S. N. Bernstein Type Estimations in the Mean on the Curves in a Complex Plane

Abstract

The present paper discusses in the metric 𝐿_𝑝 S. N. Bernstein type inequalities of the most general kind on very general accessible classes of curves in a complex plane. The obtained estimations, generally speaking, are not improvable.

1. Introduction

The estimations connecting the norms of derivatives of polynomials with the norm of the polynomial itself are usually called the Markov-Bernstein type estimations. Therewith, the similar global estimation in the metric 𝐶_[−1,1] was obtained by Markov. Bernstein considered the similar estimation in the metric 𝐶_[0,2𝜋] for trigonometric polynomials and also local estimation in the metric 𝐶_[−1,1]. Bernstein type local estimation, which is precise in the sense of order in the metric 𝐶_[−1,1], was obtained by Dzjadyk. Further, Dzjadyk considered this estimation in a complex plane. Earlier, such a global estimation in a complex plane in the metric 𝐶 was obtained by Mergelyan [1]. Validity of such estimations on arbitrary compacts in a complex plane in the metric 𝐶 was shown in the papers of Lebedev and Tamrazov [2]. Similar problems in the mean, namely, in the metric 𝐿_𝑝 have their own specification that does not allow to consider such estimations on wide classes of sets in a complex plane. For a long time, the validity of such estimations was known on very narrow classes of curves of a complex plane. These results, in particular, are given in [3].

One of the theorems with appropriate Bernstein inequality is announced in [4]. Some auxiliary statements, by means of which such inequalities are proved, are in [4].

In the sequel, we will need the following facts.

2. Preliminary Notes

Let Γ be a closed curve in a complex plane with parametric representation 𝑧=𝑧(𝑡) (0≤𝑡≤𝑙, 𝑙 is the length of Γ) of diameter 𝑑^∗(𝑑^∗=sup_{𝑡,𝜏∈Γ}|𝑡−𝜏|) the function 𝑧=𝜓(𝜔) maps the exterior of a unit circle 𝛾₀ onto the exterior of Γ, and 𝑧=𝜓₀(𝜔) maps the interior of 𝛾₀ on the interior of Γ; the functions 𝑤=𝜑(𝑧) and 𝑤=𝜑₀(𝑧) are inverse to the functions 𝑧=𝜓(𝜔) and 𝑧=𝜓₀(𝜔), respectively; Γ_1+𝜌 is a level line of the curve Γ corresponding to the equation |𝜑(𝑧)|=1+𝜌(𝜌>0).

Let 𝑡 be some fixed point on Γ_1+𝜌(𝜌>0), let 𝑑(𝑡,Γ)=𝑑 be the distance from the point 𝑡 to the curve Γ,Γ^∗_𝛿(𝑡)={𝑧∈Γ∶|𝑧−𝑡|<𝛿}, and𝜃^∗_𝑡(𝛿)=𝜃^∗_𝑡(𝛿,Γ)=mesΓ^∗_𝛿(𝑡).

Let us consider a class of curves Γ, for which 𝜃^∗_𝑡(𝛿)≤𝐶(Γ)𝛿 for 𝛿≥2𝑑. We denote this class of curves by 𝑆^∗_𝜃. It is easy to show that the class 𝑆^∗_𝜃 coincides with the class 𝑆_𝜃 introduced by Salayev [5].

Recall that the curve Γ belongs to the class 𝑆_𝜃, (Salayev’s class) if there exists a constant 𝐶(Γ)≥1, such that 𝜃(𝛿)≤𝐶(Γ)𝛿, where Γ_𝛿(𝑡)={𝜏∈Γ∶|𝑡−𝜏|≤𝛿}, (0<𝛿≤𝑑)𝜃_𝑡(𝛿)=mesΓ_𝛿(𝑡) (Lebesgue measure), and 𝜃(𝛿)=sup_𝑡∈Γ𝜃_𝑡(𝛿).

So, the following statement [4] is true.

Statement 2.1 (𝑆_𝜃=𝑆^∗_𝜃). By 𝐽_𝛾 we denote a class of Jordan rectifiable curves Γ, for which the following relation [4] 𝑑^𝛾−11(𝑡,𝑛)⋅_Γ|𝑑𝑧||𝑧−𝑡|^𝛾≤𝐶(Γ,𝛾) (2.1) is valid (𝑑 is a distance from the point 𝑡∈Γ_1+1/𝑛to the curve Γ) for the given 𝛾>1 and all 𝑡∈Γ_1+1/𝑛.

The following statement [4] is also valid.

Statement 2.2 (If 1<𝛾₁<𝛾₂, then 𝐽_𝛾1⊂𝐽_𝛾2). We will say that the function 𝜃_𝑡(𝛿,Γ)/𝛿 is almost increasing in 𝛿 uniformly in 𝑡, if there exists a constant 𝐶(Γ) not depending on 𝑡 such that for any 𝛿₁<𝛿₂ the following inequality 𝜃_𝑡(𝛿₁,Γ)/𝛿₁≥𝐶(Γ)(𝜃_𝑡(𝛿₂,Γ)/𝛿₂) is fulfilled.

Note that many known classes of rectifiable curves, in particular the curves of the class 𝑆_𝜃 (Salayev's class), satisfy the condition that 𝜃_𝑡(𝛿,Γ)/𝛿 is almost decreasing.

By 𝐽^∗_𝛾 we denote a subclass of the class of curves 𝐽_𝛾, for which 𝜃_𝑡(𝛿,Γ)/𝛿 almost decreases. For the classes of curves 𝐽_𝛾 and 𝐽^∗_𝛾, the following statement is valid [4].

Statement 2.3. There hold the embeddings 𝑆_𝜃⊂𝐽_𝛾,𝐽^∗_𝛾⊂𝑆_𝜃. (2.2)

Now, let us consider the quantity

1𝛿(𝑧,𝑛)=(_Γ1+1/𝑛|𝑑𝑡||𝑧−𝑡|²)⁻¹,𝑧∈Γ. (2.3)

3. Main Results

In particular, using the previously mentioned statement, we can prove the following theorems.

Theorem 3.1. Let Γ be an arbitrary rectifiable Jordan curve on which for any 𝑠∈(0,∞) and any natural 𝑗 the following estimation is valid (signs ≼ and ≍ define an ordinal relation. Namely, 𝐴≼𝐵 means 𝐴≤const𝐵. And 𝐴≍𝐵 means const𝐴≤𝐵≤const𝐴): 𝛿^𝑠1(𝑡,𝑛)_Γ𝛿^𝑗−𝑠(𝑧,1/𝑛)|𝑑𝑧||𝑧−𝑡|^𝑗+1≼1,𝑡∈Γ_1+1/𝑛, (3.1) where 1𝛿(𝑡,𝑛)=(_Γ|𝑑𝑧||𝑧−𝑡|²)⁻¹. (3.2)
Then ‖𝛿^𝑗−𝑠1(𝑧,𝑛)𝑃_𝑛^(𝑗)(𝑧)‖_𝐿𝑝(Γ)≼‖𝛿^−𝑠1(𝑧,𝑛)𝑃_𝑛(𝑧)‖_𝐿𝑝(Γ), (3.3) where 𝑃_𝑛(𝑧) is an algebraic polynomial of degree 𝑛∈𝑁, 𝑝≥1.

