) A n = n 1 p ∥ P n ∥ p ( J 2 1 + J 2 2 + J 2 3 ) ,

where

( J 2 k ) q ≔ ∫ E R 1 1 k ∣ Ψ ′ ( τ ) ∣ 2 − q ∣ d τ ∣ ∣ Ψ ( τ ) − Ψ ( w 1 ) ∣ γ ( q − 1 ) , k = 1 , 2 , 3 .

So, for any k = 1 , 2 , 3 , we will estimate the integrals J 2 k .

Given the possible values q ( q > 2 and q < 2 ) and γ ( − 2 < γ < 0 and γ ≥ 0 ), we will consider the cases separately.

Case 1. Let 1 < q ≤ 2 ( p ≥ 2 ) .

1.1. Let γ ≥ 0 . Applying Lemma 2, we We denote:

(64) ( J 2 1 ) q = ∫ E R 1 11 ∣ Ψ ′ ( τ ) ∣ 2 − q ∣ d τ ∣ ∣ Ψ ( τ ) − Ψ ( w 1 ) ∣ γ ( q − 1 ) ≼ ∫ E R 1 , 2 11 d ( Ψ ( τ ) , L ) ∣ τ ∣ − 1 2 − q ∣ d τ ∣ ∣ Ψ ( τ ) − Ψ ( w 1 ) ∣ γ ( q − 1 ) ≼ n ( 2 − q ) ∫ E R 1 11 ∣ d τ ∣ ∣ τ − w 1 ∣ [ γ ( q − 1 ) − ( 2 − q ) ] ( 1 + κ ) ≼ n ( 2 − q ) + [ γ ( q − 1 ) − ( 2 − q ) ] ( 1 + κ ) mes E R 1 , 2 11 ≼ n ( 2 − q ) + [ γ ( q − 1 ) − ( 2 − q ) ] ( 1 + κ ) − 1 , J 2 1 ≼ n γ p ( 1 + κ ) − 1 + 1 − 2 p κ − 1 p .

(65) ( J 2 2 ) q ≤ ∫ E R 1 12 ∣ Ψ ′ ( τ ) ∣ 2 − q ∣ d τ ∣ ∣ Ψ ( τ ) − Ψ ( w 1 ) ∣ γ ( q − 1 ) ≼ n ( 2 − q ) ∫ E R 1 12 ∣ d τ ∣ ∣ τ − w 1 ∣ [ γ ( q − 1 ) − ( 2 − q ) ] ( 1 + κ ) ≼ n ( 2 − q ) n [ γ ( q − 1 ) − ( 2 − q ) ] ( 1 + κ ) − 1 , [ γ ( q − 1 ) − ( 2 − q ) ] ( 1 + κ ) > 1 , ln n , [ γ ( q − 1 ) − ( 2 − q ) ] ( 1 + κ ) = 1 , 1 , [ γ ( q − 1 ) − ( 2 − q ) ] ( 1 + κ ) < 1 , J 2 2 ≼ n γ p ( 1 + κ ) − 1 + 1 − 2 p κ − 1 p , 2 ≤ p < 1 2 + κ + ( 2 + γ ) 1 + κ 2 + κ , γ > 1 1 + κ , n 1 − 2 p ( ln n ) ( 1 − 1 p ) , p = 1 2 + κ + ( 2 + γ ) 1 + κ 2 + κ , γ > 1 1 + κ , n 1 − 2 p , p > 1 2 + κ + ( 2 + γ ) 1 + κ 2 + κ , γ > 1 1 + κ , n 1 − 2 p , p ≥ 2 , − 2 < γ ≤ 1 1 + κ .

For τ ∈ E R 1 13 , we see that η < ∣ τ − w 1 ∣ < 2 π R 1 . Therefore, ∣ Ψ ( τ ) − Ψ ( w 1 ) ∣ ≽ 1 , from Lemma 1 and applying (18), we We denote:

(66) ( J 2 3 ) q ≼ ∫ E R 1 13 ∣ Ψ ′ ( τ ) ∣ 2 − q ∣ d τ ∣ ≼ ∫ E R 1 13 d ( Ψ ( τ ) , L ) ∣ τ ∣ − 1 2 − q ∣ d τ ∣ ≼ n κ ( 2 − q ) , J 2 3 ≼ n κ 1 − 2 p .

Combining (64)–(66), for p ≥ 2 , γ ≥ 0 and z ∈ Ω R , we denote:

(67) ∑ k = 1 3 J 2 k ≼ n γ p ( 1 + κ ) − 1 + 1 − 2 p κ − 1 p , 2 ≤ p < 1 2 + κ + ( 2 + γ ) 1 + κ 2 + κ , γ > 1 1 + κ , n 1 − 2 p ( ln n ) 1 − 1 p , p = 1 2 + κ + ( 2 + γ ) 1 + κ 2 + κ , γ > 1 1 + κ , n 1 − 2 p , p > 1 2 + κ + ( 2 + γ ) 1 + κ 2 + κ , γ > 1 1 + κ , n 1 − 2 p , p ≥ 2 , 0 < γ ≤ 1 1 + κ .

From (63)–(67), we obtain:

(68) A n ≼ ∥ P n ∥ p n γ + 2 p − 1 ( 1 + κ ) , 2 ≤ p < 1 2 + κ + ( 2 + γ ) 1 + κ 2 + κ , γ > 1 1 + κ , n ( 1 − 1 p ) ( ln n ) 1 − 1 p , p = 1 2 + κ + ( 2 + γ ) 1 + κ 2 + κ , γ > 1 1 + κ , n 1 − 1 p , p > 1 2 + κ + ( 2 + γ ) 1 + κ 2 + κ , γ > 1 1 + κ , n 1 − 1 p , p ≥ 2 , − 2 , 0 < γ ≤ 1 1 + κ .

