PUBLICATIONS DE L’INSTITUT MATHÉMATIQUE Nouvelle série, tome 100(114) (2016), 209–227 DOI: 10.2298/PIM1614209A

POLYNOMIAL INEQUALITIES IN LAVRENTIEV REGIONS WITH INTERIOR AND EXTERIOR ZERO ANGLES IN THE WEIGHTED LEBESGUE SPACE

F. G. Abdullayev and N. P. Özkartepe

Abstract. We study estimation of the modulus of algebraic polynomials in the bounded and unbounded regions with piecewise-quasismooth boundary, having interior and exterior zero angles, in the weighted Lebesgue space.

1. Introduction and main results

Let ℂ be a complex plane, ℂ̄ := ℂ ∪ {∞}; G ⊂ ℂ be a bounded Jordan region, with 0 ∈ G and the boundary L := ∂G be a simple closed rectifiable Jordan curve, Ω := ℂ̄ ∖ Ḡ = extL; Δ := {w : |w| > 1}. Let w = Φ(z) be the univalent conformal mapping of Ω onto the Δ normalized by Φ(∞) = ∞, lim_{z→∞} \frac{Φ(z)}{z} > 0. For t ⩾ 1, let us set L_{t} := {z : |Φ(z)| = t}, L_{1} ≡ L, G_{t} := intL_{t}, Ω_{t} := extL_{t}. For z ∈ ℂ and S ⊂ ℂ let d(z, S) := dist(z, S) = inf{|ζ − z| : ζ ∈ S}. Let h(z) be a weight function defined in G_{R_{0}} for some fixed R_{0} > 1 and let ℘_{n} denote the class of arbitrary algebraic polynomials P_{n}(z) of degree at most n ∈ ℕ := {1, 2, …}. For any p > 0 we denote

∥P_{n}∥_{p} := ∥P_{n}∥_{L_{p} (h,L)} := ( ∫_{L} h(z)|P_{n}(z)|^{p}|dz|)^{1/p}, 0 < p < ∞;
∥P_{n}∥_{∞} := ∥P_{n}∥_{L_{∞} (1,L)} := max_{z∈Ḡ}|P_{n}(z)|, p = ∞.

Here, we investigate the following two problems: for a given region G, find estimates of the following types, for the points z ∈ G and z ∈ Ω, respectively:

(1.1)  ∥P_{n}∥_{∞} ⩽ constν_{n}∥P_{n}∥_{p},
(1.2)  |P_{n}(z)| ⩽ constη_{n}∥P_{n}∥_{p}|Φ(z)|^{n+1} ,

where ν_{n} := ν_{n}(G, h, p) → ∞ and η_{n} := η_{n}(G, h, p, z) → ∞ in general, as n → ∞, depending on the geometrical properties of the region G and weight function h.

1991 Mathematics Subject Classification: 30A10, 30C10, 30C62, 30E10, 41A17.

Key words and phrases: algebraic polynomials, Nikol’skii-type inequalities, quasismooth curve.

Authors thank all the referees for their valuable contributions.

Communicated by Gradimir Milovanović.

Inequalities analogously to (1.1) in the literature are often found under the “Nikol’skii-type inequality”. One of the first results analogous to (1.1), in the case h(z) ≡ 1 for L = {z : |z| = 1} and 0 < p < ∞ was found by Jackson [12] as follows:

max_{|z|=1}|P_{n}(z)| ⩽ 2n^{1/p}( ∫_{0}^{2π} |P_{n}(e^{iθ})|^{p}dθ)^{1/p}.

Another classical results similar to (1.1) belong to Szegö and Zigmund [21]. Suetin [23], [24] investigated this problem with a sufficiently smooth Jordan curve and some special weight function. The estimate of type (1.1) for 0 < p < ∞ when L is a rectifiable Jordan curve, was investigated by Mamedhanov [14, 15], Nikol’skii [17, pp. 122–133], Pritsker [19] (h(z) ≡ 1), Andrievskii [10, Theorem 6], authors [7] (h(z) ≠ 1) and others. More general and detailed references regarding the Nikol’skii type inequality, we can find in Milovanovic et al. [16, Sect. 5.3].

Results analogous to (1.2) for some different norms and unbounded regions were obtained in [22], [2–7] and others.

In this work, we study problems (1.1) and (1.2) for some general regions having interior and exterior cusps, and finally, we obtain the estimate for |P_{n}(z)| in the whole complex plane, depending on the given region G and weight function h.

We give the necessary definitions and notations for the formulation of the main results.

Following [18, p.163], we say that a bounded Jordan curve L is λ-quasismooth or Lavrentiev curve, if for every pair z_{1}, z_{2} ∈ L, where l(z_{1}, z_{2}) denote the shorter subarc of L, joining z_{1} ∈ L and z_{2} ∈ L and |l(z_{1}, z_{2})| is the linear measure (length) of l(z_{1}, z_{2}), there exists a constant λ := λ(L) ⩾ 1, such that

|l(z_{1}, z_{2})| ⩽ λ|z_{1} − z_{2}|, z_{1}, z_{2} ∈ L.

In this case, the inner region intL of a Lavrentiev curve L is called a Lavrentiev region. Any subarc of a λ-quasismooth curve is called a λ-quasismooth arc. We denote this class of curves and arcs as QS(λ) and say that a Jordan region G ∈ QS(λ), if ∂G ∈ QS(λ), λ ⩾ 1. Furthermore, we denote that L(or G) ∈ QS, if L(or G) ∈ QS(λ) for some λ ⩾ 1.

We say that a bounded Jordan curve or arc L is locally λ-quasismooth at the point z ∈ L, if there exists a closed subarc ℓ ⊂ L containig z, such that every open subarc of the ℓ containing z is the λ-quasismooth.

Now, we shall define a new class of regions with piecewise quasismooth boundary, which may have at the boundary points finite number of interior and exterior cusps.

For any j = 1, 2, … and sufficiently small ε_{1} > 0, we denote by f_{j}, g_{j} : [0, ε_{1}] → ℝ twice continuously differentiable functions such that f_{j} (0) = g_{j} (0) = 0 and f_{j}^{(k)}(x) > 0, g_{j}^{(k)}(x) > 0, for x > 0 and k = 0, 1, 2.

Note that, 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 depend on G in general.

Definition 1.1. We say that a Jordan region G ∈ P QS(λ; f_{i}, g_{j}), λ ⩾ 1,

f_{i} = f_{i}(x), i = \overline{1, m_{1}}, g_{j} = g_{j} (x), j = \overline{m_{1} + 1, m} , if L := ∂G = ⋃ _{j=0}^{m} L_{j} is the union of the finite number of λ-quasismooth arcs L_{j}, connecting at the points {z_{j} }_{j=0}^{m} ∈ L, and such that L is a locally λ-quasismooth arc at the z_{0} ∈ L ∖ {z_{j} }_{j=1}^{m} and, in the (x, y) local coordinate system with its origin at the z_{j}, 1 ⩽ j ⩽ m, the following conditions are satisfied

a) for every z_{j} ∈ L, j = \overline{1, m_{1}}, m_{1} ⩽ m,

{z = x + iy : |z| ⩽ ε_{1}, c_{1}f_{i}(x) ⩽ y ⩽ c_{2}f_{i}(x)} ⊂ Ḡ,
{z = x + iy : |z| ⩽ ε_{1}, |y| ⩾ ε_{2}x} ⊂ Ω̄;
b) for every z_{j} ∈ L, j = \overline{m_{1} + 1, m} ,
{z = x + iy : |z| < ε_{3}, c_{3}g_{j} (x) ⩽ y ⩽ c_{3}g_{j} (x), 0 ⩽ x ⩽ ε_{3}} ⊂ Ω̄,
{z = x + iy : |z| < ε_{3}, |y| ⩾ ε_{3}x, 0 ⩽ x ⩽ ε_{3}} ⊂ Ḡ,

for some constants −∞ < c_{1} < c_{2} < ∞, −∞ < c_{3} < c_{4} < ∞, ε_{i} > 0, i = \overline{1, 4}.

It is clear from Definition 1.1 that each region G ∈ P QS(λ; f_{i}, g_{j}) may have m_{1} interior and m−m_{1} exterior zero angles (with respect to Ḡ). If a region G does not have interior zero angles (m_{1} = 0) (exterior zero angles (m_{1} = m)), then it is written as G ∈ P QS(λ; 0, g_{j}) (G ∈ P QS(λ; f_{i}, 0)). If a domain G does not have such angles (m = 0), then G is bounded by a λ-quasismooth circle and in this case we set P QS(λ, 0, 0) ≡ QS(λ).

Let {z_{j} }_{j=1}^{m} be a fixed system of distinct points on the curve L located in the positive direction. Consider a so-called generalized Jacobi weight function h(z) being defined by

(1.3)  h(z) := ∏_{j=1}^{m} |z − z_{j} |^{γ_{j}} , z ∈ G_{R_{0}} ,

for some R_{0} > 1, where γ_{j} > −1 for all j = 1, 2, …, m.

Here and in further, for any k ⩾ 0 and m > k, the notation j = \overline{k, m} denotes j = k, k + 1, …, m.