We can also prove a theorem of independent character used in the proof of Theorem 3.1.

Theorem 3.2. Under the conditions of Theorem 3.1 on the curve Γ, whatever was the natural number 𝑗 and 𝑠∈(−∞,∞) for the 𝑗th-order derivative of the polynomial 𝑃_𝑛(𝑧) of degree ≤𝑛, for 𝑝≥1, the following inequality ‖𝑃_𝑛^(𝑗)(𝑡)𝛿^𝑠−𝑗‖(𝑡,1/𝑛)_{𝐿𝑝(Γ1+1/𝑛)}𝑃≤𝐶(Γ,𝑝,𝑗,𝑠)‖^𝑛(𝑧)𝛿^𝑠‖(𝑧,1/𝑛)_𝐿𝑝(Γ) (3.4) is valid.

A special case of these theorems is similar theorems for concrete classes of curves, namely, for the following classes.

(a) 𝐾-quasiconformal mapping. The curve Γ, being an image of the circle under some 𝐾-quasiconformal mapping of the plane onto itself, is said to be 𝐾-quasiconformal curve. The class of curves will be denoted by 𝐴_𝑘.

(b) We will say that the set 𝐸 with rectifiable Jordan curve Γ=𝜕𝐸 belongs to the class 𝐵_𝑘 [3] for some 𝑘 (or Γ∈𝐵_𝑘), if Γ∈𝑆_𝜃 and satisfies the following conditions:

(1) |𝑧−𝑧|≍𝑑(𝑧,1/𝑛),whereforall𝑧∈Γ,𝑧=𝑧(1/𝑛)=𝜓((1+(1/𝑛))𝜑(𝑧)),𝑧_∼=𝜓((1+(1/𝑛))⁻¹𝜑(𝑧));

(2) |𝑡̃−𝑡|≼|𝑡̃−𝑧|^𝑘−1|𝑧−𝑧|,∀𝑧,𝑡∈Γ.

As Dzjadyk shows [3, page 393], the validity of the condition

|1𝑧−𝑧|≍𝑑(𝑧,𝑛) (3.5) that is equivalent to the following geometric property of domain 𝐸 [6] follows from conditions (1) and (2) of the class 𝐵_𝑘.

(3) We can connect any points of 𝑧 by the arc 𝛾(𝑧,𝜉)⊂𝐸 whose length satisfies the inequality mes𝛾(𝑧,𝜉)≼|𝑧−𝜉|. (3.6)

Furthermore, [6, Lemmas 1 and 2], the following conditions are valid for the set 𝐸 of the class 𝐵_𝑘:

(4) if 𝜉∈Ω=𝐶𝐸,𝜉_Γ=𝜓[𝜑(𝜉)|𝜑(𝜉)|⁻¹],Γ=𝜕𝐸, then 𝑑(𝜉,Γ)=^definf_𝑧∈Γ|𝜉−𝑧|≍|𝜉−𝜉_Γ|; (3.7)

(5) if 𝑧∈Γ, 𝑧_𝑅=𝜓[𝑅𝜑(𝑧)],𝑅>1, then 𝑑(𝑧,Γ_R)=^definf_{𝑡∈Γ𝑅}|𝑧−𝑡|≍|𝑧_𝑅−𝑧|. (3.8)

Note that the 𝐾-quasiconformal curves [7] satisfy conditions (1)–(5) and relation (3.5).

Consider some more general classes.

(c) We will say that 𝐸∈𝐻 (or Γ=𝜕𝐸∈𝐻), if conditions (4) and (5) are fulfilled.

(d) We will say that 𝐸 with a rectifiable boundary Γ belongs to 𝐷 (or Γ∈𝐷), if Γ∈𝑆₀, and conditions (3) or its equivalent relation (3.5) is fulfilled for it.

Obviously, the class of the sets 𝐷, possessing pure geometric description, contains the classes of the sets 𝐵_𝑘.

So, the following theorems are true.

Theorem 3.3. Let Γ be an arbitrary rectifiable 𝐾-quasiconformal curve. Then, whatever was the natural number 𝑗 and the number 𝑠∈(−∞,∞) for the 𝑗th order derivative of the polynomial 𝑃_𝑛 of power ≤𝑛 for 𝑝≥1, the following inequality is valid: ‖𝑑^𝑗−𝑠1(𝑡,𝑛)𝑃_𝑛^(𝑗)(𝑡)‖_{𝐿𝑝(Γ1+1/𝑛)}≤𝐶(Γ,𝑝,𝑗,𝑠)‖𝑑^−𝑠1(𝑧,𝑛)𝑃_𝑛(𝑧)‖_𝐿𝑝(Γ). (3.9)

Theorem 3.4. Let Γ for some natural 𝑘 belong to the class 𝐵_𝑘. Then, whatever was the natural number 𝑗 and (under some additional condition on the curve Γ, Theorem 3.4 remains valid for any 𝑠≥0 (see Remark 5.1).) 𝑠∈[0,𝑘𝑗/(𝑘−1)𝑝) for the 𝑗th-order derivative of the polynomial 𝑃_𝑛 of power ≤𝑛 for 𝑝≥1, the following inequality is valid: ‖𝑑^𝑗−𝑠1(𝑧,𝑛)𝑃_𝑛^(𝑗)(𝑧)‖_𝐿𝑝(Γ)≤𝐶(Γ,𝑝,𝑗,𝑠)‖𝑑^−𝑠1(𝑧,𝑛)𝑃_𝑛(𝑧)‖_𝐿𝑝(Γ). (3.10)

The special case of these theorems is announced in [8] and is cited in [9, 10] with incomplete proof.

Remark 3.5. The special case of Theorems 3.3 and 3.4 was also proved in [11] for curves consisting of infinitely many smooth arcs; each of these arcs has continuous curvature, and at the joint points 𝑧_𝑗(𝑖=1,𝑚) they form between themselves external angles 𝛼_𝑖𝜋 such that 1<𝛼_𝑖<2, that is, on the curves of the class 𝑊_[1,2).

In this paper, we give a complete proof of Theorems 3.3 and 3.4. Theorems 3.1 and 3.2 are proved by the same method Theorems 3.3 and 3.4 with the usage of Statements 2.1–2.3.

4. Auxiliary Lemmas

When proving Theorems 3.3 and 3.4 we’ll need the following.