1.2. If γ < 0 , analogously we have:

(69) ( J 2 1 ) q ≼ ∫ E R 1 11 ∣ Ψ ( τ ) − Ψ ( w 1 ) ∣ ( − γ ) ( q − 1 ) d ( Ψ ( τ ) , L ) ∣ τ ∣ − 1 2 − q ∣ d τ ∣ ≼ n ( 2 − q ) ∫ E R 1 11 ∣ Ψ ( τ ) − Ψ ( w 1 ) ∣ ( − γ ) ( q − 1 ) + ( 2 − q ) ∣ d τ ∣ ≼ n ( 2 − q ) ∫ E R 1 11 ∣ τ − w 1 ∣ [ ( − γ ) ( q − 1 ) + ( 2 − q ) ] ( 1 − κ ) ∣ d τ ∣ ≼ n ( 2 − q ) + [ γ ( q − 1 ) − ( 2 − q ) ] ( 1 − κ ) mes E R 1 11 ≼ n ( 2 − q ) + [ γ ( q − 1 ) − ( 2 − q ) ] ( 1 − κ ) − 1 , J 2 1 ≼ n γ p ( 1 − κ ) − 1 − 1 − 2 p κ − 1 p ≼ 1 .

For τ ∈ E R 1 12 , we denote:

(70) ( J 2 2 ) q ≼ ∫ E R 1 12 ∣ Ψ ( τ ) − Ψ ( w 1 ) ∣ ( − γ ) ( q − 1 ) d ( Ψ ( τ ) , L ) ∣ τ ∣ − 1 2 − q ∣ d τ ∣ ≼ n κ ( 2 − q ) , J 2 2 ≼ n κ 1 − 2 p ,

(71) ( J 2 3 ) q ≼ ∫ E R 1 12 ∣ Ψ ( τ ) − Ψ ( w 1 ) ∣ ( − γ ) ( q − 1 ) d ( Ψ ( τ ) , L ) ∣ τ ∣ − 1 2 − q ∣ d τ ∣ ≼ n κ ( 2 − q ) , J 2 3 ≼ n κ 1 − 2 p .

Therefore, combining (69)–(71) in case of γ < 0 for z ∈ Ω R , we have:

∑ k = 1 3 J 2 k ≼ n κ 1 − 2 p ,

and, consequently, in this case from (63), we have:

(72) A n ≼ ∥ P n ∥ p ⋅ n κ 1 − 2 p + 1 p , z ∈ Ω R .

Therefore, combining (67) and (72), for any γ > − 2 , p ≥ 2 , z ∈ Ω R , we obtain:

(73) A n ≼ ∥ P n ∥ p n ( γ + 2 p − 1 ) ( 1 + κ ) , 2 ≤ p < 1 2 + κ + ( 2 + γ ) 1 + κ 2 + κ , γ ≥ 1 1 + κ , ( n ln n ) 1 − 1 p , p = 1 2 + κ + ( 2 + γ ) 1 + κ 2 + κ , γ ≥ 1 1 + κ , n 1 − 1 p , p > 1 2 + κ + ( 2 + γ ) 1 + κ 2 + κ , γ ≥ 1 1 + κ , n 1 − 1 p , p ≥ 2 , 0 ≤ γ < 1 1 + κ , n κ 1 − 2 p + 1 p , p ≥ 2 , − 2 < γ < 0 .

Case 2. Let q > 2 ( p < 2 ) . Then, 2 − q < 0 , and so

(74) ( J 2 k ( z ) ) q ≔ ∫ E R 1 1 k ∣ d τ ∣ ∣ Ψ ′ ( τ ) ∣ q − 2 ∣ Ψ ( τ ) − Ψ ( w 1 ) ∣ γ ( q − 1 ) , k = 1 , 2 , 3 .

2.1. If γ ≥ 0 , applying Lemmas 1 and 2 and (18), we obtain:

(75) ( J 2 1 ) q ≼ ∫ E R 1 , 2 11 ∣ τ ∣ − 1 d ( Ψ ( τ ) , L ) q − 2 ∣ d τ ∣ ∣ Ψ ( τ ) − Ψ ( w 1 ) ∣ γ ( q − 1 ) ≼ n κ ( q − 2 ) ∫ E R 1 , 2 11 ∣ d τ ∣ ∣ τ − w 1 ∣ γ ( q − 1 ) ≼ n γ ( q − 1 ) ( 1 + κ ) + κ ( q − 2 ) mes E R 1 , 2 11 ≼ n γ ( q − 1 ) ( 1 + κ ) + κ ( q − 2 ) − 1 , J 2 1 ≼ n γ p ( 1 + κ ) − 1 − 2 p − 1 κ − 1 p ,

(76) ( J 2 2 ) q ≼ ∫ E R 1 12 ∣ d τ ∣ ∣ Ψ ′ ( τ ) ∣ q − 2 ∣ Ψ ( τ ) − Ψ ( w 1 ) ∣ γ ( q − 1 ) ≼ n κ ( q − 2 ) ∫ E R 1 12 ∣ d τ ∣ ∣ τ − w 1 ∣ γ ( q − 1 ) ( 1 + κ ) ≼ n γ ( q − 1 ) ( 1 + κ ) + κ ( q − 2 ) − 1 , γ ( q − 1 ) ( 1 + κ ) > 1 , n κ ( q − 2 ) ln n , γ ( q − 1 ) ( 1 + κ ) = 1 , n κ ( q − 2 ) , γ ( q − 1 ) ( 1 + κ ) < 1 , J 2 2 ≼ n γ p ( 1 + κ ) − 1 − 2 p − 1 κ − 1 p , 1 < p < 1 + γ ( 1 + κ ) , 0 < γ < 1 1 + κ , n κ 2 p − 1 ( ln n ) 1 − 1 p , p = 1 + γ ( 1 + κ ) , 0 < γ < 1 1 + κ , n κ 2 p − 1 , 1 + γ ( 1 + κ ) < p < 2 , 0 ≤ γ < 1 1 + κ , n γ p ( 1 + κ ) − 1 − 2 p − 1 κ − 1 p , 1 < p < 2 , γ ≥ 1 1 + κ .