Throughout this work, we will assume that the points {z_{j} }_{j=1}^{m} ∈ L defined in (1.3) and Definition 1.1 are identical. Without loss of generality, we assume that the points {z_{j} }_{j=1}^{m} are ordered in the positive direction on the curve L such that, G may have interior zero angles at the points {z_{j} }_{j=1}^{m_{1}} and exterior zero angles at the points {z_{j} }_{j=m_{1} +1}^{m} , and w_{j} := Φ(z_{j}).

Now we can state our new results.

Theorem 1.1. Let p > 0; G ∈ P QS(λ; f_{i}, g_{i}), for some λ ⩾ 1, f_{i}(x) = c_{i}x^{1+α_{i}} , α_{i} ⩾ 0, i = \overline{1, m_{1}}, and g_{i}(x) = c_{i}x^{1+β_{i}} , β_{i} > 0, i = \overline{m_{1} + 1, m} ; h(z) defined as in (1.3). Then, for any γ_{i} > −1, i = \overline{1, m}, and P_{n} ∈ ℘_{n}, n ∈ ℕ, there exists c_{1} = c_{1}(G, p, λ, γ_{i}, β_{i}) > 0 such that

(1.4)  ∥P_{n}∥_{∞} ⩽ c_{1}( ∑_{i=1}^{m_{1}} n^{\frac{μ̃ (γ̃_{i} +1)}{p}} + ∑_{i=m_{1} +1}^{m} n^{( \frac{γ̃_{i}}{1+β_{i}} +1) \frac{μ}{p}} ) ∥P_{n}∥_{p},

where μ := 2(1 − \frac{1}{π} arcsin \frac{1}{λ}), 1 < μ < 2, μ̃ := \{ \begin{matrix}μ, & if α_{1} = 0, \\ 2, & if α_{1} ≠ 0. \end{matrix}

We note that, if μ = 1 (i.e. λ = 1), then the terms n^{( \frac{γ̃_{i}}{1+β_{i}} +1) \frac{μ}{p}} in the second

sum of (1.4) for any γ_{i} > −1, i = \overline{m_{1} + 1, m} , can be replaced by (n^{\frac{γ̃_{i}}{1+β_{i}} +1} lnn)^{\frac{1}{p}} .

Now, for simplicity of our presentations, we assume that i = 1, 2; m_{1} = 1, m = 2; i.e., our region G may have one interior zero (or it does not exist) angle having “f_{1}-touching" with f_{1}(x) = C_{1}x^{1+α_{1}} , α_{1} ⩾ 0, at the point z_{1} and exterior zero angle having” g_{2}-touching" with g_{2}(x) = C_{2}x^{1+β_{2}} , β_{2} > 0, at the point z_{2}, for some constants −∞ < C_{1} := C_{1}(c_{1}, c_{2}) < +∞, −∞ < C_{2} := C_{2}(c_{3}, c_{4}) < +∞, where the constants c_{i}, i = \overline{1, 4} are taken from Definition 1.1. In this case, combining the terms related to the interior and exterior corners, we obtain the following

Theorem 1.2. Let p > 0; G ∈ P QS(λ; f_{1}, g_{2}), for some λ ⩾ 1, f_{1}(x) =

C_{1}x^{1+α_{1}} , α_{1} ⩾ 0, and g_{2}(x) = C_{2}x^{1+β_{2}} , β_{2} > 0; h(z) defined as in (1.3) for m = 2. Then, for any γ_{i} > −1, i = 1, 2, and P_{n} ∈ ℘_{n}, n ∈ ℕ, there exists c_{2} = c_{2}(G, p, λ, γ_{1}, γ_{2}, β_{2}) > 0 such that

(1.5)  ∥P_{n}∥_{∞} ⩽ c_{2}A_{n} ∥P_{n}∥_{p} ,

where A_{n} is defined by (1.6)

A_{n} := \{ \begin{matrix}n^{\frac{1}{p} max \{μ̃;μ\}} , & −1 < γ_{1} ⩽ 0, & −1 < γ_{2} ⩽ 0, & μ > 1, \\ n^{( \frac{γ_{2}}{1+β_{2}} +1) \frac{μ}{p}} , & 0 < γ_{1} < ( \frac{γ_{2}}{1+β_{2}} + 1) \frac{μ}{μ̃} − 1, & γ_{2} > 0, & μ > 1, \\ n^{\frac{μ̃ (γ_{1} +1)}{p}} , & γ_{1} ⩾ ( \frac{γ_{2}}{1+β_{2}} + 1) \frac{μ}{μ̃} − 1, & γ_{2} > 0, & μ > 1, \\ (n^{\frac{γ_{2}}{1+β_{2}} +1} lnn)^{\frac{1}{p}} , & 0 < γ_{1} ⩽ ( \frac{γ_{2}}{1+β_{2}} + 1) \frac{1}{μ̃} − 1, & γ_{2} > 0, & μ = 1, \\ n^{\frac{μ̃ (γ_{1} +1)}{p}} , & γ_{1} ⩾ ( \frac{γ_{2}}{1+β_{2}} + 1) \frac{1}{μ̃} − 1, & γ_{2} > 0, & μ = 1. \end{matrix}

Assume that the region G has only exterior zero angles g_{i}(x) = c_{i}x^{1+β_{i}} , β_{i} > 0, i = \overline{1, m} (i.e., m_{1} = 0). In this case, from Theorem 1.1, we obtain

Corollary 1.1. Let p > 0; G ∈ P QS(λ; 0, g_{i}), for some λ ⩾ 1, g_{i}(x) =

c_{i}x^{1+β_{i}} , β_{i} > 0, i = \overline{1, m}; h(z) defined as in (1.3). Then, for any P_{n} ∈ ℘_{n}, n ∈ ℕ, there exists c_{3} = c_{3}(G, p, λ, γ_{i}, β_{i}) > 0 such that

(1.7)  ∥P_{n}∥_{∞} ⩽ c_{3}( ∑_{i=1}^{m} n^{( \frac{γ̃_{i}}{1+β_{i}} +1) \frac{μ}{p}} ) · ∥P_{n}∥_{p},

and consequently,

(1.8)  ∥P_{n}∥_{∞} ⩽ c_{3}n^{( \frac{γ̃^{max}}{1+β_{min}} +1) \frac{μ}{p}} ∥P_{n}∥_{p},

where γ̃^{max} = max\{0; γ_{i} : i = \overline{1, m}\}, β_{min} := min\{β_{i} : i = \overline{1, m}\}.

Now, for these regions, we will state our new results corresponding to the second problem, i.e., pointwise estimations for |P_{n}(z)| for the points z ∈ Ω.

Theorem 1.3. Let p > 0; G ∈ P QS(λ; f_{i}, g_{i}), for some λ ⩾ 1, f_{i}(x) =

C_{i}x^{1+α_{i}} , α_{i} ⩾ 0, i = \overline{1, m_{1}}, and g_{i}(x) = C_{i}x^{1+β_{i}} , β_{i} > 0, i = \overline{m_{1} + 1, m} ; h(z) defined as in (1.3). Then, for any γ_{i} > −1, i = \overline{1, m}, and P_{n} ∈ ℘_{n}, n ∈ ℕ, there exists c_{4} = c_{4}(G, p, λ, γ_{i}, β_{i}) > 0 such that

(1.9)  |P_{n}(z)| ⩽ c_{4} \frac{|Φ(z)|^{n+1}}{d^{2/p}(z, L)}( ∑_{i=1}^{m_{1}} B_{n,1}^{i} + ∑_{i=m_{1} +1}^{m} B_{n,2}^{i} ) · ∥P_{n}∥_{p}, z ∈ Ω,

where

B_{n,1}^{i} := \{ \begin{matrix}n^{\frac{(γ_{i} −1)μ̃}{p}} , & if γ_{i} > 1, \\ (lnn)^{\frac{1}{p}} , & if γ_{i} = 1, \\ 1, & if − 1 < γ_{i} < 1, \end{matrix}  for all i = \overline{1, m_{1}} ,

B_{n,2}^{i} := \{ \begin{matrix}n^{\frac{γ_{i} −1}{p(1+β_{i} )} μ}, & if γ_{i} > 1, \\ (lnn)^{\frac{1}{p}} , & if γ_{i} = 1, \\ 1, & if − 1 < γ_{i} < 1, \end{matrix}  for all i = \overline{m_{1} + 1, m} .

Theorem 1.3 is local, that is, each term in the sum on the right-hand side shows the growth of |P_{n}(z)|, depending on the behavior of the weight function h(z) and the boundary L in the neighborhood of a single point \{z_{j} \} ∈ L for any j = \overline{1, m}.

Comparing the terms in the sum for each point \{z_{j} \}, j = \overline{1, m}, and using the above notations, we can obtain the following result of global character (for the simplicity, we assume that i = 1, 2; m_{1} = 1, m_{2} = 2).

