On the behavior of algebraic polynomial in unbounded regions with piecewise Dini-smooth boundary
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| Цитувати: | On the behavior of algebraic polynomial in unbounded regions with piecewise Dini-smooth boundary / F.G. Abdullayev, P. Ozkartepe // Український математичний журнал. — 2014. — Т. 66, № 5. — С. 579–597. — Бібліогр.: 14 назв. — англ. |
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Abdullayev, F.G. Ozkartepe, P. 2020-02-18T05:07:57Z 2020-02-18T05:07:57Z 2014 On the behavior of algebraic polynomial in unbounded regions with piecewise Dini-smooth boundary / F.G. Abdullayev, P. Ozkartepe // Український математичний журнал. — 2014. — Т. 66, № 5. — С. 579–597. — Бібліогр.: 14 назв. — англ. 1027-3190 https://nasplib.isofts.kiev.ua/handle/123456789/166066 517.5 en Інститут математики НАН України Український математичний журнал Статті On the behavior of algebraic polynomial in unbounded regions with piecewise Dini-smooth boundary Поведінка алгебраїчного полінома в необмежених областях з кусковими Діні-гладкими межами Article published earlier |
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On the behavior of algebraic polynomial in unbounded regions with piecewise Dini-smooth boundary |
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On the behavior of algebraic polynomial in unbounded regions with piecewise Dini-smooth boundary Abdullayev, F.G. Ozkartepe, P. Статті |
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On the behavior of algebraic polynomial in unbounded regions with piecewise Dini-smooth boundary |
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On the behavior of algebraic polynomial in unbounded regions with piecewise Dini-smooth boundary |
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On the behavior of algebraic polynomial in unbounded regions with piecewise Dini-smooth boundary |
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On the behavior of algebraic polynomial in unbounded regions with piecewise Dini-smooth boundary |
| title_sort |
on the behavior of algebraic polynomial in unbounded regions with piecewise dini-smooth boundary |
| author |
Abdullayev, F.G. Ozkartepe, P. |
| author_facet |
Abdullayev, F.G. Ozkartepe, P. |
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Статті |
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Статті |
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2014 |
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English |
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Український математичний журнал |
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Інститут математики НАН України |
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Article |
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Поведінка алгебраїчного полінома в необмежених областях з кусковими Діні-гладкими межами |
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1027-3190 |
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https://nasplib.isofts.kiev.ua/handle/123456789/166066 |
| citation_txt |
On the behavior of algebraic polynomial in unbounded regions with piecewise Dini-smooth boundary / F.G. Abdullayev, P. Ozkartepe // Український математичний журнал. — 2014. — Т. 66, № 5. — С. 579–597. — Бібліогр.: 14 назв. — англ. |
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| fulltext |
UDC 517.5
F. G. Abdullayev, P. Özkartepe (Mersin Univ., Turkey)
ON THE BEHAVIOR OF THE ALGEBRAIC POLYNOMIAL
IN UNBOUNDED REGIONS WITH PIECEWISE DINI-SMOOTH BOUNDARY
ПОВЕДIНКА АЛГЕБРАЇЧНОГО ПОЛIНОМА В НЕОБМЕЖЕНИХ ОБЛАСТЯХ
З КУСКОВИМИ ДIНI-ГЛАДКИМИ МЕЖАМИ
Let G ⊂ C be a finite region bounded by a Jordan curve L := ∂G, Ω := extG (with respect to C), ∆ := {w : |w| > 1};
w = Φ(z) be the univalent conformal mapping of Ω onto the ∆, normalized by Φ(∞) =∞, Φ′(∞) > 0. Let h(z) be a
weight function, and Ap(h,G), p > 0, denote the class of functions f which are analytic in G and satisfying the condition
‖f‖pAp(h,G) :=
∫∫
G
h(z) |f(z)|p dσz <∞,
where σ is a two-dimensional Lebesgue measure.
Let Pn(z) be an arbitrary algebraic polynomial of degree at most n ∈ N. Well known Bernstein – Walsh lemma shown
that:
|Pn(z)| ≤ |Φ(z)|n ‖Pn‖C(G) , z ∈ Ω. (∗)
In this present work we continue studying the estimation (∗), when we replace the norm ‖Pn‖C(G) by ‖Pn‖Ap(h,G) ,
p > 0, for Jacobi type weight function in regions with piecewise Dini-smooth boundary.