For τ ∈ E R 1 13 η < ∣ τ − w 1 ∣ < 2 π R ˙ 1 and from Lemma 1 ∣ Ψ ( τ ) − Ψ ( w 1 ) ∣ ≍ 1 . Then, we obtain:

(77) ( J 2 3 ) q = ∫ E R 1 13 ∣ d τ ∣ ∣ Ψ ′ ( τ ) ∣ q − 2 ∣ Ψ ( τ ) − Ψ ( w 1 ) ∣ γ ( q − 1 ) ≼ n κ ( q − 2 ) ∫ E R 1 13 ∣ d τ ∣ ≼ n κ ( q − 2 ) , J 2 3 ≼ n κ 2 p − 1 .

From (74)–(77) and (63), for γ ≥ 0 , 1 < p < 2 , z ∈ Ω R , we have:

(78) A n ≼ ∥ P n ∥ p n γ + 2 p − 1 ( 1 + κ ) , 1 < p < 1 + γ ( 1 + κ ) , 0 < γ < 1 1 + κ , n κ 2 p − 1 + 1 p ( ln n ) 1 − 1 p , p = 1 + γ ( 1 + κ ) , 0 < γ < 1 1 + κ , n κ 2 p − 1 + 1 p , 1 + γ ( 1 + κ ) < p < 2 , 0 ≤ γ < 1 1 + κ , n γ + 2 p − 1 ( 1 + κ ) , 1 < p < 2 , γ ≥ 1 1 + κ .

2.2. Let γ < 0 . For z ∈ Ω R , according to Lemma 1, we have:

(79) ( J 2 1 ) q ≼ ∫ E R 1 11 ∣ Ψ ( τ ) − Ψ ( w 1 ) ∣ ( − γ ) ( q − 1 ) ∣ τ ∣ − 1 d ( Ψ ( τ ) , L ) q − 2 ∣ d τ ∣ ≼ n κ ( q − 2 ) ∫ E R 1 11 ∣ Ψ ( τ ) − Ψ ( w 1 ) ∣ ( − γ ) ( q − 1 ) ∣ d τ ∣ ≼ n κ ( q − 2 ) ∫ E R 1 11 ∣ τ − w 1 ∣ ( − γ ) ( q − 1 ) ( 1 − κ ) ∣ d τ ∣ ≼ n κ ( q − 2 ) + γ ( q − 1 ) ( 1 − κ ) mes E R 1 11 ≼ n κ ( q − 2 ) + γ ( q − 1 ) ( 1 − κ ) − 1 ≼ n κ ( q − 2 ) − 1 , J 2 1 ≼ n 2 p − 1 κ − 1 + 1 p ,

(80) ( J 2 2 ) q ≼ ∫ E R 1 12 ∣ Ψ ( τ ) − Ψ ( w 1 ) ∣ ( − γ ) ( q − 1 ) ∣ τ ∣ − 1 d ( Ψ ( τ ) , L ) q − 2 ∣ d τ ∣ ≼ n κ ( q − 2 ) ∫ E R 1 12 ∣ Ψ ( τ ) − Ψ ( w 1 ) ∣ ( − γ ) ( q − 1 ) ∣ d τ ∣ ≼ n κ ( q − 2 ) , J 2 2 ≼ n 2 p − 1 κ ,

(81) ( J 2 3 ) q ≼ ∫ E R 1 13 ∣ Ψ ( τ ) − Ψ ( w 1 ) ∣ ( − γ ) ( q − 1 ) ∣ d τ ∣ ∣ Ψ ′ ( τ ) ∣ q − 2 ≼ n κ ( q − 2 ) , J 2 3 ≼ n 2 p − 1 κ .

So, for γ < 0 , from (63), we have:

(82) A n ≼ n 2 p − 1 κ + 1 p ∥ P n ∥ p , z ∈ Ω R .

Therefore, for any γ ≥ − 2 , 1 < p < 2 , from (78) and (82), we obtain:

(83) A n ≼ ∥ P n ∥ p n γ + 2 p − 1 ( 1 + κ ) , 1 < p < 1 + γ ( 1 + κ ) , 0 < γ < 1 1 + κ , n κ 2 p − 1 + 1 p ( ln n ) 1 − 1 p , p = 1 + γ ( 1 + κ ) , 0 < γ < 1 1 + κ , n κ 2 p − 1 + 1 p , 1 + γ ( 1 + κ ) < p < 2 , 0 ≤ γ < 1 1 + κ , n γ + 2 p − 1 ( 1 + κ ) , 1 < p < 2 , γ ≥ 1 1 + κ , n κ 2 p − 1 + 1 p , 1 < p < 2 , − 2 < γ < 0 .

Combining estimates (19)–(63), (73), and (83), we get:

∣ P n ( z ) ∣ ≼ ∣ Φ n + 1 ( z ) ∣ d ( z , L R 1 ) ∥ P n ∥ p A n ,

where for p ≥ 2

A n ≼ ∥ P n ∥ p n ( γ + 2 p − 1 ) ( 1 + κ ) , 2 ≤ p < 1 2 + κ + ( 2 + γ ) 1 + κ 2 + κ , γ ≥ 1 1 + κ , n ( 1 − 1 p ) ( ln n ) ( 1 − 1 p ) , p = 1 2 + κ + ( 2 + γ ) 1 + κ 2 + κ , γ ≥ 1 1 + κ , n ( 1 − 1 p ) , p > max 2 ; 1 2 + κ + ( 2 + γ ) 1 + κ 2 + κ , γ ≥ 0 , n κ 1 − 2 p + 1 p , p ≥ 2 , − 2 < γ < 0 ,

and for 1 < p < 2

A n ≼ ∥ P n ∥ p n γ + 2 p − 1 ( 1 + κ ) , 1 < p < min { 2 ; 1 + γ ( 1 + κ ) } , γ > 0 , n κ 2 p − 1 + 1 p ( ln n ) 1 − 1 p , p = 1 + γ ( 1 + κ ) , 0 < γ < 1 1 + κ , n κ 2 p − 1 + 1 p , max { 1 ; 1 + γ ( 1 + κ ) } < p < 2 , − 2 < γ < 1 1 + κ ,

and therefore, the proof of Theorem 4 is completed.□

Proof of Theorems 7 and 8

According to Theorems 2, 4, 5 and estimates (20), (21), (43), (44), (60), for m = 2 and p ≥ 2 from (19), we have:

∣ P n ″ ( z ) ∣ ≤ P n ( z ) Φ n + 1 ( z ) ″ + ∑ j = 1 2 C 2 j 1 Φ n + 1 ( z ) ( j ) ∣ P n ( 2 − j ) ( z ) ∣ ≤ ∣ Φ n + 1 ( z ) ∣ P n ( z ) Φ n + 1 ( z ) ″ + C 2 1 B n , 1 1 ∣ P n ′ ( z ) ∣ + C 2 2 B n , 2 1 ∣ P n ( z ) ∣ ≼ ∣ Φ n + 1 ( z ) ∣ ∥ P n ∥ p d ( z , L ) A n 1 ( z , 2 ) + C 2 1 B n , 1 1 ∣ P n ′ ( z ) ∣ + C 2 2 B n , 2 1 ∣ P n ( z ) ∣ .

Substituting estimates for the B n , j 1 , j = 1 , 2 , ∣ P n ( z ) ∣ and ∣ P n ′ ( z ) ∣ from Theorems 2, 4, and 5 correspondingly, we obtain:

∣ P n ″ ( z ) ∣ ≼ ∣ Φ 3 ( n + 1 ) ( z ) ∣ d ( z , L R 1 ) ∥ P n ∥ p n γ + 2 p + 1 ( 1 + κ ) , 2 ≤ p < 1 2 + κ + ( 2 + γ ) 1 + κ 2 + κ , γ ≥ 1 1 + κ , n 1 − 1 p + 2 ( 1 + κ ) ( ln n ) 1 − 1 p , p = 1 2 + κ + ( 2 + γ ) 1 + κ 2 + κ , γ ≥ 1 1 + κ , n 1 − 1 p + 2 ( 1 + κ ) , p > 1 2 + κ + ( 2 + γ ) 1 + κ 2 + κ , γ ≥ 1 1 + κ , n 1 − 1 p + 2 ( 1 + κ ) , 2 ≤ p < 1 + ( 2 + γ ) ( 1 + κ ) , 0 ≤ γ < 1 1 + κ , n 2 + 1 p + 3 − 2 p κ , p ≥ 2 , − 2 < γ < 0 ,

for z ∈ Ω ( δ ) , and

∣ P n ″ ( z ) ∣ ≼ ∣ Φ 3 ( n + 1 ) ( z ) ∣ d ( z , L R 1 ) ∥ P n ∥ p n γ + 2 p ( 1 + κ ) + κ , 2 ≤ p < 1 2 + κ + ( 2 + γ ) 1 + κ 2 + κ , γ ≥ 1 1 + κ , n 1 + 2 κ − 1 p ( ln n ) 1 − 1 p , p = 1 2 + κ + ( 2 + γ ) 1 + κ 2 + κ , γ ≥ 1 1 + κ , n 1 + 2 κ − 1 p , p > 1 2 + κ + ( 2 + γ ) 1 + κ 2 + κ , γ ≥ 1 1 + κ , n 1 + 2 κ − 1 p , p ≥ 2 , 0 ≤ γ < 1 1 + κ , n κ 2 − 2 p + 1 p + κ , p ≥ 2 , − 2 < γ < 0 ,

for z ∈ Ω ^ ( δ ) ,

Analogously, from Theorems 3, 4, and 6, for 1 < p < 2 , we have:

∣ P n ″ ( z ) ∣ ≼ ∣ Φ 3 ( n + 1 ) ( z ) ∣ d ( z , L R 1 ) ∥ P n ∥ p n γ + 2 p + 1 ( 1 + κ ) , 1 < p < 2 , γ ≥ 1 1 + κ , n γ + 2 p + 1 ( 1 + κ ) , 1 < p < 1 + γ ( 1 + κ ) , 0 < γ < 1 1 + κ , n κ 2 p + 1 + 1 p + 2 ( ln n ) 1 − 1 p , p = 1 + γ ( 1 + κ ) , 0 < γ < 1 1 + κ , n κ 2 p + 1 + 1 p + 2 , 1 + γ ( 1 + κ ) < p < 2 , 0 ≤ γ < 1 1 + κ , n κ 2 p + 1 + 1 p + 2 , 1 < p < 2 , − 2 < γ < 0 ,

for z ∈ Ω ( δ ) , and

∣ P n ″ ( z ) ∣ ≼ ∣ Φ 3 ( n + 1 ) ( z ) ∣ d ( z , L R 1 ) ∥ P n ∥ p n γ + 2 p + 1 ( 1 + κ ) , 1 < p < 2 , γ ≥ 1 1 + κ , n γ + 2 p + 1 ( 1 + κ ) , p < 1 + γ ( 1 + κ ) , 0 < γ < 1 1 + κ , n κ 2 p − 1 + 1 p + 2 κ ( ln n ) 1 − 1 p , p = 1 + γ ( 1 + κ ) , 0 < γ < 1 1 + κ , n κ 2 p − 1 + 1 p + 2 κ , p > 1 + γ ( 1 + κ ) , 0 < γ < 1 1 + κ , n κ 2 p − 1 + 1 p + 2 κ , 1 < p < 2 , − 2 < γ ≤ 0 ,

for z ∈ Ω ^ ( δ ) . Therefore, the proof of Theorems 7 and 8 completed.

The proofs of Theorems 5 and 6 are similarly carried out to the proofs of Theorems 7 and 8, using the corresponding estimates (11)–(13).