Theorem 1.4. Let p > 0; G ∈ P QS(λ; f_{1}, g_{2}), for some λ ⩾ 1, f_{1}(x) =

C_{1}x^{1+α_{1}} , α_{1} ⩾ 0, and g_{2}(x) = C_{2}x^{1+β_{2}} , β_{2} > 0; h(z) defined as in (1.3) for m = 2. Then, for any γ_{j} = 1, 2, and P_{n} ∈ ℘_{n}, n ∈ ℕ, there exists c_{5} = c_{5}(G, p, λ, γ_{j} , β_{2}) > 0 such that

(1.10)  |P_{n}(z)| ⩽ c_{5} \frac{|Φ(z)|^{n+1}}{d^{2/p}(z, L)}B_{n} · ∥P_{n}∥_{p}, z ∈ Ω,

where

(1.11)  B_{n} := \{ \begin{matrix}n^{\frac{γ_{2} −1}{p(1+β_{2} )} μ}, & 1 < γ_{1} < \frac{γ_{2} −1}{1+β_{2}} \frac{μ}{μ̄} + 1, & γ_{2} > 1, \\ n^{\frac{(γ_{1} −1)μ̃}{p}} , & γ_{1} ⩾ \frac{γ_{2} −1}{1+β_{2}} \frac{μ}{μ̄} + 1, & γ_{2} > 1, \\ (lnn)^{\frac{1}{p}} , & γ_{1} = 1, & γ_{2} = 1, \\ 1, & −1 < γ_{1} < 1, & −1 < γ_{2} < 1. \end{matrix}

Therefore, combining estimations (1.4) with (1.9), we obtain an estimate on the growth of |P_{n}(z)| in the whole complex plane.

Corollary 1.2. Under the conditions and notations of Theorems 1.1 and 1.3, we have

(1.12) |P_{n}(z)| ⩽ c_{6} \{ \begin{matrix}∑_{i=1}^{m_{1}} n^{\frac{μ̃ (γ̃_{i} +1)}{p}} + ∑_{i=m_{1} +1}^{m} n^{( \frac{γ̃_{i}}{1+β_{i}} +1) \frac{μ}{p}} , & z ∈ Ḡ, \\ \frac{|Φ(z)|^{n+1}}{d^{2/p}(z, L)}(∑_{i=1}^{m_{1}} B_{n,1}^{i} + ∑_{i=m_{1} +1}^{m} B_{n,2}^{i}), & z ∈ Ω, \end{matrix} \} · ∥P_{n}∥_{p},

where c_{6} = c_{6}(G, p, λ, γ_{i}, β_{i}) > 0 constant, independent of z and n.

In particular, in the case i = 1, 2; m_{1} = 1, m = 2, from Corollary 1.2 we have

Corollary 1.3. Let p > 0; G ∈ P QS(λ; f_{1}, g_{2}), for some λ ⩾ 1, f_{1}(x) = C_{1}x^{1+α_{1}} , α_{1} ⩾ 0, and g_{2}(x) = C_{2}x^{1+β_{2}} , β_{2} > 0; h(z) defined as in (1.3) for m = 2. Then, for any γ_{j}, j = 1, 2, and P_{n} ∈ ℘_{n}, n ∈ ℕ, there exists c_{7} = c_{7}(G, p, λ, γ_{j} , β_{2}) > 0 such that

|P_{n}(z)| ⩽ c_{7} \{ \begin{matrix}A_{n}, & z ∈ Ḡ, \\ \frac{|Φ(z)|^{n+1}}{d^{2/p}(z, L)}B_{n}, & z ∈ Ω, \end{matrix} \} · ∥P_{n}∥_{p},

where A_{n} and B_{n} are taken from (1.6) and (1.11), respectively.

Remark 1.1. We note that, according to the well-known Bernstein–Walsh Lemma [27], estimations (1.4)–(1.8) are also true in G_{R} with another constant for some R = 1 + \frac{c}{n}. Therefore, if we choose

R̃ := sup{R > 1 : ∥P_{n}∥_{C(Ḡ_{R} )} ⩽ c_{7}∥P_{n}∥_{C(Ḡ)}},

then estimations (1.9)–(1.10) will be significant for z ∈ ΩR̃.

The sharpness of estimations (1.4)–(1.12) for some special cases can be discussed by comparing them with the following results.

Remark 1.2. For any n ∈ ℕ and i = 1, 2, there exist polynomials P_{n}^{(i)} ∈ ℘_{n} and regions G^{i} ⊂ ℂ, such that

∥P_{n}^{(1)} ∥ ⩾ c_{8}n^{\frac{1}{p}} ∥P_{n}^{(1)} ∥_{L_{p} (L^{1} )} ,
|P_{n}^{(2)}(z)| ⩾ c_{9} |Φ(z)|^{n} ∥P_{n}^{(2)} ∥_{L_{2} (L^{2} )} , ∀z ∈ F ⋐ ℂ̄ ∖ \overline{G^{2}},

where c_{8} = c_{8}(G^{1}) > 0, c_{9} = c_{9}(G^{2}) > 0 are constants, independent of z and n, L^{i} := ∂G^{i}, i = 1, 2.

2. Some auxiliary results

For a > 0 and b > 0, we use the notation “a ⪯ b” (order inequality), if a ⩽ cb and “a ≍ b” are equivalent to c_{1}a ⩽ b ⩽ c_{2}a for some constants c, c_{1}, c_{2} (independent of a and b) respectively.

Since, any quasismooth curve is quasiconformal, then in the proofs of many facts we use some properties of quasiconformal curves. Therefore, we give the corresponding definition of the quasiconformal curves.

Definition 2.1. [13, p.97], [20] The Jordan arc (or curve) L is called Kquasiconformal (K ⩾ 1), if there is a K-quasiconformal mapping f of the region D ⊃ L such that f(L) is a line segment (or circle).

Let F (L) denote the set of all sense preserving plane homeomorphisms f of the region D ⊃ L such that f(L) is a line segment (or circle) and let define K_{L} := inf{K(f) : f ∈ F (L)}, where K(f) is the maximal dilatation of such a mapping f. L is a quasiconformal curve, if K_{L} < ∞, and L is a K-quasiconformal curve, if K_{L} ⩽ K.

Let B := {w : |w| < 1}, z = ψ(w) be the univalent conformal mapping of B onto the G normalized by ψ(0) = 0, ψ′(0) > 0 and let φ := ψ^{−1} . For 0 < t < 1, let L_{t} := {z : |φ(z)| = t}.

Lemma 2.1. [1] Let L be a K-quasiconformal curve, 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

a) The statements |z_{1}−z_{2}| ⪯ |z_{1}−z_{3}| and |w_{1}−w_{2}| ⪯ |w_{1}−w_{3}| are equivalent.

So are |z_{1} − z_{2}| ≍ |z_{1} − z_{3}| and |w_{1} − w_{2}| ≍ |w_{1} − w_{3}|.
b) If |z_{1} − z_{2}| ⪯ |z_{1} − z_{3}|, then

| \frac{w_{1} − w_{3}}{w_{1} − w_{2}} |^{ε} ⪯ | \frac{z_{1} − z_{3}}{z_{1} − z_{2}} | ⪯ | \frac{w_{1} − w_{3}}{w_{1} − w_{2}} |^{c},

where ε < 1, c > 1, 0 < r_{0} < 1 are constants, depending on G.

Lemma 2.2. [25], [26] Let G ∈ QS(λ) for some λ ⩾ 1. Then

|Ψ(w_{1}) − Ψ(w_{2})| ⪰ |w_{1} − w_{2}|^{μ},

for all w_{1}, w_{2} ∈ Ω̄′, where μ := 2(1 − \frac{1}{π} arcsin \frac{1}{λ}).

For Ψ′ the following is true (see, for example, [9, Th.2.8]):

(2.1)  |Ψ′(τ)| ≍ \frac{d(Ψ(τ), L)}{|τ| − 1} .

Let {z_{j} }_{j=1}^{m} be a fixed system of the points on L and the weight function h(z) defined as (1.3).

Lemma 2.3. Let L be a rectifiable Jordan curve; h(z) as defined in (1.3). Then, for arbitrary P_{n}(z) ∈ ℘_{n}, any R > 1 and n ∈ ℕ, we have

(2.2)  ∥P_{n}∥_{L_{p} (h,L_{R} )} ⩽ R^{n+ \frac{1+γ∗}{p}} ∥P_{n}∥_{L_{p} (h,L)} , p > 0,

where γ^{∗} = max\{0; γ_{k}, k = \overline{1, m}\}.

Remark 2.1. In the case of h(z) ≡ 1, estimate (2.2) has been proved in [11].

3. Proof of theorems

Proof of Theorem 1.1. Let G ∈ P QS(λ; f_{i}, g_{i}), for some λ ⩾ 1, f_{i}(x) = c_{i}x^{1+α_{i}} , α_{i} ⩾ 0, i = \overline{1, m_{1}}, and g_{i}(x) = c_{i}x^{1+β_{i}} , β_{i} > 0, i = \overline{m_{1} + 1, m} , be given. For some R > 1, let w = φ_{R}(z) be the univalent conformal mapping of G_{R} onto

216 ABDULLAYEV AND ÖZKARTEPE

the B normalized by φ_{R}(0) = 0, φ′_{R}(0) > 0 and let {ζ_{j} }, 1 ⩽ j ⩽ m ⩽ n, be zeros of P_{n}(z) lying on G_{R}. Let

B_{m,R} (z) := ∏_{j=1}^{m} B_{j,R} (z) = ∏_{j=1}^{m} \frac{φ_{R}(z) − φ_{R}(ζ_{j})}{1 − \overline{φ_{R}(ζ_{j})}φ_{R}(z)}

denote a Blashke function with respect to zeros {ζ_{j} }, 1 ⩽ j ⩽ m ⩽ n, of P_{n}(z).

For any p > 0 and z ∈ G_{R} let us set

T_{n}(z) := [ \frac{P_{n}(z)}{B_{m,R} (z)}]^{p/2}.