Нехай G ⊂ C — скiнченна множина, обмежена жордановою кривою L := ∂G, Ω := extG (вiдносно C), ∆ :=
:= {w : |w| > 1} ; w = Φ(z) — однолисте конформне вiдображення Ω на ∆, нормоване так, що Φ(∞) = ∞ та
Φ′(∞) > 0. Також нехай h(z) — вагова функцiя, а Ap(h,G), p > 0, — клас функцiй f, аналiтичних в G, що
задовольняють умову
‖f‖pAp(h,G) :=
∫∫
G
h(z) |f(z)|p dσz <∞,
де σ — двовимiрна мiра Лебега.
Нехай Pn(z) — довiльний алгебраїчний полiном степеня не бiльшого за n ∈ N. Вiдома лема Бернштейна –
Волша стверджує, що
|Pn(z)| ≤ |Φ(z)|n ‖Pn‖C(G) , z ∈ Ω. (∗)
У данiй роботi продовжено дослiдження оцiнки (∗), в якiй норму ‖Pn‖C(G) замiнено на ‖Pn‖Ap(h,G) , p > 0,
для вагової функцiї типу Якобi в областях з кусковими Дiнi-гладкими межами.
1. Introduction and main results. Let C be a complex plane, C := C ∪ {∞} , G ⊂ C be a
bounded Jordan region with 0 ∈ G and the boundary L := ∂G in the form of a simple closed Jordan
curve, Ω := C\G, B := B(0, 1) := {z : |z| < 1} , and ∆ := ∆(0, 1) := {w : |w| > 1}. Also let
w = Φ(z)(w = ϕ(z)) be a univalent conformal mapping of Ω(G) onto the ∆(B) normalized by
Φ(∞) =∞, limz→∞
Φ(z)
z
> 0 (ϕ(0) = 0, ϕ′(0) > 0), and Ψ := Φ−1 (ψ := ϕ−1).
By ℘n we denote the class of arbitrary algebraic polynomials Pn(z) of degree at most n ∈ N.
Let h(z) be a weight function. By Ap(h,G), p > 0, we denote a class of functions f analytic in
G and satisfying the condition
‖f‖Ap(h,G) :=
∫∫
G
h(z) |f(z)|p dσz
1/p
<∞,
where σz is the two-dimensional Lebesgue measure and Ap(1, G) ≡ Ap(G).
c© F. G. ABDULLAYEV, P. ÖZKARTEPE, 2014
ISSN 1027-3190. Укр. мат. журн., 2014, т. 66, № 5 579
580 F. G. ABDULLAYEV, P. ÖZKARTEPE
In case where L is rectifiable, let Lp(h, L), p > 0, denote a class of functions f integrable on L
and satisfying the condition
‖f‖Lp(h,L) :=
∫
L
h(z) |f(z)|p |dz|
1/p
<∞,
and Lp(1, L) ≡ Lp(L).
From well known Bernstein – Walsh lemma [14], we see that
|Pn(z)| ≤ |Φ(z)|n ‖Pn‖C(G) , z ∈ Ω. (1.1)
For R > 1, we set LR := {z : |Φ(z)| = R} , GR := intLR, ΩR := extLR. Then (1.1) can be
rewritten as follows:
‖Pn‖C(GR) ≤ R
n ‖Pn‖C(G) . (1.2)
Hence, setting R = 1 +
1
n
, according to (1.2), we see that the C-norm of polynomials Pn(z) in GR
and G is equivalent, i.e., the norm ‖Pn‖C(G) increases with no more than a constant in GR.
In the case where L is rectifiable, a similar estimate of the (1.2)-type in the space Lp(L) was
investigated in [9] and obtained in the following form:
‖Pn‖Lp(LR) ≤ R
n+
1
p ‖Pn‖Lp(L) , p > 0. (1.3)
To give an inequality similar to (1.3) for the case of theAp-norm, we first present the following
definitions and notation:
Definition 1.1 [10, p. 97, 12]. A Jordan arc (curve) L is called K-quasiconformal (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).
We denote by F (L) 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
KL := inf {K(f) : f ∈ F (L)} ,
where K(f) is the maximal dilatation of a mapping f of this kind, L is a quasiconformal curve if
KL <∞, and L is a K-quasiconformal curve if KL ≤ K.
We know that there exist quasiconformal curves that are not rectifiable [10, p. 104].
Let {zj}mj=1 be a fixed system of distinct points in the curve L located in the positive direction.