In conclusion, note that in proofs, everywhere there is a quantity d ( z , L R 1 ) . Let us show that d ( z , L R 1 ) ≽ d ( z , L ) holds for all z ∈ Ω R . For the points z ∉ Ω ( L R 1 , d ( L R 1 , L R ) ) , we have: d ( z , L R 1 ) ≽ δ ≽ d ( z , L ) . Now, let z ∈ Ω ( L R 1 , d ( L R 1 , L R ) ) . Denote by ξ 1 ∈ L R 1 the point such that d ( z , L R 1 ) = ∣ z − ξ 1 ∣ , and point ξ 2 ∈ L , such that d ( z , L ) = ∣ z − ξ 2 ∣ , and for w = Φ ( z ) , t 1 = Φ ( ξ 1 ) , t 2 = Φ ( ξ 2 ) , we have: ∣ w − w 1 ∣ ≥ ∣ ∣ w − w 2 ∣ − ∣ w 2 − w 1 ∣ ∣ ≥ ∣ w − w 2 ∣ − 1 2 ∣ w − w 2 ∣ ≥ 1 2 ∣ w − w 2 ∣ . Then, according to Lemma 1, we obtain d ( z , L R 1 ) ≽ d ( z , L ) .□

Acknowledgements

I am grateful to all referees for their careful review of the manuscript and valuable comments.

  1. Conflict of interest: The author states no conflict of interest.

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Received: 2021-04-17
Revised: 2021-09-12
Accepted: 2021-09-27
Published Online: 2021-12-31

© 2021 Fahreddin G. Abdullayev, published by De Gruyter

This work is licensed under the Creative Commons Attribution 4.0 International License.