The Cauchy integral representation for the region G_{R} gives

T_{n}(z) = \frac{1}{2πi} ∫_{L_{R}} T_{n}(ζ)\frac{dζ}{ζ − z} , z ∈ G_{R}.

Then

|[ \frac{P_{n}(z)}{B_{m,R} (z)}]^{p/2} | ⩽ \frac{1}{2π} ∫_{L_{R}} | \frac{P_{n}(ζ)}{B_{m,R} (ζ)}|^{p/2} \frac{|dζ|}{|ζ − z|} ⩽ \frac{1}{2π} ∫_{L_{R}} |P_{n}(ζ)|^{p/2} \frac{|dζ|}{|ζ − z|} ,

since |B_{m,R} (ζ)| = 1, for ζ ∈ L_{R}. Multiplying the numerator and determinator of the integrand by h^{1/2} (ζ), according to the Hölder inequality, we obtain

| \frac{P_{n}(z)}{B_{m,R} (z)} |^{p/2} ⩽ \frac{1}{2π} ( ∫_{L_{R}} h(ζ)|P_{n}(ζ)|^{p}|dζ|)^{1/2} × ( ∫_{L_{R}} \frac{|dζ|}{∏ _{j=1}^{m} |ζ − z_{j} |^{γ_{j}} |ζ − z|^{2}} )^{1/2}

=: \frac{1}{2π} J_{n,1} × J_{n,2} ,

where

J_{n,1} := ∥P_{n}∥_{L_{p} (h,L_{R} )}^{p/2} , J_{n,2} := ( ∫_{L_{R}} \frac{|dζ|}{∏ _{j=1}^{m} |ζ − z_{j} |^{γ_{j}} |ζ − z|^{2}} )^{1/2} .

Let z ∈ L. Then, since |B_{m,R} (z)| < 1 for z ∈ L, from Lemma 2.3 we have

(3.1)  |P_{n}(z)| ⪯ (J_{n,1} · J_{n,2} )^{2/p} ⪯ ∥P_{n}∥_{p} · (J_{n,2} )^{2/p}.

To estimate the integral J_{n,2} , we introduce:

w_{j} := Φ(z_{j}), φ_{j} := argw_{j} , L_{R}^{j} := L_{R} ∩ Ω̄^{j} , j = \overline{1, m},

where

Ω^{j} := Ψ(Δ′_{j}), Δ′_{1} := \{t = Re^{iθ} : R > 1, \frac{φ_{m} + φ_{1}}{2} ⩽ θ < \frac{φ_{1} + φ_{2}}{2} \},

Δ′_{m} := \{t = Re^{iθ} : R > 1, \frac{φ_{m−1} + φ_{m}}{2} ⩽ θ < \frac{φ_{m} + φ_{1}}{2} \},

Δ′_{j} := \{t = Re^{iθ} : R > 1, \frac{φ_{j−1} + φ_{j}}{2} ⩽ θ < \frac{φ_{j} + φ_{j+1}}{2} \}, for j = \overline{2, m − 1} .

Then, we get

(3.2)  (J_{n,2} )^{2} = ∑_{i=1}^{m} ∫_{L_{R}^{i}} \frac{|dζ|}{∏ _{j=1}^{m} |ζ − z_{j} |^{γ_{j}} |ζ − z|^{2}}

≍ ∑_{i=1}^{m} ∫_{L_{R}^{i}} \frac{|dζ|}{|ζ − z_{i}|^{γ_{i}} |ζ − z|^{2}} =: ∑_{i=1}^{m} J_{n,2}^{i} ,

where

(3.3)  J_{n,2}^{i} := ∫_{L_{R}^{i}} \frac{|dζ|}{|ζ − z_{i}|^{γ_{i}} |ζ − z|^{2}} , i = \overline{1, m},

since the points {z_{i}} ∈ L are distinct. It remains to estimate the integrals J_{n,2}^{i} for each i = \overline{1, m}.

For simplicity of our next calculations, we assume that: i = 1, 2; m_{1} = 1, m = 2; z_{1} = −1, z_{2} = 1; (−1, 1) ⊂ G; R = 1 + \frac{ε_{0}}{n} , and let the local co-ordinate axis in Definition 1.1 be parallel to OX and OY in the OXY co-ordinate system; L = L^{+} ∪ L^{−} , where L^{+} := {z ∈ L : Imz ⩾ 0}, L^{−} := {z ∈ L : Imz < 0}. Let w^{±} := \{w = e^{iθ} : θ = \frac{φ_{1} ±φ_{2}}{2} \}, z^{±} ∈ Ψ(w^{±} ) and L^{i} be arcs, connecting the points z^{+} , z_{i}, z^{−} ∈ L; L^{i,±} := L^{i} ∩ L^{±} , i = 1, 2. Let z_{0} be taken as an arbitrary point on L^{+} (or on L^{−} subject to the chosen direction). For simplicity, without loss of generality, we assume that z_{0} = z^{+} (z_{0} = z^{−} ). Analogously, we introduce: L_{R} = L_{R}^{+} ∪ L_{R}^{−}, where L_{R}^{+} := {z ∈ L_{R} : Imz ⩾ 0}, L_{R}^{−} := {z ∈ L_{R} : Imz < 0}. Let w_{R}^{±} := \{w = Re^{iθ} : θ = \frac{φ_{1} ±φ_{2}}{2} \}, z_{R}^{±} ∈ Ψ(w_{R}^{±}). We set: z_{i,R} ∈ L_{R}, such that d_{i,R} = |z_{i} − z_{i,R} | and ζ^{±} ∈ L^{±} , such that d(z_{2,R} , L^{2} ∩ L^{±} ) := d(z_{2,R} , L^{±} ); z_{i}^{±} := {ζ ∈ L^{i} : |ζ − z_{i}| = c_{i}d(z_{i}, L_{R})}, z_{i,R}^{±} := {ζ ∈ L_{R}^{i} : |ζ − z_{i,R} | = c_{i}d(z_{i,R} , L_{R})}, w_{i,R}^{±} = Φ(z_{i,R}^{±} ). Let L_{R}^{i}, i = 1, 2, denote arcs, connecting the points z_{R}^{+}, z_{i,R} , z_{R}^{−} ∈ L_{R}, L_{R}^{i,±} := L_{R}^{i} ∩ L_{R}^{±} and l_{i,R}^{±} (z_{i,R}^{±} , z_{R}^{±}) denote arcs, connecting the points z_{i,R}^{±} with z_{R}^{±}, respectively and |l_{i,R}^{±} | := mesl_{i,R}^{±} (z_{i,R}^{±} , z_{R}^{±}), i = 1, 2. We denote:

E_{1,R}^{i,±} : = {ζ ∈ L_{R}^{i,±} : |ζ − z_{i}| < c_{i}d_{i,R} },
E_{2,R}^{i,±} : = {ζ ∈ L_{R}^{i,±} : c_{i}d_{i,R} ⩽ |ζ − z_{i}| ⩽ |l_{i,R}^{±} |}, F_{j,R}^{i,±} := Φ(E_{j,R}^{i,±} );
E_{1}^{i,±} : = {ζ ∈ L^{i,±} : |ζ − z_{i}| < c_{i}d_{i,R} },
E_{2}^{i,±} : = {ζ ∈ L^{i,±} : c_{i}d_{i,R} ⩽ |ζ − z_{i}| ⩽ |l_{i,R}^{±} |}, F_{j}^{i,±} := Φ(E_{j}^{i,±} ), i, j = 1, 2.

Taking into consideration these designations, from (3.3), we have

J_{n,2}^{i} ≍ ∑_{i,j=1}^{2} ∫_{E_{j,R}^{i,+} ∪E_{j,R}^{i,−}} \frac{|dζ|}{|ζ − z_{i}|^{γ_{i}} |ζ − z|^{2}} =: ∑_{i,j=1}^{2} [J(E_{j,R}^{i,+} ) + J(E_{j,R}^{i,−} )].

So, we need to evaluate the integrals J(E_{j,R}^{i,+} ) and J(E_{j,R}^{i,−} ) for each i, j = 1, 2. Let

(3.4)  ∥P_{n}∥_{∞} =: |P_{n}(z′)|, z′ ∈ L = L^{1} ∪ L^{2}.

There are two possible cases: the point z′ may lie on L^{1} or L^{2}.

1) Suppose first that z′ ∈ L^{1}. Consider the individual cases.