(1^∘) A nonnegative function 𝜌(𝑧) given on the plane 𝑧 will be said to be admissible if 𝐴(𝜌)=𝜌²𝑑𝑥𝑑𝑦<+∞. (4.1)

If 𝚃 is a family of locally rectifiable curves on the plane, we put 𝐿_𝜌(𝚃)=inf_{𝛾∈𝚃𝛾}𝜌|𝑑𝑧| (4.2) (if 𝜌 is not measurable on 𝛾, we assume that∫_𝛾𝜌|𝑑𝑧|=∞). If 𝙿 is a class of admissible functions, then the quantity 𝜆(𝚃)=sup_𝜌∈𝙿𝐿²_𝜌(𝚃)𝐴(𝜌) (4.3) is said to be external length of 𝚃, and its inverse quantity 𝜆⁻¹(𝚃)=^def𝑚(𝚃) (4.4) a modulus of 𝚃 is 𝑚(𝚃)=𝜆⁻¹(𝚃)=inf_𝜌∈𝙿𝐴(𝜌)𝐿²_𝜌(𝚃). (4.5)

Let Ω be an arbitrary one-connected domain of a complex domain containing the point 𝑧=∞; let 𝐵 be a complement to Ω; let Γ=𝜕Ω=𝜕𝐵 be their common boundary; let 𝑤=𝜑(𝑧) be a function that conformally and univalently maps Ω onto Ω^′ exterior of a unit circle and is normed by the following condition:

𝜑(∞)=∞,lim_𝑧→∞𝜑(𝑧)𝑧>0;𝑧=𝜓(𝑤)=𝜑⁻¹(𝑤); (4.6)

Γ=_1+𝜎^def{𝑡∶|𝜑(𝑡)|=1+𝜎≥1} be a level line of the continuum 𝐵; let 𝑑(𝑧,𝜎)=^definf_{𝑡∈Γ1+𝜎}|𝑧−𝑡|, for 𝑧∈Γ; let 𝑑(𝑡,𝜎)=^definf_𝑧∈Γ|𝑧−𝑡|, for 𝑡∈Γ_1+𝜎.

The following statements are valid.

Lemma A [see [12, Lemma 1.2]] Let 𝐵 be an arbitrary continuum with connected complement Ω, 𝑧₀=Γ=𝜕𝐵, 𝑧₁,𝑧₂∈Ω.

If |𝑧₁−𝑧₀|>|𝑧₂−𝑧₀|, |𝜑(𝑧₁)−𝜑(𝑧₂)|≤𝐶₁|𝜑(𝑧₂)−𝜑(𝑧₀)|, then

12𝜋ln|𝑧₁−𝑧₀||𝑧₂−𝑧₀|≤𝑚(𝚃)=𝑚(𝚃^′)≤𝐶(𝐶₁,Γ), (4.7) where 𝚃 is a family of curves isolating the points 𝑧₁ and 𝑧_∗ in Ω (in simplest cases 𝑧_∗=𝑧₀) from the points 𝑧₂ and ∞ and 𝚃^′=𝜑(𝚃).

Lemma B [see [13, Theorem 1]] Let 𝐵 be an arbitrary continuum with connected complement. Then for 𝑤∈Ω^′|𝜓^′(𝑤)|≍𝑑(𝜓(𝑤),𝐵)|𝑤|−1, (4.8) or |𝜑^′(𝑤)|≍|𝜑(𝑧)|−1𝑑(𝑧,𝐵),𝑧=𝜓(𝑤), (4.9) where 𝑑(𝑧,𝐵) is a distance from the point 𝑧=𝜓(𝑤) to 𝐵.

Lemma C [see [14, Lemma 1]] Let 𝑤=𝐹(𝑧) realizes 𝐾-quasiconformal mapping of plane onto itself, 𝐹(∞)=∞. 𝐶_𝑧,𝐶_𝑤 are, respectively, 𝑧- and 𝑤-complex planes; 𝑧_𝑗∈𝐶_𝑧,𝐹(𝑧_𝑗)=𝑤_𝑗∈𝐶_𝑤,(𝑗=1,2,3).

Then we have the following:

(1) the conditions |𝑧₁−𝑧₂|≼|𝑧₁−𝑧₃| and |𝑤₁−𝑤₂|≼|𝑤₁−𝑤₃| are equivalent, and consequently the conditions |𝑧₁−𝑧₂|≍|𝑧₁−𝑧₃|,|𝑤₁−𝑤₂|≍|𝑤₁−𝑤₃| (4.10) are also equivalent;

(2) if |𝑧₁−𝑧₂|≼|𝑧₁−𝑧₃|, then |𝑤₁−𝑤₃||𝑤₁−𝑤₂|^𝐴≼|𝑧₁−𝑧₃||𝑧₁−𝑧₂|≼|𝑤₁−𝑤₃||𝑤₁−𝑤₂|^𝐵, (4.11) where 𝐴=𝐾⁻¹,𝐵=𝐾.

Lemma D [see [11]] Let 𝐺 be a domain with a rectifiable boundary Γ, and Ω=𝐶𝐺(∞∈Ω). If 𝑓∈𝐸_𝑝(Ω)(𝑝≥1), then for any 𝑅>1 and for all 𝜌∈(1,𝑅] the following inequality holds: ‖𝑓‖_{𝐿𝑝(Γ𝜌)}≤𝑅^2/𝑝‖𝑓‖_𝐿𝑝(Γ). (4.12)

(2^∘) Let 𝜉 be some arbitrary fixed point lying outside of Γ, and let 𝑑=𝑑(𝜉,Γ) be a distance from the point 𝜉 to Γ, Γ_𝛿(𝜉)={𝑧∈Γ∶|𝑧−𝜉|<𝛿}, and 𝜃^∗_𝜉(𝛿)=𝜃^∗_𝜉(𝛿,Γ)=mesΓ_𝛿(𝜉).

To prove these theorems we will need the following lemmas.

Lemma 4.1. Let a rectifiable curve Γ∈𝐻, then for a polynomial 𝑃_𝑛 of power ≤𝑛 for 𝑝≥1 the following inequality is valid for 𝑠∈(−∞,∞): ‖𝑑^−𝑠1(𝑡,𝑛)𝑃_𝑛(𝑡)‖_{𝐿𝑝(Γ1+1/𝑛)}≤𝐶(Γ,𝑝,𝑠)‖𝑑^−𝑠1(𝑧,𝑛)𝑃_𝑛(𝑧)‖_𝐿𝑝(Γ). (4.13)

Lemma 4.2. Under conditions of Lemma 4.1 on the curve Γ, for a polynomial 𝑃_𝑛 of power ≤𝑛 for 𝑝≥1, 𝑠∈(−∞,∞) and 𝜌≤1/𝑛 the following inequality is valid: ‖𝑑^−𝑠(𝜉,Γ_𝑅)𝑃_𝑛(𝜉)‖_{𝐿𝑝(Γ1+𝜌)}≤𝐶(Γ,𝑝,𝑠)‖𝑑^−𝑠1(𝑧,𝑛)𝑃_𝑛(𝑧)‖_𝐿𝑝(Γ), (4.14) where under 𝑑^−𝑠(𝜉,Γ_𝑅),𝜉∈Γ_1+𝜌 one understands a distance from the point 𝜉,𝜉=𝜓(𝜏) to the level line Γ_𝑅, where 𝑅=|𝜏|(1+1/𝑛),𝜏=𝜑(𝜉).