Consider a so-called generalized Jacobi weight function h(z) defined as follows:
h(z) :=
m∏
j=1
|z − zj |γj , z ∈ GR, (1.4)
where γj > −2 for all j = 1,m (i.e., j = 1, 2, . . . ,m).
The Bernstein – Walsh type estimation for the regions G with quasiconformal boundary and for
the weight function h(z) of (1.4) type in the space Ap(h,G), p > 0, was contained in [3]. In
particular, for h(z) ≡ 1,
ISSN 1027-3190. Укр. мат. журн., 2014, т. 66, № 5
ON THE BEHAVIOR OF THE ALGEBRAIC POLYNOMIAL IN UNBOUNDED REGIONS . . . 581
‖Pn‖
Ap(GR)
≤ c2 R
∗n+1
p ‖Pn‖
Ap(G)
, p > 0, (1.5)
where R∗ := 1 + c3(R − 1). Therefore, if we choose R = 1 +
c1
n
, then (1.5) can be shown like
that Ap-norm of polynomials Pn(z) in GR and G is equivalent. Note that, here and throughout this
paper we denote by c, c0, c1, c2, . . . positive constants (in general, different in different relations) that
depend on G in general and on parameters inessential for the argument; otherwise, such dependence
will be explicitly stated.
N. Stylianopoulos in [13] replaced the norm ‖Pn‖C(G) with norm ‖Pn‖A2(G) on the right-hand
side of (1.1) and so has found a new version of the Bernstein – Walsh lemma. Before we give the
corresponding result of N. Stylianopoulos, we will give the following:
Definition 1.2. A bounded Jordan region G is called a k-quasidisk, 0 ≤ k < 1, if any conformal
mapping ψ can be extended to a K-quasiconformal, K =
1 + k
1− k
, homeomorphism of the plane C on
the C. In this case, the curve L := ∂G is called a k-quasicircle. The region G (curve L) is called a
quasidisk (quasicircle), if it is k-quasidick (k-quasicircle) with some 0 ≤ k < 1.
Remark 1.1. It is well known that if we are not interested in the coefficients of quasiconformality
of the curve, then the definitions of “quasicircle” and “quasiconformal curve” (in the case of D = C)
are identical. In the case where we are also interested in the coefficients of quasiconformality of the
given curve, we consider that if the curve L is K-quasiconformal, then it is a k-quasicircle with
k =
K2 − 1
K2 + 1
.
Following Remark 1.1, for the sake of simplicity, we use both terms, depending on the situation.
Lemma A [13]. Assume that L is quasicircle and rectifiable. Then there exists a constant c =
= c(L) > 0 depending only on L such that
|Pn(z)| ≤ c
√
n
d(z, L)
‖Pn‖A2(G) |Φ(z)|n+1 , z ∈ Ω, (1.6)
where d(z, L) := inf {|ζ − z| : ζ ∈ L} , holds for every Pn ∈ ℘n.
In the present work, we study a problem similar to (1.6) for regions with piecewise Dini-smooth
boundary and for generalized Jacobi weight function h(z) defined as in (1.4) in Ap(h,G), p > 1.
Let us give corresponding definition and some notation that will be used in what follows.
Definition 1.3 [11, p. 48] (see also [7, p. 32]). A Jordan curve L is called Dini-smooth if it has
a parametrization z = z(s), 0 ≤ s ≤ |L| := mesL, such that z′(s) 6= 0, 0 ≤ s ≤ |L| and
|z′(s2)− z′(s1)| < g(s2 − s1), s1 < s2, where g is an increasing function for which
1∫
0
g(x)
x
dx <∞.
Definition 1.4. We say that a Jordan region G has a piecewise Dini-smooth boundary if L :=
:= ∂G consists of the union of finite Dini-smooth arcs Lj , j = 1,m, such that they have exterior
(with respect to G) angles λjπ, 0 < λj < 2 at the corner points {zj}, j = 1,m, where two arcs
meet.
ISSN 1027-3190. Укр. мат. журн., 2014, т. 66, № 5
582 F. G. ABDULLAYEV, P. ÖZKARTEPE
According to the “three-point” criterion [10, p. 100], every piecewise Dini-smooth curve (without
cusps) is quasiconformal.