(1.1) If z′ ∈ E_{1}^{1,±} ∪ E_{2}^{1,±} , then

J(E_{1,R}^{1,+} ) + J(E_{1,R}^{1,−} ) = ∫_{E_{1,R}^{1,+} ∪E_{1,R}^{1,−}}  \frac{|dζ|}{|ζ − z_{1}|^{γ_{1}} |ζ − z′|^{2}}

218 ABDULLAYEV AND ÖZKARTEPE

⪯ ∫_{E_{1,R}^{1,+} ∪E_{1,R}^{1,−}} \frac{|dζ|}{[min\{|ζ − z_{1}|; |ζ − z′|\}]^{γ_{1} +2}}

⪯ ∫_{d_{1,R}}^{cd_{1,R}} \frac{ds}{s^{γ_{1} +2}} ⪯ \frac{1}{d_{1,R}^{γ_{1} +1}} ,

for γ_{1} > 0, and

J(E_{1,R}^{1,+} )+J(E_{1,R}^{1,−} ) = ∫_{E_{1,R}^{1,+} ∪E_{1,R}^{1,−}} \frac{|ζ − z_{1}|^{(−γ_{1} )} |dζ|}{|ζ − z′|^{2}} ⪯ (cd_{1,R})^{(−γ_{1} )} ∫_{d_{1,R}}^{cd_{1,R}} \frac{ds}{s^{2}} ⪯ \frac{1}{d_{1,R}^{γ_{1} +1}} ,

for −1 < γ_{1} ⩽ 0

(1.2) If z′ ∈ E_{1}^{1,±} , then

J(E_{2,R}^{1,+} ) + J(E_{2,R}^{1,−} ) = ∫_{E_{2,R}^{1,+} ∪E_{2,R}^{1,−}}  \frac{|dζ|}{|ζ − z_{1}|^{γ_{1}} |ζ − z′|^{2}}

⪯ ∫_{E_{2,R}^{1,+} ∪E_{2,R}^{1,−}} \frac{|dζ|}{[min\{|ζ − z_{1}|; |ζ − z′|\}]^{γ_{1} +2}}

⪯ ∫_{cd_{1,R}}^{|l_{1,R}^{±} |} \frac{ds}{s^{γ_{1} +2}} ⪯ \frac{1}{d_{1,R}^{γ_{1} +1}} ,

for γ_{1} > 0 and

J(E_{2,R}^{1,+} ) + J(E_{2,R}^{1,−} ) = ∫_{E_{2,R}^{1,+} ∪E_{2,R}^{1,−}}  \frac{|ζ − z_{1}|^{(−γ_{1} )} |dζ|}{|ζ − z′|^{2}}

⪯ ∫_{E_{2,R}^{1,+} ∪E_{2,R}^{1,−}} \frac{|dζ|}{|ζ − z′|^{2}} ⪯ ∫_{cd_{1,R}}^{|l_{1,R}^{±} |} \frac{ds}{s^{2}} ⪯ \frac{1}{d_{1,R}} ,

for −1 < γ_{1} ⩽ 0; (1.3) If z′ ∈ E_{2}^{1,±} , then

J(E_{2,R}^{1,+} ) + J(E_{2,R}^{1,−} ) = ∫_{E_{2,R}^{1,+} ∪E_{2,R}^{1,−}}  \frac{|dζ|}{|ζ − z_{1}|^{γ_{1}} |ζ − z′|^{2}}

⪯ \frac{1}{d_{1,R}^{γ_{1}}} ∫_{E_{2,R}^{1,+} ∪E_{2,R}^{1,−}} \frac{|dζ|}{|ζ − z′|^{2}} ⪯ \frac{1}{d_{1,R}^{γ_{1}}} ∫_{cd_{1,R}}^{|l_{1,R}^{±} |} \frac{ds}{s^{2}} ⪯ \frac{1}{d_{1,R}^{(γ_{1} +1)}} ,

for γ_{1} > 0, and

J(E_{2,R}^{1,+} ) + J(E_{2,R}^{1,−} ) = ∫_{E_{2,R}^{1,+} ∪E_{2,R}^{1,−}}  \frac{|ζ − z_{1}|^{(−γ_{1} )} |dζ|}{|ζ − z′|^{2}}

⪯ ∫_{E_{2,R}^{1,+} ∪E_{2,R}^{1,−}} \frac{|dζ|}{|ζ − z′|^{2}} ⪯ ∫_{cd_{1,R}}^{|l_{1,R}^{±} |} \frac{ds}{s^{2}} ⪯ \frac{1}{d_{1,R}} ,

for −1 < γ_{1} ⩽ 0. Combining cases (1.1)–(1.3), we obtain

(3.5)  ∑_{i=1}^{2} [J(E_{i,R}^{1,+} ) + J(E_{i,R}^{1,−} )] ⪯ \frac{1}{d_{1,R}^{γ̃_{1} +1}} .

According to Lemma 2.2 and [8, p.10], from (3.5), we have

(3.6)  d_{1,R} ⪰ \{ \begin{matrix}n^{−μ} & if α_{1} = 0; \\ n^{−2} & if α_{1} ≠ 0, \end{matrix}

and, consequently, from (3.5) and (3.6), we get

(3.7)  ∑_{i=1}^{2} [J(E_{i,R}^{1,+} ) + J(E_{i,R}^{1,−} )] ⪯ \{ \begin{matrix}n^{2(γ̃_{1} +1)} , & if α_{1} ≠ 0; \\ n^{μ(γ̃_{1} +1)} , & if α_{1} = 0. \end{matrix}

2) Now, suppose that z′ ∈ L^{2}. In this case, replacing the variable τ = Φ(ζ), according to (2.1), we have

(3.8)  J_{n,2}^{i} ≍ ∑_{i,j=1}^{2} ∫_{F_{j,R}^{i,±}} \frac{|Ψ′(τ)||dτ|}{|Ψ(τ) − Ψ(w_{2})|^{γ_{2}} |Ψ(τ) − Ψ(w′)|^{2}}

=: ∑_{i,j=1}^{2} [J(F_{j,R}^{i,+} ) + J(F_{j,R}^{i,−} )].

(2.1) If z′ ∈ E_{1}^{2,±} , then
(3.9) J(F_{1,R}^{2,+} ) + J(F_{1,R}^{2,−} ) = ∫_{F_{1,R}^{2,+} ∪F_{1,R}^{2,−}}  \frac{|Ψ′(τ)||dτ|}{|Ψ(τ) − Ψ(w_{2})|^{γ_{2}} |Ψ(τ) − Ψ(w′)|^{2}}

≍ ∫_{F_{1,R}^{2,+} ∪F_{1,R}^{2,−}} \frac{d(Ψ(τ), L)|dτ|}{|Ψ(τ) − Ψ(w_{2})|^{γ_{2}} |Ψ(τ) − Ψ(w′)|^{2}(|τ| − 1)}

⪯ n ∫_{F_{1,R}^{2,+}} \frac{|dτ|}{|Ψ(τ) − Ψ(w_{2})|^{γ_{2}} |Ψ(τ) − Ψ(w′)|}

+ n ∫_{F_{1,R}^{2,−}} \frac{|dτ|}{|Ψ(τ) − Ψ(w_{2})|^{γ_{2}} |Ψ(τ) − Ψ(w′)|} ,

for all γ_{2} > −1. The last two integrals are evaluated identically. Therefore, we evaluate one of them, say the first. When τ ∈ F_{1,R}^{2,+} for the |Ψ(τ) − Ψ(w′)|, we obtain

|Ψ(τ) − Ψ(w′)| ⪰ max{|Ψ(τ) − Ψ(w_{2})|; |Ψ(τ) − z_{2}^{+} |}
⪰ |Ψ(τ) − Ψ(w_{2})| ⪰ |Ψ(τ) − z_{2}^{+} |^{\frac{1}{1+β_{2}}} .

Then, from (3.9), we get

J(F_{1,R}^{2,+} ) ⪯ n ∫_{F_{1,R}^{2,+}} \frac{|dτ|}{|Ψ(τ) − z_{2}^{+} |^{\frac{γ_{2} +1}{1+β_{2}}}} ⪯ n ∫_{F_{1,R}^{2,+}} \frac{|dτ|}{|Ψ(τ) − z_{2}^{+} |^{\frac{γ_{2} +1}{1+β_{2}}}} ⪯ n ∫_{F_{1,R}^{2,+}} \frac{|dτ|}{|τ − w_{2}^{+} |^{\frac{γ_{2} +1}{1+β_{2}} μ}}

⪯ \{ \begin{matrix}n^{\frac{γ_{2} +1}{1+β_{2}} μ}, & \frac{γ_{2} +1}{1+β_{2}} μ > 1, \\ n lnn, & \frac{γ_{2} +1}{1+β_{2}} μ = 1, \\ n, & \frac{γ_{2} +1}{1+β_{2}} μ < 1, \end{matrix}

220 ABDULLAYEV AND ÖZKARTEPE

if γ_{2} > 0, and

J(F_{1,R}^{2,+} ) ⪯ n ∫_{F_{1,R}^{2,+}} \frac{|Ψ(τ) − Ψ(w_{2})|^{(−γ_{2} )} |dτ|}{|Ψ(τ) − z_{2}^{+} |^{\frac{1}{1+β_{2}}}}

⪯ n ∫_{F_{1,R}^{2,+}} \frac{|dτ|}{|τ − w_{2}^{+} |^{\frac{μ}{1+β_{2}}}} ⪯ \{ \begin{matrix}n^{\frac{μ}{1+β_{2}}} , & \frac{μ}{1+β_{2}} > 1, \\ n lnn, & \frac{μ}{1+β_{2}} = 1, \\ n, & \frac{μ}{1+β_{2}} < 1, \end{matrix}

if −1 < γ_{2} ⩽ 0, and so, in this case we get

(3.10)  J(F_{1,R}^{2,+} ) + J(F_{1,R}^{2,−} ) ⪯ \{ \begin{matrix}n^{\frac{γ̃_{2} +1}{1+β_{2}} μ}, & \frac{γ̃_{2} +1}{1+β_{2}} μ > 1, \\ n lnn, & \frac{γ̃_{2} +1}{1+β_{2}} μ = 1, \\ n, & \frac{γ̃_{2} +1}{1+β_{2}} μ < 1. \end{matrix}
(2.2) If z′ ∈ E_{2}^{2,±} , then
J(F_{1,R}^{2,+} ) + J(F_{1,R}^{2,−} ) = ∫_{F_{1,R}^{2,+} ∪F_{1,R}^{2,−}}  \frac{d(Ψ(τ), L)|dτ|}{|Ψ(τ) − Ψ(w_{2})|^{γ_{2}} |Ψ(τ) − Ψ(w′)|^{2}(|τ| − 1)}