Articles in the same Issue

  1. Regular Articles
  2. Sharp conditions for the convergence of greedy expansions with prescribed coefficients
  3. Range-kernel weak orthogonality of some elementary operators
  4. Stability analysis for Selkov-Schnakenberg reaction-diffusion system
  5. On non-normal cyclic subgroups of prime order or order 4 of finite groups
  6. Some results on semigroups of transformations with restricted range
  7. Quasi-ideal Ehresmann transversals: The spined product structure
  8. On the regulator problem for linear systems over rings and algebras
  9. Solvability of the abstract evolution equations in Ls-spaces with critical temporal weights
  10. Resolving resolution dimensions in triangulated categories
  11. Entire functions that share two pairs of small functions
  12. On stochastic inverse problem of construction of stable program motion
  13. Pentagonal quasigroups, their translatability and parastrophes
  14. Counting certain quadratic partitions of zero modulo a prime number
  15. Global attractors for a class of semilinear degenerate parabolic equations
  16. A new implicit symmetric method of sixth algebraic order with vanished phase-lag and its first derivative for solving Schrödinger's equation
  17. On sub-class sizes of mutually permutable products
  18. Asymptotic solution of the Cauchy problem for the singularly perturbed partial integro-differential equation with rapidly oscillating coefficients and with rapidly oscillating heterogeneity
  19. Existence and asymptotical behavior of solutions for a quasilinear Choquard equation with singularity
  20. On kernels by rainbow paths in arc-coloured digraphs
  21. Fully degenerate Bell polynomials associated with degenerate Poisson random variables
  22. Multiple solutions and ground state solutions for a class of generalized Kadomtsev-Petviashvili equation
  23. A note on maximal operators related to Laplace-Bessel differential operators on variable exponent Lebesgue spaces
  24. Weak and strong estimates for linear and multilinear fractional Hausdorff operators on the Heisenberg group
  25. Partial sums and inclusion relations for analytic functions involving (p, q)-differential operator
  26. Hodge-Deligne polynomials of character varieties of free abelian groups
  27. Diophantine approximation with one prime, two squares of primes and one kth power of a prime
  28. The equivalent parameter conditions for constructing multiple integral half-discrete Hilbert-type inequalities with a class of nonhomogeneous kernels and their applications
  29. Boundedness of vector-valued sublinear operators on weighted Herz-Morrey spaces with variable exponents
  30. On some new quantum midpoint-type inequalities for twice quantum differentiable convex functions
  31. Quantum Ostrowski-type inequalities for twice quantum differentiable functions in quantum calculus
  32. Asymptotic measure-expansiveness for generic diffeomorphisms
  33. Infinitesimals via Cauchy sequences: Refining the classical equivalence
  34. The (1, 2)-step competition graph of a hypertournament
  35. Properties of multiplication operators on the space of functions of bounded φ-variation
  36. Disproving a conjecture of Thornton on Bohemian matrices
  37. Some estimates for the commutators of multilinear maximal function on Morrey-type space
  38. Inviscid, zero Froude number limit of the viscous shallow water system
  39. Inequalities between height and deviation of polynomials
  40. New criteria-based ℋ-tensors for identifying the positive definiteness of multivariate homogeneous forms
  41. Determinantal inequalities of Hua-Marcus-Zhang type for quaternion matrices
  42. On a new generalization of some Hilbert-type inequalities
  43. On split quaternion equivalents for Quaternaccis, shortly Split Quaternaccis
  44. On split regular BiHom-Poisson color algebras
  45. Asymptotic stability of the time-changed stochastic delay differential equations with Markovian switching
  46. The mixed metric dimension of flower snarks and wheels
  47. Oscillatory bifurcation problems for ODEs with logarithmic nonlinearity
  48. The B-topology on S∗-doubly quasicontinuous posets
  49. Hyers-Ulam stability of isometries on bounded domains
  50. Inhomogeneous conformable abstract Cauchy problem
  51. Path homology theory of edge-colored graphs
  52. Refinements of quantum Hermite-Hadamard-type inequalities
  53. Symmetric graphs of valency seven and their basic normal quotient graphs
  54. Mean oscillation and boundedness of multilinear operator related to multiplier operator
  55. Numerical methods for time-fractional convection-diffusion problems with high-order accuracy
  56. Several explicit formulas for (degenerate) Narumi and Cauchy polynomials and numbers
  57. Finite groups whose intersection power graphs are toroidal and projective-planar
  58. On primitive solutions of the Diophantine equation x2 + y2 = M
  59. A note on polyexponential and unipoly Bernoulli polynomials of the second kind
  60. On the type 2 poly-Bernoulli polynomials associated with umbral calculus
  61. Some estimates for commutators of Littlewood-Paley g-functions
  62. Construction of a family of non-stationary combined ternary subdivision schemes reproducing exponential polynomials
  63. On the evolutionary bifurcation curves for the one-dimensional prescribed mean curvature equation with logistic type
  64. On intersections of two non-incident subgroups of finite p-groups
  65. Global existence and boundedness in a two-species chemotaxis system with nonlinear diffusion
  66. Finite groups with 4p2q elements of maximal order
  67. Positive solutions of a discrete nonlinear third-order three-point eigenvalue problem with sign-changing Green's function
  68. Power moments of automorphic L-functions related to Maass forms for SL3(ℤ)
  69. Entire solutions for several general quadratic trinomial differential difference equations
  70. Strong consistency of regression function estimator with martingale difference errors
  71. Fractional Hermite-Hadamard-type inequalities for interval-valued co-ordinated convex functions
  72. Montgomery identity and Ostrowski-type inequalities via quantum calculus
  73. Universal inequalities of the poly-drifting Laplacian on smooth metric measure spaces
  74. On reducible non-Weierstrass semigroups
  75. so-metrizable spaces and images of metric spaces