For 0 < δj < δ0 :=
1
4
min
{
|zi − zj | : i, j = 1, 2, . . . ,m, i 6= j
}
, let Ω(zj , δj) := Ω ∩
∩
{
z : |z−zj | ≤ δj
}
; δ := min1≤j≤m δj , Ω(δ) :=
m⋃
j=1
Ω(zj , δ), Ω̂ := Ω \ Ω(δ); ∆j := Φ(Ω(zj , δ)),
∆(δ) :=
m⋃
j=1
Φ(Ω(zj , δ)), ∆̂(δ) := ∆\∆(δ). Let wj := Φ(zj) and, for ϕj := argwj , j = 1,m,
we set ∆′j :=
{
t = Reiθ : R > 1,
ϕj−1 + ϕj
2
≤ θ <
ϕj + ϕj+1
2
}
, where ϕ0 ≡ ϕm, ϕ1 ≡ ϕm+1;
Ωj := Ψ(∆′j), L
j
R1
:= LR1 ∩ Ωj . Clearly, Ω =
m⋃
j=1
Ωj .
Let the points {zj}mj=1 be on the curve L in the positive direction. For k ≤ m, we de-
fine λ∗k := max
{
λj : j = 1, k
}
, λk∗ := min
{
λj : j = 1, k
}
, λ∗ := λ∗m, λ∗ := λm∗, λ̃k :=
:=
{
λk∗, if p ≥ 2,
λ∗k, if p < 2,
and λ̃ :=
{
λ∗, if p ≥ 2,
λ∗, if p < 2.
For any j = 1,m, let us µj :=
1
λj
+ (p − 2);
ηj :=
1
λj
− (2 − p), ωj :=
p− 1
λj
, γ∗k := max
{
γj , j = 1, k
}
, γ∗ := γ∗m. Γ :=
{
γj , j = 1,m
}
,
Γj,k :=
{
γj ∈ Γ: γj ≤ µk, k, j = 1,m
}
, Γ̃j,k := Γ\Γj,k. Let wj := Φ(zj).
We can now state our new results.
Theorem 1.1. Let p > 1. Assume that a Jordan region G has a piecewise Dini-smooth boundary
L := ∂G and h(z) is defined as in (1.4). Then, for any Pn ∈ ℘n and R1 = 1 +
1
n
|Pn(z)| ≤ c1
Dn,1
d(z, LR1)
‖Pn‖Ap(h,G) |Φ(z)|n+1 , z ∈ ΩR1 , (1.7)
where c1 = c1(G, p) > 0 and
Dn,1 =
n
1
p ,
if p ≥ 2, 0 < λj < 2, −2 < γj <
1
λj
+ (p− 2),
or p < 2, 1 ≤ λj < 2, −2 < γj <
1
λj
− (2− p),
or p < 2, 0 < λj < 1, −2 < γj <
p− 1
λj
for all j = 1,m;
m∑
j=1
n
γjλj
p +
(
2
p−1
)
λj ,
if p ≥ 2, 0 < λj < 2, γj ≥
1
λj
+ (p− 2),
or p < 2, 1 ≤ λj < 2, γj ≥
1
λj
− (2− p)
for all j = 1,m;
m∑
j=1
n
γjλj
p +
(
2
p−1
)
,
if p < 2, 0 < λj < 1, γj ≥
p− 1
λj
for all j = 1,m.
ISSN 1027-3190. Укр. мат. журн., 2014, т. 66, № 5
ON THE BEHAVIOR OF THE ALGEBRAIC POLYNOMIAL IN UNBOUNDED REGIONS . . . 583
Theorem 1.1 is local, i.e., each term in the sum on the right-hand side shows the growth of
|Pn(z)| , depending on the behavior of the weight function h(z) and the outside corner λj in the
neighborhood of a single point zj for any j = 1,m.
Comparing the terms in the sum for each point {zj}, j = 1,m, and using the above notation, we
can obtain following result of the global character.