⪯ n ∫_{F_{1,R}^{2,+} ∪F_{1,R}^{2,−}} \frac{|dτ|}{|Ψ(τ) − Ψ(w_{2})|^{γ_{2}} |Ψ(τ) − Ψ(w′)|} , (3.11)

for all γ_{2} > −1. When τ ∈ F_{1,R}^{2,+} for the |Ψ(τ)−Ψ(w′)|, we obtain |Ψ(τ)−Ψ(w′)| ⪰ |Ψ(τ) − z_{2}^{+} | and, analogously to the previous case, we get

J(F_{1,R}^{2,+} ) ⪯ n ∫_{F_{1,R}^{2,+}} \frac{|dτ|}{|Ψ(τ) − Ψ(w_{2})|^{γ_{2}} |Ψ(τ) − z_{2}^{+} |} ⪯ n ∫_{F_{1,R}^{2,+}} \frac{|dτ|}{|Ψ(τ) − z_{2}^{+} |^{\frac{γ_{2}}{1+β_{2}} +1}}

⪯ n ∫_{F_{1,R}^{2,+}} \frac{|dτ|}{|τ − w_{2}^{+} |^{( \frac{γ_{2}}{1+β_{2}} +1)μ}} ⪯ \{ \begin{matrix}n^{( \frac{γ_{2}}{1+β_{2}} +1)μ} , & (\frac{γ_{2}}{1+β_{2}} + 1)μ > 1, \\ n lnn, & (\frac{γ_{2}}{1+β_{2}} + 1)μ = 1, \end{matrix} (3.12)

if γ_{2} > 0, and

(3.13)  J(F_{1,R}^{2,+} ) ⪯ n ∫_{F_{1,R}^{2,+}} \frac{|Ψ(τ) − Ψ(w_{2})|^{(−γ_{2} )} |dτ|}{|Ψ(τ) − z_{2}^{+} |} ⪯ n ∫_{F_{1,R}^{2,+}} \frac{|dτ|}{|Ψ(τ) − z_{2}^{+} |}

⪯ n ∫_{F_{1,R}^{2,+}} \frac{|dτ|}{|τ − w_{2}^{+} |^{μ}} ⪯ \{ \begin{matrix}n^{μ}, & μ > 1, \\ n lnn, & μ = 1, \\ n, & μ < 1, \end{matrix}

if −1 < γ_{2} ⩽ 0. So, in this case, from (3.11)–(3.13), we have

(3.14)  J(F_{1,R}^{2,+} ) + J(F_{1,R}^{2,−} ) ⪯ \{ \begin{matrix}n^{( \frac{γ̃_{2}}{1+β_{2}} +1)μ} , & ( \frac{γ̃_{2}}{1+β_{2}} + 1)μ > 1, \\ n lnn, & ( \frac{γ̃_{2}}{1+β_{2}} + 1)μ = 1. \end{matrix}
(2.3) If z′ ∈ E_{1}^{2,±} , then
(3.15)  J(F_{2,R}^{2,+} ) + J(F_{2,R}^{2,−} ) = ∫_{F_{2,R}^{2,+} ∪F_{2,R}^{2,−}}  \frac{|Ψ′(τ)||dτ|}{|Ψ(τ) − Ψ(w_{2})|^{γ_{2}} |Ψ(τ) − Ψ(w′)|^{2}}

⪯ n ∫_{F_{2,R}^{2,+} ∪F_{2,R}^{2,−}} \frac{|dτ|}{|Ψ(τ) − Ψ(w_{2})|^{γ_{2}} |Ψ(τ) − Ψ(w′)|}

⪯ n ∫_{F_{2,R}^{2,+}} \frac{|dτ|}{|Ψ(τ) − Ψ(w_{2})|^{γ_{2}} |Ψ(τ) − Ψ(w′)|}

+ n ∫_{F_{2,R}^{2,−}} \frac{|dτ|}{|Ψ(τ) − Ψ(w_{2})|^{γ_{2}} |Ψ(τ) − Ψ(w′)|} ,

for γ_{2} > 0. The last two integrals are evaluated in the same way. Let us evaluate the first integral.

For τ ∈ F_{2,R}^{2,+} and z′ ∈ E_{1}^{2,±} , we have

|Ψ(τ) − Ψ(w′)| ⪰ |Ψ(τ) − z_{2}^{+} |;
|Ψ(τ) − Ψ(w_{2})| ⪰ d_{2,R} ⪰ |z_{2,R} − z_{2}^{+} |^{\frac{1}{1+β_{2}}} ⪰ ( \frac{1}{n}) ^{\frac{1}{1+β_{2}} μ}.

Then

J(F_{2,R}^{2,+} ) ⪯ n ∫_{F_{2,R}^{2,+}} \frac{|dτ|}{|Ψ(τ) − z_{2}^{+} |^{γ_{2}} |Ψ(τ) − z_{2}^{+} |} ⪯ n^{\frac{γ_{2}}{1+β_{2}} μ+1} ∫_{F_{2,R}^{2,+}} \frac{|dτ|}{|τ − w_{2}^{+} |^{μ}}

⪯ \{ \begin{matrix}n^{\frac{γ_{2}}{1+β_{2}} μ+μ} , & μ > 1, \\ n^{\frac{γ_{2}}{1+β_{2}} μ+1} lnn, & μ = 1, \end{matrix}

and so, for γ_{2} > 0 we obtain

J(F_{2,R}^{2,+} ) + J(F_{2,R}^{2,−} ) ⪯ \{ \begin{matrix}n^{\frac{γ_{2}}{1+β_{2}} μ+μ} , & μ > 1, \\ n^{\frac{γ_{2}}{1+β_{2}} μ+1} lnn, & μ = 1. \end{matrix}

For −1 < γ_{2} ⩽ 0 we get

(3.16) J(F_{2,R}^{2,+} ) + J(F_{2,R}^{2,−} ) = ∫_{F_{2,R}^{2,+} ∪F_{2,R}^{2,−}}  \frac{|Ψ(τ) − Ψ(w_{2})|^{(−γ_{2} )} |Ψ′(τ)||dτ|}{|Ψ(τ) − Ψ(w′)|^{2}}

⪯ n ∫_{F_{2,R}^{2,+}} \frac{|dτ|}{|Ψ(τ) − Ψ(w′)|} ⪯ n ∫_{F_{2,R}^{2,+}} \frac{|dτ|}{|Ψ(τ) − z_{2}^{+} |}

⪯ n ∫_{F_{2,R}^{2,+}} \frac{|dτ|}{|τ − w_{2}^{+} |^{μ}} ⪯ \{ \begin{matrix}n^{μ}, & μ > 1, \\ n lnn, & μ = 1. \end{matrix}

Then, in this case, we have

(3.17)  J(F_{2,R}^{2,+} ) + J(F_{2,R}^{2,−} ) ⪯ \{ \begin{matrix}n^{[ \frac{γ̃_{2}}{1+β_{2}} +1]μ} , & μ > 1, \\ n^{\frac{γ̃_{2}}{1+β_{2}} μ+1} lnn, & μ = 1. \end{matrix}

(2.4) If z′ ∈ E_{2}^{2,+} , then for γ_{2} > 0

J(F_{2,R}^{2,+} ) = ∫_{F_{2,R}^{2,+}} \frac{|Ψ(τ)||dτ|}{|Ψ(τ) − Ψ(w_{2})|^{γ_{2}} |Ψ(τ) − Ψ(w′)|^{2}}  ⪯ \frac{n}{d_{2,R}^{γ_{2}}} ∫_{F_{2,R}^{2,+}} \frac{|dτ|}{|Ψ(τ) − Ψ(w′)|}

222 ABDULLAYEV AND ÖZKARTEPE

⪯ n^{1+ \frac{γ_{2}}{1+β_{2}} μ} ∫_{F_{2,R}^{2,+}} \frac{|dτ|}{|τ − w′|^{μ}} ⪯ \{ \begin{matrix}n^{[ \frac{γ_{2}}{1+β_{2}} +1]μ} , & μ > 1, \\ n^{\frac{γ_{2}}{1+β_{2}} μ+1} lnn, & μ = 1, \end{matrix} (3.18)

and

J(F_{2,R}^{2,−} ) = ∫_{F_{2,R}^{2,−}} \frac{|Ψ′(τ)||dτ|}{|Ψ(τ) − Ψ(w_{2})|^{γ_{2}} |Ψ(τ) − Ψ(w′)|^{2}}  ⪯ \frac{n}{d_{2,R}^{γ_{2}}} ∫_{F_{2,R}^{2,−}} \frac{|dτ|}{|Ψ(τ) − Ψ(w′)|}

⪯ n^{\frac{γ_{2}}{1+β_{2}} μ+!} ∫_{F_{2,R}^{2,−}} \frac{|dτ|}{|Ψ(τ) − Ψ(w′)|} ⪯ n^{\frac{γ_{2}}{1+β_{2}} μ+1} ∫_{F_{2,R}^{2,−}} \frac{|dτ|}{|τ − w′|^{μ}}