  76. Some new parameterized inequalities for co-ordinated convex functions involving generalized fractional integrals
  77. The concept of cone b-Banach space and fixed point theorems
  78. Complete consistency for the estimator of nonparametric regression model based on m-END errors
  79. A posteriori error estimates based on superconvergence of FEM for fractional evolution equations
  80. Solution of integral equations via coupled fixed point theorems in 𝔉-complete metric spaces
  81. Symmetric pairs and pseudosymmetry of Θ-Yetter-Drinfeld categories for Hom-Hopf algebras
  82. A new characterization of the automorphism groups of Mathieu groups
  83. The role of w-tilting modules in relative Gorenstein (co)homology
  84. Primitive and decomposable elements in homology of ΩΣℂP∞
  85. The G-sequence shadowing property and G-equicontinuity of the inverse limit spaces under group action
  86. Classification of f-biharmonic submanifolds in Lorentz space forms
  87. Some new results on the weaving of K-g-frames in Hilbert spaces
  88. Matrix representation of a cross product and related curl-based differential operators in all space dimensions
  89. Global optimization and applications to a variational inequality problem
  90. Functional equations related to higher derivations in semiprime rings
  91. A partial order on transformation semigroups with restricted range that preserve double direction equivalence
  92. On multi-step methods for singular fractional q-integro-differential equations
  93. Compact perturbations of operators with property (t)
  94. Entire solutions for several complex partial differential-difference equations of Fermat type in ℂ2
  95. Random attractors for stochastic plate equations with memory in unbounded domains
  96. On the convergence of two-step modulus-based matrix splitting iteration method
  97. On the separation method in stochastic reconstruction problem
  98. Robust estimation for partial functional linear regression models based on FPCA and weighted composite quantile regression
  99. Structure of coincidence isometry groups
  100. Sharp function estimates and boundedness for Toeplitz-type operators associated with general fractional integral operators
  101. Oscillatory hyper-Hilbert transform on Wiener amalgam spaces
  102. Euler-type sums involving multiple harmonic sums and binomial coefficients
  103. Poly-falling factorial sequences and poly-rising factorial sequences
  104. Geometric approximations to transition densities of Jump-type Markov processes
  105. Multiple solutions for a quasilinear Choquard equation with critical nonlinearity
  106. Bifurcations and exact traveling wave solutions for the regularized Schamel equation
  107. Almost factorizable weakly type B semigroups
  108. The finite spectrum of Sturm-Liouville problems with n transmission conditions and quadratic eigenparameter-dependent boundary conditions
  109. Ground state sign-changing solutions for a class of quasilinear Schrödinger equations
  110. Epi-quasi normality
  111. Derivative and higher-order Cauchy integral formula of matrix functions
  112. Commutators of multilinear strongly singular integrals on nonhomogeneous metric measure spaces
  113. Solutions to a multi-phase model of sea ice growth
  114. Existence and simulation of positive solutions for m-point fractional differential equations with derivative terms
  115. Bernstein-Walsh type inequalities for derivatives of algebraic polynomials in quasidisks
  116. Review Article
  117. Semiprimeness of semigroup algebras
  118. Special Issue on Problems, Methods and Applications of Nonlinear Analysis (Part II)
  119. Third-order differential equations with three-point boundary conditions
  120. Fractional calculus, zeta functions and Shannon entropy
  121. Uniqueness of positive solutions for boundary value problems associated with indefinite ϕ-Laplacian-type equations
  122. Synchronization of Caputo fractional neural networks with bounded time variable delays
  123. On quasilinear elliptic problems with finite or infinite potential wells
  124. Deterministic and random approximation by the combination of algebraic polynomials and trigonometric polynomials
  125. On a fractional Schrödinger-Poisson system with strong singularity
  126. Parabolic inequalities in Orlicz spaces with data in L1
  127. Special Issue on Evolution Equations, Theory and Applications (Part II)
  128. Impulsive Caputo-Fabrizio fractional differential equations in b-metric spaces
  129. Existence of a solution of Hilfer fractional hybrid problems via new Krasnoselskii-type fixed point theorems
  130. On a nonlinear system of Riemann-Liouville fractional differential equations with semi-coupled integro-multipoint boundary conditions
  131. Blow-up results of the positive solution for a class of degenerate parabolic equations
  132. Long time decay for 3D Navier-Stokes equations in Fourier-Lei-Lin spaces
  133. On the extinction problem for a p-Laplacian equation with a nonlinear gradient source
  134. General decay rate for a viscoelastic wave equation with distributed delay and Balakrishnan-Taylor damping
  135. On hyponormality on a weighted annulus
  136. Exponential stability of Timoshenko system in thermoelasticity of second sound with a memory and distributed delay term
  137. Convergence results on Picard-Krasnoselskii hybrid iterative process in CAT(0) spaces
  138. Special Issue on Boundary Value Problems and their Applications on Biosciences and Engineering (Part I)
  139. Marangoni convection in layers of water-based nanofluids under the effect of rotation
  140. A transient analysis to the M(τ)/M(τ)/k queue with time-dependent parameters
  141. Existence of random attractors and the upper semicontinuity for small random perturbations of 2D Navier-Stokes equations with linear damping
  142. Degenerate binomial and Poisson random variables associated with degenerate Lah-Bell polynomials
  143. Special Issue on Fractional Problems with Variable-Order or Variable Exponents (Part I)
  144. On the mixed fractional quantum and Hadamard derivatives for impulsive boundary value problems
  145. The Lp dual Minkowski problem about 0 < p < 1 and q > 0
https://doi.org/10.1515/math-2021-0138
Keywords for this article
algebraic polynomial; quasiconformal mapping; quasicircle
Creative Commons
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Articles in the same Issue