Lemma 4.3. Let Γ∈𝐵_𝑘. Then whatever was a natural number 𝑗 and 𝑠∈[0,𝑘𝑗/(𝑘−1)), the inequality 𝑑^𝑠1(𝑡,𝑛)_Γ𝑑^𝑗−𝑠(𝑧,1/𝑛)|𝑑𝑧||𝑧−𝑡|^𝑗+1≤𝐶(Γ,𝑠,𝑗),∀𝑡∈Γ_1+1/𝑛 (4.15) is valid.

Lemma 4.4 (see [9]). Let Γ∈𝑆_𝜃. Then for 𝛾>1 and all 𝑡∈Γ_1+1/𝑛 the relation (2.1) is valid; that is, the imbedding 𝑆_𝜃⊂𝐽_𝛾(𝛾>1) is valid.

Lemma 4.5 (see [9]). Let Γ∈𝐷. Then for 𝛾>1 and all 𝑧∈Γ the inequality 𝑑^𝛾−11(𝜉,𝑛)_Γ1+1/𝑛|𝑑𝑡||𝑡−𝑧|^𝛾≤𝐶(Γ,𝛾) (4.16) is valid.

Proof of Lemma 4.1. Let an arbitrary rectifiable curve Γ∈𝐻. At first we consider the case 𝑠≥0. İntroduce some auxiliary function 𝑆(𝑧)=[𝜑^′(𝑧)]^𝑠𝑃_𝑛(𝑧)[𝜑(𝑧)]^𝑛, (4.17) where 𝑧=𝑧(1/𝑛)=^def𝜓((1+1/𝑛)𝜑(𝑧)).
Obviously, 𝑆(𝑧)→0, as 𝑧→∞ and each of its branchs is holomorphic in 𝐶𝐺(Γ=𝜕𝐺) and continuous in 𝐶𝐺. Therefore 𝑆∈𝐸_𝑝(𝐶𝐺). Consequently, we can apply to 𝑆(𝑧) Lemma D where by estimation of Lemma B we will have ‖𝑃_𝑛(𝑡)𝑑^𝑠(𝑡,̃Γ)𝜑^𝑛‖(𝑡)_{𝐿𝑝(Γ1+1/𝑛)}𝑃≤𝐶(𝑃)‖^𝑛(𝑧)𝑑^𝑠(𝑧,1/𝑛)𝜑^𝑛‖(𝑧)_𝐿𝑝(Γ). (4.18)

Now, if we consider that |𝜑(𝑡)|^𝑛≍1, for 𝑡∈Γ_1+1/𝑛 and the relations 𝑑(𝑧,1/𝑛)≍|𝑧−𝑧|≍𝑑(𝑧,1/𝑛), which is valid for any Γ∈𝐻, then for the proof of (4.13) it suffices to prove the validity of the relation

𝑑(𝑡,̃Γ)≍𝑑(𝑡,1𝑛), (4.19) where 𝑡∈Γ_1+1/𝑛.

Let 𝑡∈Γ_1+1/𝑛,𝑡=_∼^def𝜓((1+1/𝑛)⁻¹𝜑(𝑡)). Obviously _∼𝑡∈Γ. By the property of curves of the class 𝐻, we have

1𝑑(𝑡,𝑛)≍|𝑡−𝑡_∼|≍𝑑(𝑡_∼,1𝑛̃̃),𝑑(𝑡,Γ)≍|𝑡−𝑡_∼|≍𝑑(𝑡_∼,2+1/𝑛𝑛). (4.20) Prove that

𝑑(𝑡_∼,1𝑛)≍𝑑(𝑡,2+1/𝑛𝑛). (4.21)

Obviously, it suffices to prove that

𝑑(𝑡_∼,1𝑛)⋟𝑑(𝑡_∼,2+1/𝑛𝑛). (4.22) Let 𝑡₁∈Γ_1+1/𝑛, 𝑡₂∈Γ_{1+((2+1/𝑛)/𝑛)} be such that

𝑑(𝑡_∼,1𝑛)=|𝑡_∼−𝑡₁|,𝑑(𝑡_∼,2+1/𝑛𝑛)=|𝑡_∼−𝑡₂𝑤|;₁=𝜑(𝑡₁),𝑤₂=𝜑(𝑡₂),𝑤=𝜑(𝑡). (4.23)

Following Belyi [7], we take in the ring

11+𝑛≤|𝑤|≤1+2+1/𝑛𝑛 (4.24) a segment and an arc of a circle connecting the points 𝑤₁ and 𝑤₂. Let 𝑙=𝑙(𝑤₁,𝑤₂). Construct a family of circles with a center at the point 𝑡, intersecting 𝑙. Each of these has an annular arc in Ω=𝐶𝐺, intersecting 𝑙. We denote a family of such arcs by 𝚃. Obviously, the family 𝚃 separates in Ω the point 𝑡₁ and some point _∼𝑡^∗ (in the simplest cases _∼𝑡^∗=𝑡_∼) from 𝑡₂ and ∞. Therefore, by Lemma A we have

12𝜋ln𝑑(𝑡_∼,(2+1/𝑛)/𝑛)𝑑(𝑡_∼≍1,1/𝑛)2𝜋ln|𝑡_∼−𝑡₂||𝑡_∼−𝑡₁|≤𝑚(𝚃)=𝑚(𝚃^′)≤𝐶(Γ). (4.25)

Hence (4.22) and relation (4.21) together with (4.20) prove (4.19),

So, Lemma 4.1 is proved in the case 𝑠≥0.

The proof in the case 𝑠<0 is conducted by means of analytic reasoning after introducing the auxiliary function

𝑆₁(𝑧)=[𝜑^′(𝑧)]^𝑠𝑃_𝑛(𝑧)𝜑^𝑛+|𝑠|(𝑧). (4.26) The proof of Lemma 4.2 is conducted in the same way.

Indeed, in the case 𝑠≥0, instead of relation (4.18) from Lemma D we’ll have

‖𝑃_𝑛(𝜉)𝑑^𝑠(𝜉̃,Γ)𝜑^𝑛‖(𝜉)_{𝐿𝑝(Γ1+1/𝑛)}𝑃≤𝐶(𝑃)‖^𝑛(𝑧)𝑑^𝑠(𝑧,1/𝑛)𝜑^𝑛‖(𝑧)_𝐿𝑝(Γ). (4.27)

Therefore, in order to prove the statement of Lemma 4.2, obviously, it suffices to see the validity of the relation

𝑑(𝜉̃,Γ)≍𝑑(𝜉,Γ_𝑅),𝜉∈Γ_1+𝜌,𝜉̃=𝜓((1+1𝑛1)𝜑(𝜉)),𝑅=|𝜏|(1+𝑛), (4.28) and since the estimation 𝑑(𝜉,Γ_𝑅)≤𝑑(𝜉̃,Γ) is obvious, we have to show that

𝑑(𝜉̃,Γ)≼𝑑(𝜉,Γ_𝑅),𝜉∈Γ_1+𝜌,𝑅=|𝜏|(1+1𝑛),(|𝜏|=1+𝜌). (4.29) This relation is proved exactly in the same way as relation (4.19) in Lemma 4.1.