⪯ \{ \begin{matrix}n^{[ \frac{γ_{2}}{1+β_{2}} +1]μ} , & μ > 1, \\ n^{\frac{γ_{2}}{1+β_{2}} μ+1} lnn, & μ = 1. \end{matrix} (3.19)

The case of z′ ∈ E_{2}^{2,−} is absolutely identical to the case z′ ∈ E_{2}^{2,+} . If −1 < γ_{2} ⩽ 0, then

(3.20)  J(F_{2,R}^{2,+} ) = ∫_{F_{2,R}^{2,+}} \frac{|Ψ(τ) − Ψ(w_{2})|^{(−γ_{2} )} |Ψ′(τ)||dτ|}{|Ψ(τ) − Ψ(w′)|^{2}}

⪯ n ∫_{F_{2,R}^{2,+}} \frac{|dτ|}{|Ψ(τ) − Ψ(w′)|} ⪯ \{ \begin{matrix}n^{μ}, & μ > 1, \\ n lnn, & μ = 1, \end{matrix}

and

(3.21)  J(F_{2,R}^{2,−} ) = ∫_{F_{2,R}^{2,−}} \frac{|Ψ(τ) − Ψ(w_{2})|^{(−γ_{2} )} |Ψ′(τ)||dτ|}{|Ψ(τ) − Ψ(w′)|^{2}}

⪯ n ∫_{F_{2,R}^{2,−}} \frac{|dτ|}{|Ψ(τ) − Ψ(w′)|} ⪯ \{ \begin{matrix}n^{μ}, & μ > 1, \\ n lnn, & μ = 1, \end{matrix}

Combining relations (3.7)–(3.21), for m_{1} = 1, m_{2} = 1, and any p > 0, we get

(3.22)  J_{n,2}^{1} + J_{n,2}^{2} ⪯ \{ \begin{matrix}n^{2}, & if α_{1} ≠ 0; \\ n^{μ} & if α_{1} = 0; \end{matrix} + \{ \begin{matrix}n^{μ}, & μ > 1, \\ n lnn, & μ = 1, \end{matrix}

for each −1 < γ_{1} ⩽ 0 and −1 < γ_{2} ⩽ 0;

(3.23)  J_{n,2}^{1} + J_{n,2}^{2} ⪯ n^{μ̃(γ_{1} +1)} + \{ \begin{matrix}n^{( \frac{γ_{2}}{1+β_{2}} +1)μ} , & μ > 1, \\ n^{\frac{γ_{2}}{1+β_{2}} +1} lnn, & μ = 1, \end{matrix}

for each γ_{1} > 0 and γ_{2} > 0, where μ̃ := \{ \begin{matrix}μ, & if α_{1} = 0, \\ 2, & if α_{1} ≠ 0, \end{matrix} and p > 0. Then, from (3.1)–(3.4), (3.22) and (3.23), for all z ∈ L, we obtain

|P_{n}(z)| ⪯ ∥P_{n}∥_{p} · [n^{\frac{μ̃ (γ̃_{1} +1)}{p}} + \{ \begin{matrix}n^{( \frac{γ̃_{2}}{1+β_{2}} +1) \frac{μ}{p}} , & μ > 1, \\ (n^{\frac{γ̃_{2}}{1+β_{2}} +1} lnn)^{\frac{1}{p}} , & μ = 1, \end{matrix} ]

⪯ ∥P_{n}∥_{p} · \{ \begin{matrix}n^{\frac{1}{p} max \{2,μ\}} , & μ̃ = 2, α_{1} ≠ 0, −1 < γ_{1}, γ_{2} ⩽ 0; \\ n^{\frac{μ}{p}} , & μ̃ = μ, α_{1} = 0, −1 < γ_{1}, γ_{2} ⩽ 0; \\ n^{( \frac{γ_{2}}{1+β_{2}} +1) \frac{μ}{p}} , & μ̃ = 2, α_{1} ≠ 0, 0 < γ_{1} ⩽ ( \frac{γ_{2}}{1+β_{2}} + 1) \frac{μ}{2} − 1; \\ n^{\frac{2(γ_{1} +1)}{p}} , & μ̃ = 2, α_{1} ≠ 0, γ_{1} > ( \frac{γ_{2}}{1+β_{2}} + 1) \frac{μ}{2} − 1; \\ n^{( \frac{γ_{2}}{1+β_{2}} +1) \frac{μ}{p}} , & μ̃ = μ, α_{1} = 0, 0 < γ_{1} ⩽ \frac{γ_{2}}{1+β_{2}} ; \\ n^{\frac{μ(γ_{1} +1)}{p}} , & μ̃ = μ, α_{1} = 0, γ_{1} > \frac{γ_{2}}{1+β_{2}} ; \\ (n^{\frac{γ̃_{2}}{1+β_{2}} +1} lnn)^{\frac{1}{p}} & μ = 1, −1 < γ_{1}, γ_{2}, \end{matrix}

which completes the proof. □

Proof of Theorem 1.3. Suppose G ∈ P QS(λ; f_{i}, g_{i}), for some λ ⩾ 1, f_{i}(x) = c_{i}x^{1+α_{i}} , α_{i} ⩾ 0, i = \overline{1, m_{1}}, and g_{i}(x) = c_{i}x^{1+β_{i}} , β_{i} > 0, i = \overline{m_{1} + 1, m} ; h(z) defined as in (1.3). Let {ζ_{j} }, 1 ⩽ j ⩽ m ⩽ n, zeros of P_{n}(z) lying on Ω and let

B_{m}(z) := ∏_{j=1}^{m} B_{j} (z) = ∏_{j=1}^{m} \frac{Φ(z) − Φ(ζ_{j})}{1 − \overline{Φ(ζ_{j})}Φ(z)}

denote a Blashke function with respect to zeros {ζ_{j} }, 1 ⩽ j ⩽ m ⩽ n, of P_{n}(z). For any p > 0 and z ∈ Ω, let us set

(3.24)  G_{n}(z) := [ \frac{P_{n}(z)}{B_{m}(z)Φ^{n+1} (z)} ]^{p/2}.

The Cauchy integral representation for the unbounded region Ω gives

(3.25)  G_{n}(z) = −\frac{1}{2πi} ∫_{L} G_{n}(ζ)\frac{dζ}{ζ − z} , z ∈ Ω.

Since |B_{m}(ζ)| = 1, for ζ ∈ L, then, for arbitrary ε, 0 < ε < ε_{1}, there exists a circle |w| = 1 + \frac{ε_{1}}{n} , such that for any j = \overline{1, m} the following is satisfied:

|B_{j} (Ψ(w)| > 1 − ε.

Then, |B_{m}(ζ)| > (1 − ε)^{m} ⪰ 1, for each ε ⩽ n^{−1} . On the other hand, |Φ(ζ)| = 1, for ζ ∈ L. Therefore, for any z ∈ Ω, from (3.24) and (3.25), we have

(3.26)  | \frac{P_{n}(z)}{B_{m}(z)Φ^{n+1} (z)} |^{p/2} ⩽ \frac{1}{2π} ∫_{L} | \frac{P_{n}(ζ)}{B_{m}(ζ)Φ^{n+1} (ζ)} |^{p/2} \frac{|dζ|}{|ζ − z|}

⪯ \frac{1}{d(z, L)} ∫_{L} |P_{n}(ζ)|^{p/2} |dζ| =: \frac{1}{d(z, L)}A_{n}.

Similarly to the notations of the previous proof, we have

(3.27)  A_{n} = ∑_{i=1}^{m} ∫_{L^{i}} |P_{n}(ζ)|^{p/2} |dζ|.

Multiplying the numerator and denominator of the integrand by h^{1/2} (ζ), after applying the Hölder inequality, we obtain

(3.28)  A_{n} ⩽ ∑_{i=1}^{m} ( ∫_{L^{i}} h(ζ)|P_{n}(ζ)|^{p}|dζ|)^{1/2} × ( ∫_{L^{i}} \frac{|dζ|}{∏ _{j=1}^{m} |ζ − z_{j} |^{γ_{j}}} )^{1/2}

=: ∑_{i=1}^{m} (J̃_{n,1}^{i} · J̃_{n,2}^{i} )^{1/2} .

According to Lemma 2.3, for the J̃_{n,1}^{i} we get

(3.29)  J̃_{n,1}^{i} ⪯ ∥P_{n}∥p^{p/2} , i = \overline{1, m}.

Then, from (3.28) and (3.29), we have

A_{n} ⪯ ∥P_{n}∥_{p}^{p/2} ∑_{i=1}^{m} (J̃_{n,2}^{i} )^{1/2} .

For the integral J̃_{n,2}^{i} we obtain
(3.30)  J̃_{n,2}^{i} := ∫_{L^{i}} \frac{|dζ|}{∏ _{j=1}^{m} |ζ − z_{j} |^{γ_{i}}} ≍ ∫_{L^{i}} \frac{|dζ|}{|ζ − z_{i}|^{γ_{i}}} , i = 1, 2,

since the points {z_{j} }_{j=1}^{m} are distinct on L. Then, from (3.30), we have

(3.31)  A_{n} ⪯ ∥P_{n}∥_{p}^{p/2} ∑_{i=1}^{2} (J̃_{n,2}^{i} )^{1/2} ,

where

(3.32)  J̃_{n,2}^{1} = ∫_{L^{1}} \frac{|dζ|}{|ζ − z_{1}|^{γ_{1}}} ; J̃_{n,2}^{2} = ∫_{L^{2}} \frac{|dζ|}{|ζ − z_{2}|^{γ_{2}}} .