  1. Regular Articles
  2. Sharp conditions for the convergence of greedy expansions with prescribed coefficients
  3. Range-kernel weak orthogonality of some elementary operators
  4. Stability analysis for Selkov-Schnakenberg reaction-diffusion system
  5. On non-normal cyclic subgroups of prime order or order 4 of finite groups
  6. Some results on semigroups of transformations with restricted range
  7. Quasi-ideal Ehresmann transversals: The spined product structure
  8. On the regulator problem for linear systems over rings and algebras
  9. Solvability of the abstract evolution equations in Ls-spaces with critical temporal weights
  10. Resolving resolution dimensions in triangulated categories
  11. Entire functions that share two pairs of small functions
  12. On stochastic inverse problem of construction of stable program motion
  13. Pentagonal quasigroups, their translatability and parastrophes
  14. Counting certain quadratic partitions of zero modulo a prime number
  15. Global attractors for a class of semilinear degenerate parabolic equations
  16. A new implicit symmetric method of sixth algebraic order with vanished phase-lag and its first derivative for solving Schrödinger's equation
  17. On sub-class sizes of mutually permutable products
  18. Asymptotic solution of the Cauchy problem for the singularly perturbed partial integro-differential equation with rapidly oscillating coefficients and with rapidly oscillating heterogeneity
  19. Existence and asymptotical behavior of solutions for a quasilinear Choquard equation with singularity
  20. On kernels by rainbow paths in arc-coloured digraphs
  21. Fully degenerate Bell polynomials associated with degenerate Poisson random variables
  22. Multiple solutions and ground state solutions for a class of generalized Kadomtsev-Petviashvili equation
  23. A note on maximal operators related to Laplace-Bessel differential operators on variable exponent Lebesgue spaces
  24. Weak and strong estimates for linear and multilinear fractional Hausdorff operators on the Heisenberg group
  25. Partial sums and inclusion relations for analytic functions involving (p, q)-differential operator
  26. Hodge-Deligne polynomials of character varieties of free abelian groups
  27. Diophantine approximation with one prime, two squares of primes and one kth power of a prime
  28. The equivalent parameter conditions for constructing multiple integral half-discrete Hilbert-type inequalities with a class of nonhomogeneous kernels and their applications
  29. Boundedness of vector-valued sublinear operators on weighted Herz-Morrey spaces with variable exponents
  30. On some new quantum midpoint-type inequalities for twice quantum differentiable convex functions
  31. Quantum Ostrowski-type inequalities for twice quantum differentiable functions in quantum calculus
  32. Asymptotic measure-expansiveness for generic diffeomorphisms
  33. Infinitesimals via Cauchy sequences: Refining the classical equivalence
  34. The (1, 2)-step competition graph of a hypertournament
  35. Properties of multiplication operators on the space of functions of bounded φ-variation
  36. Disproving a conjecture of Thornton on Bohemian matrices
  37. Some estimates for the commutators of multilinear maximal function on Morrey-type space
  38. Inviscid, zero Froude number limit of the viscous shallow water system
  39. Inequalities between height and deviation of polynomials
  40. New criteria-based ℋ-tensors for identifying the positive definiteness of multivariate homogeneous forms
  41. Determinantal inequalities of Hua-Marcus-Zhang type for quaternion matrices
  42. On a new generalization of some Hilbert-type inequalities
  43. On split quaternion equivalents for Quaternaccis, shortly Split Quaternaccis
  44. On split regular BiHom-Poisson color algebras
  45. Asymptotic stability of the time-changed stochastic delay differential equations with Markovian switching
  46. The mixed metric dimension of flower snarks and wheels
  47. Oscillatory bifurcation problems for ODEs with logarithmic nonlinearity
  48. The B-topology on S∗-doubly quasicontinuous posets
  49. Hyers-Ulam stability of isometries on bounded domains
  50. Inhomogeneous conformable abstract Cauchy problem
  51. Path homology theory of edge-colored graphs
  52. Refinements of quantum Hermite-Hadamard-type inequalities
  53. Symmetric graphs of valency seven and their basic normal quotient graphs
  54. Mean oscillation and boundedness of multilinear operator related to multiplier operator
  55. Numerical methods for time-fractional convection-diffusion problems with high-order accuracy
  56. Several explicit formulas for (degenerate) Narumi and Cauchy polynomials and numbers
  57. Finite groups whose intersection power graphs are toroidal and projective-planar
  58. On primitive solutions of the Diophantine equation x2 + y2 = M
  59. A note on polyexponential and unipoly Bernoulli polynomials of the second kind
  60. On the type 2 poly-Bernoulli polynomials associated with umbral calculus
  61. Some estimates for commutators of Littlewood-Paley g-functions
  62. Construction of a family of non-stationary combined ternary subdivision schemes reproducing exponential polynomials
  63. On the evolutionary bifurcation curves for the one-dimensional prescribed mean curvature equation with logistic type
  64. On intersections of two non-incident subgroups of finite p-groups
  65. Global existence and boundedness in a two-species chemotaxis system with nonlinear diffusion
  66. Finite groups with 4p2q elements of maximal order
  67. Positive solutions of a discrete nonlinear third-order three-point eigenvalue problem with sign-changing Green's function
  68. Power moments of automorphic L-functions related to Maass forms for SL3(ℤ)
  69. Entire solutions for several general quadratic trinomial differential difference equations
  70. Strong consistency of regression function estimator with martingale difference errors
  71. Fractional Hermite-Hadamard-type inequalities for interval-valued co-ordinated convex functions
  72. Montgomery identity and Ostrowski-type inequalities via quantum calculus
  73. Universal inequalities of the poly-drifting Laplacian on smooth metric measure spaces
  74. On reducible non-Weierstrass semigroups
  75. so-metrizable spaces and images of metric spaces
  76. Some new parameterized inequalities for co-ordinated convex functions involving generalized fractional integrals
  77. The concept of cone b-Banach space and fixed point theorems
  78. Complete consistency for the estimator of nonparametric regression model based on m-END errors
  79. A posteriori error estimates based on superconvergence of FEM for fractional evolution equations
  80. Solution of integral equations via coupled fixed point theorems in 𝔉-complete metric spaces
  81. Symmetric pairs and pseudosymmetry of Θ-Yetter-Drinfeld categories for Hom-Hopf algebras
  82. A new characterization of the automorphism groups of Mathieu groups
  83. The role of w-tilting modules in relative Gorenstein (co)homology
  84. Primitive and decomposable elements in homology of ΩΣℂP∞
  85. The G-sequence shadowing property and G-equicontinuity of the inverse limit spaces under group action
  86. Classification of f-biharmonic submanifolds in Lorentz space forms
  87. Some new results on the weaving of K-g-frames in Hilbert spaces
  88. Matrix representation of a cross product and related curl-based differential operators in all space dimensions
  89. Global optimization and applications to a variational inequality problem
  90. Functional equations related to higher derivations in semiprime rings
  91. A partial order on transformation semigroups with restricted range that preserve double direction equivalence
  92. On multi-step methods for singular fractional q-integro-differential equations
  93. Compact perturbations of operators with property (t)
  94. Entire solutions for several complex partial differential-difference equations of Fermat type in ℂ2
  95. Random attractors for stochastic plate equations with memory in unbounded domains
  96. On the convergence of two-step modulus-based matrix splitting iteration method
  97. On the separation method in stochastic reconstruction problem
  98. Robust estimation for partial functional linear regression models based on FPCA and weighted composite quantile regression
  99. Structure of coincidence isometry groups
  100. Sharp function estimates and boundedness for Toeplitz-type operators associated with general fractional integral operators
  101. Oscillatory hyper-Hilbert transform on Wiener amalgam spaces
  102. Euler-type sums involving multiple harmonic sums and binomial coefficients
  103. Poly-falling factorial sequences and poly-rising factorial sequences
  104. Geometric approximations to transition densities of Jump-type Markov processes
  105. Multiple solutions for a quasilinear Choquard equation with critical nonlinearity
  106. Bifurcations and exact traveling wave solutions for the regularized Schamel equation
  107. Almost factorizable weakly type B semigroups
  108. The finite spectrum of Sturm-Liouville problems with n transmission conditions and quadratic eigenparameter-dependent boundary conditions
  109. Ground state sign-changing solutions for a class of quasilinear Schrödinger equations
  110. Epi-quasi normality
  111. Derivative and higher-order Cauchy integral formula of matrix functions
  112. Commutators of multilinear strongly singular integrals on nonhomogeneous metric measure spaces
  113. Solutions to a multi-phase model of sea ice growth
  114. Existence and simulation of positive solutions for m-point fractional differential equations with derivative terms
  115. Bernstein-Walsh type inequalities for derivatives of algebraic polynomials in quasidisks
  116. Review Article
  117. Semiprimeness of semigroup algebras
  118. Special Issue on Problems, Methods and Applications of Nonlinear Analysis (Part II)
  119. Third-order differential equations with three-point boundary conditions
  120. Fractional calculus, zeta functions and Shannon entropy
  121. Uniqueness of positive solutions for boundary value problems associated with indefinite ϕ-Laplacian-type equations
  122. Synchronization of Caputo fractional neural networks with bounded time variable delays
  123. On quasilinear elliptic problems with finite or infinite potential wells
  124. Deterministic and random approximation by the combination of algebraic polynomials and trigonometric polynomials
  125. On a fractional Schrödinger-Poisson system with strong singularity
  126. Parabolic inequalities in Orlicz spaces with data in L1
  127. Special Issue on Evolution Equations, Theory and Applications (Part II)
  128. Impulsive Caputo-Fabrizio fractional differential equations in b-metric spaces
  129. Existence of a solution of Hilfer fractional hybrid problems via new Krasnoselskii-type fixed point theorems
  130. On a nonlinear system of Riemann-Liouville fractional differential equations with semi-coupled integro-multipoint boundary conditions
  131. Blow-up results of the positive solution for a class of degenerate parabolic equations
  132. Long time decay for 3D Navier-Stokes equations in Fourier-Lei-Lin spaces
  133. On the extinction problem for a p-Laplacian equation with a nonlinear gradient source
  134. General decay rate for a viscoelastic wave equation with distributed delay and Balakrishnan-Taylor damping
  135. On hyponormality on a weighted annulus
  136. Exponential stability of Timoshenko system in thermoelasticity of second sound with a memory and distributed delay term
  137. Convergence results on Picard-Krasnoselskii hybrid iterative process in CAT(0) spaces
  138. Special Issue on Boundary Value Problems and their Applications on Biosciences and Engineering (Part I)
  139. Marangoni convection in layers of water-based nanofluids under the effect of rotation
  140. A transient analysis to the M(τ)/M(τ)/k queue with time-dependent parameters
  141. Existence of random attractors and the upper semicontinuity for small random perturbations of 2D Navier-Stokes equations with linear damping
  142. Degenerate binomial and Poisson random variables associated with degenerate Lah-Bell polynomials
  143. Special Issue on Fractional Problems with Variable-Order or Variable Exponents (Part I)
  144. On the mixed fractional quantum and Hadamard derivatives for impulsive boundary value problems
  145. The Lp dual Minkowski problem about 0 < p < 1 and q > 0
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