The case 𝑠≤0 is proved similarly.

Proof of Lemma 4.3. Let Γ∈𝐵_𝑘. Consider two possible cases.
(1) We have 𝑠≤𝑗. The case 𝑠=𝑗 follows from Lemma 4.4.
Let 𝑡=𝜓((1+1/𝑛)𝜑(𝑡_∼)), where 𝑡∈Γ_1+1/𝑛, and _∼𝑡∈Γ. Then _∼𝑡̃=𝜓((1+1/𝑛)𝜑(𝑡_∼))=𝑡. By the property of the class 𝐵_𝑘, we will have 1𝑑(𝑧,𝑛)≍|𝑧−𝑧|≼|𝑧−𝑡_∼|^{(𝑘−1)/𝑘}|𝑡_∼̃−𝑡_∼|^1/𝑘 (4.30) and (see [3, page 393]) |𝑡−𝑡_∼1|≍𝑑(𝑡,𝑛). (4.31) Now, by (4.30) and (4.31), we will get 1𝑑(𝑧,𝑛)≼|𝑧−𝑡_∼|^{(𝑘−1)/𝑘}𝑑^1/𝑘1(𝑡,𝑛),_∼𝑡̃=𝑡. (4.32) Hence we will get 𝐵=^def𝑑^𝑠1(𝑡,𝑛)_Γ𝑑^𝑗−𝑠(𝑧,1/𝑛)|𝑑𝑧||𝑧−𝑡|^𝑗+1≼𝑑^𝑠1(𝑡,𝑛)𝑑^{(𝑗−𝑠)/𝑘}1(𝑡,𝑛)_Γ|𝑧−𝑡_∼|^{((𝑘−1)/𝑘)(𝑗−𝑠)}|𝑑𝑧||𝑧−𝑡|^𝑗+1. (4.33) Now, if we take into account |𝑧−𝑡_∼|≼|𝑧−𝑡| and |𝑧−𝑡_∼|≤|𝑧−𝑧|+|𝑧−𝑡_∼1|≼𝑑(𝑧,𝑛)+|𝑧−𝑡|+|𝑡−𝑡_∼1|≤|𝑧−𝑡|+|𝑧−𝑡|+𝑑(𝑡,𝑛)≼|𝑧−𝑡|, (4.34) then by Lemma 4.4, for Γ∈𝑆_𝜃(𝐵_𝑘⊂𝑆_𝜃) we will get 𝐵≼𝑑^{𝑠+(𝑗−𝑠)/𝑘}1(𝑡,𝑛)_Γ|𝑑𝑧||𝑧−𝑡|^{1+𝑠+(𝑗−𝑠)/𝑘}≼1. (4.35)
(2) We have 𝑗<𝑠<𝑘𝑗/(𝑘−1). By the property of the class of curves 𝐵_𝑘, we will have |𝑡−𝑡_∼|^𝑘≼|𝑡−𝑧|^𝑘−1|𝑧−𝑧|, (4.36) hence 1𝑑(𝑧,𝑛)⋟|𝑧−𝑧|⋟|𝑡−𝑡_∼|^𝑘|𝑡−𝑧|^𝑘−1⋟𝑑^𝑘(𝑡,1/𝑛)|𝑡−𝑧|^𝑘−1. (4.37) Hence, using Lemma 4.4𝐵=^def𝑑^𝑠1(𝑡,𝑛)_Γ|𝑑𝑧|𝑑^𝑠−𝑗(𝑧,1/𝑛)|𝑧−𝑡|^𝑗+1≼𝑑^{𝑠−𝑘(𝑠−𝑗)}1(𝑡,𝑛)_Γ|𝑑𝑧||𝑧−𝑡|^{1+𝑠−𝑘(𝑠−𝑗)}≼1. (4.38) So, Lemma 4.3 is proved.