Taking into consideration the above notations, from (3.31) we get

(3.33)  J̃_{n,2}^{i} =: I_{n,1}^{i} (E_{1}^{i,±} ) + I_{n,2}^{i} (E_{2}^{i,±} ) =: I_{n,1}^{i,±} + I_{n,2}^{i,±} , i = 1, 2,

where

I_{n,k}^{i,±} := I_{n,k}^{i} (E_{k}^{i,±} ) := ∫_{E_{k}^{i,±}} \frac{|dζ|}{|ζ − z_{i}|^{γ_{i}}} ; i, k = 1, 2.

According to (3.26) and (3.27), it is sufficient to estimate the integrals I_{n,k}^{i,±} for each i = 1, 2 and k = 1, 2.

Let us start with the evaluation of the integrals J̃_{n,2}^{1} from (3.32) and (3.33)

(3.34)  J̃_{n,2}^{1} = ∫_{L^{1}} \frac{|dζ|}{|ζ − z_{1}|^{γ_{1}}} = ∑_{k=1}^{2} ∫_{E_{k}^{1,±}} \frac{|dζ|}{|ζ − z_{i}|^{γ_{1}}} =: I_{n,1}^{1,±} + I_{n,2}^{1,±} .

Given the possible values of γ_{i}(−1 < γ_{i} < 0, γ_{i} ⩾ 0, i = 1, 2), we will consider the estimates for the J̃_{n,2}^{1} separately.

POLYNOMIAL INEQUALITIES IN LAVRENTIEV REGIONS... 225

Let γ_{1} ⩾ 0 and γ_{2} ⩾ 0. In this case for the integral J̃_{n,2}^{1} , we get

I_{n,1}^{1,±} ⪯ ∫_{E_{1}^{1,±}} \frac{|dζ|}{|ζ − z_{1}|^{γ_{1}}} ⪯ ∫_{0}^{c_{1} d_{1,R}} \frac{ds}{s^{γ_{1}}} ⪯ \{ \begin{matrix}d_{1,R}^{1−γ_{1}} , & γ_{1} > 1, \\ 1, & γ_{1} ⩽ 1, \end{matrix} c_{1} > 1; (3.35)

I_{n,2}^{1,±} ⪯ ∫_{E_{2}^{1,±}} \frac{|dζ|}{|ζ − z_{1}|^{γ_{1}}} ⪯ ∫_{c_{1} d_{1,R}}^{|l_{1}^{±} |} \frac{ds}{s^{γ_{1}}} ⪯ \{ \begin{matrix}d_{1,R}^{1−γ_{1}} , & γ_{1} > 1, \\ ln \frac{1}{d_{1,R}} , & γ_{1} = 1, \\ 1, & γ_{1} < 1. \end{matrix}

A similar estimate for the integral J̃_{n,2}^{2} is

I_{n,1}^{2,±} ⪯ ∫_{E_{1}^{2,±}} \frac{|dζ|}{|ζ − z_{2}|^{γ_{2}}} ⪯ ∫_{0}^{c_{2} d_{2,R}} \frac{ds}{s^{γ_{2}}} ⪯ \{ \begin{matrix}d_{2,R}^{1−γ_{2}} , & γ_{2} > 1, \\ 1, & γ_{2} ⩽ 1, \end{matrix} c_{2} > 1; (3.36)

I_{n,2}^{2,±} ⪯ ∫_{E_{2}^{2,±}} \frac{|dζ|}{|ζ − z_{2}|^{γ_{2}}} ⪯ ∫_{c_{2} d_{2,R}}^{|l_{2}^{±} |} \frac{ds}{s^{γ_{2}}} ⪯ \{ \begin{matrix}d_{2,R}^{1−γ_{2}} , & γ_{2} > 1, \\ ln \frac{1}{d_{2,R}} , & γ_{2} = 1, \\ 1, & γ_{2} < 1. \end{matrix}

Let γ_{1} < 0 and γ_{2} < 0. Then, analogously to (3.35) and (3.36)

(3.37)  I_{n,1}^{1,±} ⪯ ∫_{E_{1}^{1,±}} |ζ − z_{1}|^{(−γ_{1} )} |dζ| ⪯ d_{1,R}^{(−γ_{1} )} mesE_{1}^{1} ⪯ 1,
I_{n,2}^{1} ⪯ ∫_{E_{2}^{1,±}} |ζ − z_{1}|^{(−γ_{1} )} |dζ| ⪯ |l_{1}^{±} |^{(−γ_{1} )+1} ⪯ 1,

and

(3.38)  I_{n,1}^{2,±} ⪯ ∫_{E_{1}^{2,±}} |ζ − z_{2}|^{(−γ_{2} )} |dζ| ⪯ d_{2,R}^{(−γ_{2} )} mesE_{1}^{2,±} ⪯ 1,
I_{n,2}^{2,±} ⪯ ∫_{E_{2}^{2,±}} |ζ − z_{2}|^{(−γ_{2} )} |dζ| ⪯ |l_{2}^{±} |^{(−γ_{2} )+1} ⪯ 1.

Therefore, in this case, from (3.31)–(3.38), we obtain

(3.39)  A_{n} ⪯ ∥P_{n}∥_{L_{p} (h,L)} \{ \begin{matrix}d1,R^{\frac{1−γ_{1}}{2}} + d2,R^{\frac{1−γ_{2}}{2}} , & γ_{1}, γ_{2} > 1, \\ ( ln \frac{1}{d_{1,R}} )^{\frac{1}{2}} + ( ln \frac{1}{d_{2,R}} )^{\frac{1}{2}} , & γ_{1} = γ_{2} = 1, \\ 1, & −1 < γ_{1}, γ_{2} < 1. \end{matrix}

Comparing (3.26), (3.27) and (3.39), we have

(3.40)  |P_{n}(z)| ⩽ c\frac{B_{n,1}^{0}}{d^{2/p}(z, L)}∥P_{n}∥_{L_{p} (h,L)} |Φ(z)|^{n+1} ,

where c = c(G, p, γ_{i}) > 0, i = \overline{1, p}, is the constant independent of n and z, and

(3.41)  B_{n,1}^{0} := \{ \begin{matrix}d1,R^{\frac{1−γ_{1}}{p}} + d2,R^{\frac{1−γ_{2}}{p}} , & γ_{1}, γ_{2} > 1, \\ ( ln \frac{1}{d_{1,R}} )^{1/p} + ( ln \frac{1}{d_{2,R}} )^{1/p}, & γ_{1} = γ_{2} = 1, \\ 1, & 0 < γ_{1}, γ_{2} < 1. \end{matrix}

According to (3.6), for the point z_{1}, we get

(3.42)  d_{1,R} ⪰ \{ \begin{matrix}n^{−μ} , & if α_{1} = 0; \\ n^{−2} , & if α_{1} ≠ 0 \end{matrix}

For the estimate d_{2,R}, let us set z_{R} ∈ L_{R}, such that d_{2,R} = |z_{2} − z_{R}|; ζ^{±} ∈ L^{±} , such that d(z_{R}, L^{2} ∩ L^{±} ) := d(z_{R}, L^{+} ); z_{2}^{±} := ζ ∈ L^{2} : |ζ − z_{2}| = c_{2}d_{2,R}. Under these notations, from Lemma 2.1, we obtain

(3.43)  d_{R}^{±} := d(z_{R}, L^{2} ∩ L^{±} ) ≍ |z_{R} − z_{2}^{±} | ≍ d_{2,R}^{1+β_{2}} .

Hence, d_{2,R} = (d_{R}^{±})^{\frac{1}{1+β_{2}}} . On the other hand, according to Lemma 2.2 and [8, Corollary 2], we get d_{R}^{±} ⪰ n^{−μ} . Therefore,

(3.44)  d_{2,R} ⪰ n^{\frac{−μ}{1+β_{2}}} .

Comparing (3.40)–(3.44), we get

|P_{n}(z)| ⪯ \frac{B_{n,1}}{d^{2/p}(z, L)}∥P_{n}∥_{p}|Φ(z)|^{n+1} ,

where

B_{n,1} := \{ \begin{matrix}n^{\frac{(γ_{1} −1)μ̃}{p}} + n^{\frac{γ_{2} −1}{p(1+β_{2} )} μ}, & γ_{1}, γ_{2} > 1, \\ (lnn)^{\frac{1}{p}} , & γ_{1} = γ_{2} = 1, \\ 1, & γ_{1}, γ_{2} < 1. \end{matrix}

Given the above mentioned notation, we complete the proof. □

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Mersin University (Received 26 08 2014) Faculty of Arts and Science (Revised 12 10 2015) Department of Mathematics Mersin Turkey

Kyrgyzistan–Turkey Manas University Faculty of Science Department of Applied Mathematics and Informatics Bishkek Kyrgyzistan

fabdul@mersin.edu.tr
fahreddinabdullayev@gmail.com
fahreddin.abdullayev@manas.edu.kg

Mersin University Faculty of Arts and Science Department of Mathematics Mersin Turkey

pelinozkartepe@gmail.com