5. Proofs of Theorems

Proof of Theorem 3.3. Consider the case 𝑝=1. Let Γ be an arbitrary rectifiable 𝐾-quasiconformal curve. By the Cauchy formula, we will have 𝐴^def𝑃=‖^𝑛(𝑗)(𝑡)𝑑^𝑠−𝑗‖(𝑡,1/𝑛)_{𝐿1(Γ1+1/𝑛)}=𝑗!2𝜋_Γ1+1/𝑛|𝑑𝑡|𝑑^𝑠−𝑗|(𝑡,1/𝑛)^𝛾𝑡𝑃_𝑛(𝜉)𝑑^𝑠(𝜉−𝑡)^𝑗+1|≤𝑗!2𝜋_Γ1+1/𝑛|𝑑𝑡|𝑑^𝑠−𝑗(𝑡,1/𝑛)^𝛾𝑡|𝑃_𝑛(𝜉)||𝑑𝜉|, (5.1) where 𝛾_𝑡 denotes a closed curve containing the point 𝑡 interior to itself, and that is defined in the following way.
Let the point 𝑡∈Γ_1+1/𝑛 under the mapping 𝑤=𝜑(𝑡) go over to the point 𝑢 (Figure 1).Draw a circle 𝛾_𝑢 with a center at the point 𝑢 of radius 1/𝑛. Denote preimage of this circle under the mapping 𝑧=𝜓(𝑤)(𝑤=𝜑(𝑧)) by 𝛾_𝑡.
With such a construction of 𝛾_𝑡 it is easy to see that by Lemma C, for all 𝜉∈𝛾_𝑡, the relation 1|𝜉−𝑡|≍𝑑(𝑡,𝑛) (5.2) will be valid.
Really, since the relation |𝜏−𝑢|=|𝑢−𝑢_∼|=1/𝑛,𝑢_∼=𝜑(𝑡_∼),𝜓(𝜏)=𝜉,𝜓(𝑢)=𝑡 is valid for all 𝜏∈𝛾_𝑢, then by Lemma C we will have |𝜉−𝑡|≍|𝑡−𝑡_∼|. (5.3) And since |𝑡−𝑡_∼1|≍𝑑(𝑡,𝑛) (5.4) (see [7]), then |𝜉−𝑡|≍𝑑(𝑡,1/𝑛).
Therefore, by Lemma C from relation (5.1) we find 𝐴≼_Γ1+1/𝑛|𝑑𝑡|𝑑^𝑠+1(𝑡,1/𝑛)^𝛾𝑡|𝑃_𝑛=(𝜉)||𝑑𝜉|_{|𝑢|=1+1/𝑛}|𝜓^′(𝑢)||𝑑𝑢|𝑑^𝑠+1×(𝜓(𝑢),1/𝑛)^𝛾𝑢|𝑃_𝑛(𝜓(𝜏))||𝜓^′(𝜏)|𝑑𝜏||≍𝑛_{|𝑢|=1+1/𝑛}|𝑑𝑢|_𝛾𝑢|𝑃_𝑛(𝜓(𝜏))||𝜓^′(𝜏)|𝑑^𝑠(𝜓(𝜏),Γ_𝑅)|𝑑𝜏|, (5.5) and under 𝑑(𝜓(𝜏),Γ_𝑅) we understand a distance from the point 𝜉=𝜓(𝜏) to the level line Γ_𝑅, where 𝑅=|𝜏|(1+1/𝑛). Therewith, by Lemma C, we take into account that this distance has the same order of 𝑑(𝑡,1/𝑛), that is, 𝑑(𝜓(𝜏),Γ_𝑅1)≍𝑑(𝜓(𝑢),𝑛). (5.6) Really, 1|𝜏−𝜏(1+𝑛)|=|𝜏−𝑢|≍|𝑢−𝑢_∼|(𝑢_∼1=(1+𝑛)⁻¹𝑢) (5.7) is obvious.
Hence, by Lemma C it follows that 1|𝜓(𝜏)−𝜓(𝜏(1+𝑛))|=|𝜓(𝜏)−𝜓(𝑢)|≍|𝜓(𝑢)−𝜓(𝑢_∼)|=|𝑡−𝑡_∼|. (5.8) And since |𝑡−𝑡_∼1|≍𝑑(𝑡,𝑛1)=𝑑(𝜓(𝑢),𝑛) (5.9) (see [7]), then 1|𝜓(𝜏)−𝜓(𝜏(1+𝑛1))|≍𝑑(𝜓(𝑢),𝑛). (5.10) It remains to show that 𝑑(𝜉,Γ_𝑅)=𝑑(𝜓(𝜏),Γ_𝑅)≍|1𝜉−𝜉|=|𝜓(𝜏(1+𝑛))−𝜓(𝜏)|. (5.11) And since the relation 𝑑(𝜓(𝜏),Γ_𝑅)≤|𝜓(𝜏)−𝜓(𝜏)| (5.12) is obvious, it suffices to show that 𝑑(𝜓(𝜏),Γ_𝑅)⋟|𝜓(𝜏)−𝜓(𝜏)|. (5.13)
Let 𝜉₀∈Γ_𝑅(𝑅=|𝜏|(1+1/𝑛)),𝜉₀=𝜓(𝜏₀),|𝜏₀|=|𝜏|(1+1/𝑛) be a point for which |𝜉₀−𝜉|=𝑑(𝜉,Γ_𝑅). Obviously, |𝜏₀−𝜏|≥|𝜏−𝜏|. Hence, by Lemma C, it follows the estimation 𝑑(𝜓(𝜏),Γ_𝑅)=𝑑(𝜉,Γ_𝑅)=|𝜉̃−𝜉|=|𝜓(𝜏)−𝜓(𝜏)| (5.14) that proves relation (5.11) and (5.13); hence the relation (5.6) that we need follows.
Now, in order to estimate the right-hand side of relation (5.5), we divide the circle 𝛾_𝑢 into the arc 𝛾₁, situated interior to the circle |𝑤|=1+1/𝑛 with the ends at the points 𝐵₁ and 𝐵₂ (see Figure 2) and the arc 𝛾₂=𝛾_𝑢⧵𝛾₁. In its turn, we divide the arc 𝛾₁ into 𝛾^′₁ and 𝛾₁^′′, where 𝛾^′₁, part of the arc 𝛾₁, are situated from the left of the ray 𝑜𝑢, connecting the origin of coordinates with the point 𝑢 and 𝛾₁^′′ from the right of this ray.
Obviously, we will have 1𝐴≼𝑛(1+𝑛)₀^2𝜋𝑑𝜑{_𝛾1+_𝛾2}|𝑃_𝑛(𝜓(𝜏))||𝜓^′(𝜏)|𝑑^𝑠(𝜓(𝜏),1/𝑛)|𝑑𝜏|=𝐴₁+𝐴₂. (5.15) Estimate the quantity 𝐴₁ that will be represented in the form 𝐴₁1=𝑛(1+𝑛)₀^2𝜋𝑑𝜑{_𝛾′1+_𝛾1′′}|𝑃_𝑛(𝜓(𝜏))||𝜓^′(𝜏)|𝑑^𝑠(𝜓(𝜏),1/𝑛)|𝑑𝜏|=𝐴^′₁+𝐴₁^′′. (5.16) Obviously, for the estimation of 𝐴₁, it suffies to estimate the quantity 𝐴^′₁, since the obtained estimation remains valid for the quantity 𝐴₁^′′ as well, because of symmetric arrangement of arcs 𝛾^′₁ and 𝛾₁^′′ with respect to the arc 𝑜𝑢.
Let 𝜏∈𝛾^′₁. Then obviously, it will lie on some circle 𝛾_𝜌 with center in 𝑜 and radius equal 1+𝜌, where 𝜌∈[0,1/𝑛].
Since |𝜏−𝑢|=1/𝑛, then 𝜏=𝑢+(1/𝑛)𝑒^𝑖𝜃(𝑢=(1+(1/𝑛)𝑒^𝑖𝜃)) (see Figure 2), where 𝜃 is an angle between the ray 𝜏𝑢 and a real axis. Obviously, 𝜃=𝜋+𝜑−𝜶, where 𝜶 is an angle between the radii 𝜏𝑢 and 𝑜𝑢 (see Figure 2) that may be determined by the cosines theorem from the triangle 𝑜𝜏𝑢𝛼=arccos1/𝑛+𝜌²𝑛+2𝜌𝑛+2𝜌2(1+1/𝑛)=𝑓(𝜌). (5.17) Hence, we directly have 1𝜏=𝑢+𝑛𝑒^{𝑖(𝜑+𝜋−𝑓(𝜌))}𝑖,𝑑𝜏=𝑛𝑒^{𝑖(𝜑+𝜋−𝑓(𝜌))}(−𝑓^′(𝜌))𝑑𝜌. (5.18) Estimating the quantity 𝐴^′₁, we’ll get 𝐴^′₁1=𝑛(1+𝑛)₀^2𝜋𝑑𝜑_𝛾′1|𝑃_𝑛(𝜓(𝜏))||𝜓^′(𝜏)|𝑑^𝑠1(𝜓(𝜏),1/𝑛)𝑑𝑡=(1+𝑛)₀^2𝜋𝑑𝜑₀^1/𝑛|𝑃_𝑛(𝜓(𝑢+(1/𝑛)𝑒^{𝑖(𝜋+𝜑−𝑓(𝜌))}))|𝜓^′(𝑢+(1/𝑛)𝑒^{𝑖(𝜋+𝜑−𝑓(𝜌))})|𝑓^′(𝜌)|𝑑𝜌𝑑^𝑠(𝜓(𝑢+(1/𝑛)𝑒^{𝑖(𝜋+𝜑−𝑓(𝜌))}1),1/𝑛)=(1+𝑛)₀^1/𝑛|𝑓^′(𝜌)|𝑑𝜌₀^2𝜋|𝑃_𝑛(𝜓(𝑢+(1/𝑛)𝑒^{𝑖(𝜋+𝜑−𝑓(𝜌))}))||𝜓^′(𝑢+(1/𝑛)𝑒^{𝑖(𝜋+𝜑−𝑓(𝜌))})|𝑑𝜑𝑑^𝑠(𝜓(𝑢+(1/𝑛)𝑒^{𝑖(𝜋+𝜑−𝑓(𝜌))}.),1/𝑛) (5.19)
Now, making substitution 𝜏=(1+1/𝜌−𝜌)𝑒^𝑖𝜆 and considering that 𝜆−𝜑=𝑐(𝜌) (we can determire this from the triangle 𝑜𝜏𝑢 where the sides 𝑜𝑢 and 𝜏𝑢 are constant by the sines theorem) we will have 𝐴^′₁1=(1+𝑛)₀^1/𝑛𝑓^′(𝜌)𝑑𝜌₀^2𝜋|𝑃_𝑛(𝜓(1+1/𝑛−𝜌)𝑒^𝑖𝜆)||𝜓^′(1+1/𝑛−𝜌)𝑒^𝑖𝜆|𝑑𝜆𝑑^𝑠(𝜓(1+1/𝑛−𝜌)𝑒^𝑖𝜆1,1/𝑛)=(1+𝑛)₀^1/𝑛|𝑓^′(𝜌)|𝑑𝜌1+1/𝑛−𝜌_{|𝜏|=1+1/𝑛−𝜌}|𝑃_𝑛(𝜓(𝜏))||𝑑𝜏|𝑑^𝑠≍(𝜓(𝜏),1/𝑛)_01/𝑛|𝑓^′(𝜌)|𝑑𝜌_{Γ1+1/𝑛−𝜌}|𝑃_𝑛(𝜉)|𝑑^𝑠(𝜉,1/𝑛)|𝑑𝜉|. (5.20) Hence, by Lemma 4.2, we will find 𝐴^′₁=𝐶(Γ)₀^1/𝑛|𝑓^′𝑃(𝜌)|𝑑𝜌‖^𝑛(𝑧)𝑑^𝑠‖(𝑧,1/𝑛)_𝐿1(Γ)𝑃≤𝐶(Γ)‖^𝑛(𝑧)𝑑^𝑠‖(𝑧,1/𝑛)_𝐿1(Γ). (5.21)
As it was said above, this estimation remains valid for the quantity 𝐴₁^′′, as well.
The same estimation is similarly proved for the quantity 𝐴₂, as well that allows us to see validity of the relation 𝑃𝐴≤𝐶(Γ)‖^𝑛(𝑧)𝑑^𝑠‖(𝑧,1/𝑛)_𝐿1(Γ), (5.22) and hence, considering (5.1), the statement of Theorem 3.3 follows for 𝑝=1 when Γ is an arbitrary restifiable 𝐾-quasiconformal curve. The case 𝑝>1 is proved similarly. Really, by Lemmas B and C, 4.2, relation (5.6) and relation (5.5), and the Holder inequality, we get 𝐴_𝑝^def𝑃=‖^(𝑗)𝑛(𝑡)𝑑^𝑠‖(𝑡,1/𝑛)_{𝐿𝑝(Γ1+1/𝑛)}=𝑗!{2𝜋_Γ1+1/𝑛|𝑑𝑡|𝑑^{(𝑠−𝑗)𝑝}|(𝑡,1/𝑛)^𝛾𝑡𝑃_𝑛(𝜉)𝑑𝜉(𝜉−𝑡)^𝑗+1|^𝑝}^1/𝑝≼{_Γ1+1/𝑛|𝑑𝑡|𝑑^(𝑠+1)𝑝((𝑡,1/𝑛)^𝛾𝑡|𝑃_𝑛(𝜉)||𝑑𝜉|)^𝑝}^1/𝑝≍{_Γ1+1/𝑛|𝑑𝑡|(_𝛾𝑡|𝑃_𝑛(𝜉)||𝑑𝜉|𝑑^𝑠+1(𝜉,Γ_𝑅))^𝑝}^1/𝑝={_{|𝑢|=1+1/𝑛}|𝜓^′(𝑢)||𝑑𝑢|(_𝛾𝑢|𝑃_𝑛(𝜓(𝜏))||𝜓^′(𝜏)||𝑑𝜏|𝑑^𝑠+1(𝜓(𝜏),Γ_𝑅))^𝑝}^1/𝑝≼{𝑛_{|𝑢|=1+1/𝑛}|𝜓^′(𝑢)||𝑑𝑢|(_𝛾𝑢|𝑃_𝑛(𝜓(𝜏))||𝑑𝜏|𝑑^𝑠(𝜓(𝜏),Γ_𝑅))^𝑝}^1/𝑝≼𝑛^1/𝑝{_{|𝑢|=1+1/𝑛}|𝜓^′(𝑢)||𝑑𝑢|_𝛾𝑢|𝑃_𝑛(𝜓(𝜏))𝑑^𝑠(𝜓(𝜏),Γ_𝑅)|^𝑝|𝑑𝜏|}^1/𝑝. (5.23)
Later on, by Lemmas B and C and relation (5.6) it is easy to see the validity of the relation |𝜓^′(𝑢)|≍|𝜓^′1(𝜏)|,|𝑢|=1+𝑛,𝜏∈𝛾_𝑢, (5.24) where 𝛾_𝑢 is a circle with a center at the point 𝑢 and of radius equal 1/2𝑛.
Hence, we directly get 𝐴_𝑝≼𝑛^1/𝑝{_{|𝑢|=1+1/𝑛}|𝑑𝑢|_𝛾𝑢|𝑃_𝑛(𝜓(𝜏))|^𝑝|𝜓^′(𝜏)|𝑑^𝑠(𝜓(𝜏),Γ_𝑅)|𝑑𝜏|}^1/𝑝. (5.25)
Further, the proof is completed in the same way as in the case 𝑝=1 .

Figure 1 

Figure 2 

So, Theorem 3.3 is proved for the case when Γ∈𝐴_𝑘. The same reasoning allow us to affirm that Theorem 3.3 will be valid in the case Γ∈𝐵_𝑘, as well.

Finally, we give the proof of Theorem 3.4.

Proof of Theorem 3.4. Let Γ∈𝐵_𝑘 and 𝑠∈[0,𝑘𝑗/(𝑘−1)𝑝). Consider the case 𝑝>1.
Apply the Holder inequality to inner integral of the right-hand side of the relation 𝐴=_𝑝^def‖𝑑^𝑗−𝑠1(𝑧,𝑛)𝑃^(𝑗)_𝑛(𝑧)‖_𝐿𝑝(Γ)=𝑗!{2𝜋_Γ|𝑑𝑧|𝑑^{(𝑠−𝑗)}|(𝑧,1/𝑛)^Γ_1+1/𝑛𝑃_𝑛(𝑡)𝑑𝑡(𝑡−𝑧)^{(𝑗+1)/𝑝}(𝑡−𝑧)^{(𝑗+1)/𝑞}|^𝑝}^1/𝑝, (5.26) where 1/𝑝+1/𝑞=1.
By Lemma 4.5𝐴_𝑝≼{_Γ𝑑^{𝑗−𝑝𝑠}1(𝑧,𝑛)|𝑑𝑧|_Γ1+1/𝑛|𝑃_𝑛(𝑡)|^𝑝|𝑑𝑡||𝑧−𝑡|^𝑗+1}^1/𝑝. (5.27)
Hence, changing the integration order and applying the statements of Lemmas 4.3 and 4.1, we get the required inequality (3.10) in the case 𝑝>1.
In order to see validity of Theorem 3.4 in the case 𝑝=1, in the right-hand side of the obvious relation _Γ|𝑑^𝑗−𝑠1(𝑧,𝑛)𝑃_𝑛^(𝑗)(𝑧)||𝑑𝑧|≼_Γ𝑑^𝑗−𝑠1(𝑧,𝑛)_Γ1+1/𝑛|𝑃_𝑛(𝑡)||𝑑𝑡||𝑡−𝑧|^𝑗+1|𝑑𝑧|, (5.28) it siffies to change the integration order and apply the statements of Lemmas 4.3 and 4.1.
Remark 5.1. It is easy to show that Theorem 3.4 is valid for any 𝑠∈[0,∞), if Γ is fulfiled as the condition (obviously, this condition is always fulfilled if Γ is a boundary of an arbitrary convex domain) |𝜓^′(𝑤)|≼|𝜓^′(1+1/𝑛)𝑤| for all 𝑤∶|𝑤|=1.
Really, let 𝑠≥𝑘𝑗/(𝑘−1)𝑝. Choose 𝑚>𝑗 such that the condition 𝑠<𝑘𝑚/(𝑘−1)𝑝 is fulfilled. Then repeating the reasoning mentioned above in the case 𝑠<𝑘𝑗/(𝑘−1)𝑝, we get ‖𝑃_𝑛^(𝑚)(𝑧)𝑑^𝑠−𝑚‖(𝑧,1/𝑛)_𝐿𝑝(Γ)𝑃≤𝐶(Γ,𝑝,𝑚,𝑠)‖^𝑛(𝑧)𝑑^𝑠‖(𝑧,1/𝑛)_𝐿𝑝(Γ). (5.29)
Now, expand the function 𝑃_𝑛^(𝑗)(𝑧) in Taylor’s series in the vicinity of the point 𝑧=𝑧(1/𝑛)∈Γ_1+1/𝑛∶𝑃_𝑛^(𝑗)(𝑧)=𝑃_𝑛^(𝑗)(𝑃𝑧)+^𝑛(𝑗+1)(𝑧)(𝑧−𝑧)𝑃1!+⋯+^{𝑛(𝑚−1)}(𝑧)((𝑚−𝑗−1)!𝑧−𝑧)^{𝑚−𝑗−1}+1(𝑚−𝑗−1)!^𝑧_𝑧(𝝃−𝑧)^{𝑚−𝑗−1}𝑃_𝑛^(𝑚)(𝝃)𝑑𝝃. (5.30)
Further, divide both parts of this equality into 𝑑^𝑠−𝑗(𝑧,1/𝑛), and consider that 𝑑(𝑧,1/𝑛)⋟𝑑(𝝃,Γ_𝑅) (see (5.6)) raise to the 𝑝th power, integrate with respect to Γ and take the 𝑝th power root. We will have 𝐴=_𝑝^def(_Γ|𝑃_𝑛^(𝑗)(𝑧)𝑑^𝑠−𝑗|(𝑧,1/𝑛)^𝑝|𝑑𝑧|)^1/𝑝≼(_Γ|𝑃_𝑛^(𝑗)(𝑧)𝑑^𝑠−𝑗|(𝑧,1/𝑛)^𝑝|𝑑𝑧|)^1/𝑝+(_Γ|𝑃_𝑛^(𝑗+1)(𝑧)𝑑^{𝑠−(𝑗+1)}|(𝑧,1/𝑛)^𝑝|𝑑𝑧|)^1/𝑝+⋯+(_Γ|𝑃_𝑛^(𝑚−1)(𝑧)𝑑^{𝑠−(𝑚−1)}|(𝑧,1/𝑛)^𝑝|𝑑𝑧|)^1/𝑝+(_Γ|^𝑧_𝑧𝑃_𝑛^(𝑚)(𝝃)𝑑𝝃𝑑^{𝑠−𝑚+1}|(𝝃,1/𝑛)^𝑝|𝑑𝑧|)^1/𝑝=𝐴_𝑝^(𝑗)+⋯+𝐴_𝑝^(𝑚). (5.31)
Now considering Lemmas B and C, 4.1, and Theorem 3.3 and making substitution 𝜂=𝑧,𝑧=𝜓((1+1/𝑛)⁻¹𝜑(𝜂))=𝜂_∼, we get (here in our reasoning we assume, |𝜓^′((1+1/𝑛)⁻¹𝜑(𝑡))|≼|𝜓^′(𝜑(𝑡))| for all 𝑡∈Γ_1+1/𝑛): 𝐴=_𝑝^(𝑗)def(_Γ|𝑃_𝑛^(𝑗)(𝑧)𝑑^𝑠−𝑗|(𝑧,1/𝑛)^𝑝|𝑑𝑧|)^1/𝑝≼(_Γ1+1/𝑛|𝑃_𝑛^(𝑗)(𝜂)𝑑^𝑠−𝑗|(𝜂,1/𝑛)^𝑝|𝑑𝜂|)^1/𝑝≼(_Γ|𝑃_𝑛(𝑧)𝑑^𝑠|(𝑧,1/𝑛)^𝑝|𝑑𝑧|)^1/𝑝. (5.32) All remaining integrals on the right-hand side of relation (5.31) are similarly estimated except for the last one, for which following the proof of Theorem 3.3 we find 𝐴=_𝑝^(𝑚)def(_Γ|^𝑧_𝑧𝑃_𝑛^(𝑚)(𝜉)𝑑𝜉𝑑^{𝑠−𝑚+1}|(𝜉,1/𝑛)^𝑝|𝑑𝑧|)^1/𝑝≼𝑛^1/𝑝(_|𝑤|=1|𝑑𝑤|^𝑤_𝑤|𝑃_𝑛^(𝑚)(𝜓(𝜏))𝑑^𝑠−𝑚(𝜓(𝜏),Γ_𝑅)|^𝑝|𝜓^′(𝜏)||𝑑𝜏|)^1/𝑝. (5.33)
Reasoning in the same way as in obtaining estimation (5.5), we’ll have 𝐴^(𝑚)_𝑝𝑃≼‖^𝑛(𝑧)𝑑^𝑠‖(𝑧,1/𝑛)_𝐿𝑝(Γ). (5.34)
Hence by (5.31) the statement of Theorem 3.4 will follow in the case 𝑠≥𝑘𝑗/(𝑘−1)𝑝.
So, Theorem 3.4 is proved.

Remark 5.2. Note that by Lemma 4.4 and the inverse to it of result 1<𝛾≤2 proved in the paper [15], we will have 𝑆_𝜃=𝐽_𝛾(1<𝛾≤2). Obviously, this result will allow us to derive from Theorems 3.1 and 3.2 the validity of these theorems on arbitrary curves Γ∈𝑆_𝜃 as a corollary.

Acknowledgment

The authors would like to thank the reviewers for carefully checking the manuscript and for his/her valuable comments and suggestions.