Theorem 1.2. Let p > 1. Assume that a Jordan region G has a piecewise Dini-smooth boundary
L := ∂G and h(z) is defined as in (1.4). Then, for any Pn ∈ ℘n and R1 = 1 +
1
n
|Pn(z)| ≤ c2
Dn,2
d(z, LR1)
‖Pn‖Ap(h,G) |Φ(z)|n+1 , z ∈ ΩR1 , (1.8)
where c2 = c2(G, p,m) > 0 and
Dn,2 =
n
1
p ,
if p ≥ 2, 0 < λj < 2, −2 < γj < µ1,
or p < 2, 1 ≤ λj < 2, −2 < γj < η1,
or p < 2, 0 < λj < 1, −2 < γj < ω1
for all j = 1,m;
n
γ∗λ∗
p +
(
2
p−1
)
λ̃
,
if p ≥ 2, 0 < λj < 2, γj ≥ µm,
or p < 2, 1 ≤ λj < 2, γj ≥ ηm
for all j = 1,m;
n
γ∗kλ
∗
k
p +
(
2
p−1
)
λ̃k ,
if p ≥ 2, 0 < λj < 2, µk ≤ γj < µk+1,
or p < 2, 1 ≤ λj < 2, ηk ≤ γj < ηk+1
for all k = 1,m− 1 and j = 1,m;
n
γ∗λ∗
p +
(
2
p−1
)
,
if p < 2, 0 < λj < 1, γj ≥ ωm
for all j = 1, 2, . . . ,m;
n
γ∗kλ
∗
k
p +
(
2
p−1
)
,
if p < 2, 0 < λj < 1, ωk ≤ γj < ωk+1
for all k = 1, 2, . . . ,m− 1 and j = 1, 2, . . . ,m.
In particular, in the case of one singular point (m = 1) on the boundary curve L, we assume, for
simplicity, that λ1 =: λ and obtain the following corollary.
Corollary 1.1. Let p > 1, m = 1. Assume that a Jordan region G has a piecewise Dini-smooth
boundary L := ∂G and h(z) is defined as in (1.4) for m = 1. Then, for any Pn ∈ ℘n and R1 = 1+
1
n
we have
|Pn(z)| ≤ c3
Dn,3
d(z, LR1)
‖Pn‖Ap(h,G) |Φ(z)|n+1 , z ∈ ΩR1 , (1.9)
where c3 = c3(G, p) > 0 and
ISSN 1027-3190. Укр. мат. журн., 2014, т. 66, № 5
584 F. G. ABDULLAYEV, P. ÖZKARTEPE
Dn,3 =
n
1
p ,
if p ≥ 2, 0 < λ < 2, −2 < γ <
1
λ
+ (p− 2),
or p < 2, 1 ≤ λ < 2, −2 < γ <
1
λ
− (2− p),
or p < 2, 0 < λ < 1, −2 < γ <
p− 1
λ
;
n
γλ
p +
(
2
p−1
)
λ
,
if p ≥ 2, 0 < λ < 2, γ ≥ 1
λ
+ (p− 2),
or p < 2, 1 ≤ λ < 2, γ ≥ 1
λ
− (2− p);
n
γλ
p +
(
2
p−1
)
, if p < 2, 0 < λ < 1, γ ≥ p− 1
λ
.
Corollary 1.2. Let p = 2, m = 1. Assume that a Jordan region G has a piecewise Dini-smooth
boundary L := ∂G and h(z) is defined as in (1.4). Then, for any Pn ∈ ℘n we have
|Pn(z)| ≤ c4
Dn,4
d(z, LR1)
‖Pn‖A2(h,G) |Φ(z)|n+1 , z ∈ ΩR1 , (1.10)
where c4 = c4(G) > 0 and
Dn,4 <
n
1
2 , −2 < γ <
1
λ
, 0 < λ < 2,
n
γλ
2 , γ ≥ 1
λ
, 0 < λ < 2.
The sharpness of (1.7) – (1.10) can be seen from the following remark.
Remark 1.2. For any n ∈ N there exists P ∗n ∈ ℘n and G∗ ⊂ C such that
|P ∗n(z)| ≥ c5
√
n
d(z, L)
‖P ∗n‖A2(G∗) |Φ(z)|n+1 , z ∈ F b Ω∗ := CG
∗
, (1.11)
where c5 = c5(G∗) > 0.
2. Some auxiliary results. Let G ⊂ C be a finite region bounded by Jordan curve L. The
interior level curve can be defined for t > 0 as Lt :=
{
z : |ϕ(z)| = t, if t < 1
}
, L1 ≡ L, and let
Gt := intLt, Ωt := extLt.
Throughout this paper we also denote by ε, ε1, ε2, . . . sufficiently small positive constants (in
general, different in different relations) that depend on G in general and on parameters inessential for
the argument. For the a > 0 and b > 0, we use the expression “a � b” (order inequality), if a ≤ cb
and the expression “a � b” means that c1a ≤ b ≤ c2a for some constants c, c1, c2 (independent of a
andb) respectively.
Let L is a K-quasiconformal curve. Then [5] there exists a quasiconformal reflection y(·) across
L such that y(G) = Ω, y(Ω) = G and y(·) fixes the points of L. The quasiconformal reflection y(·)
can be chosen such that it satisfies the following conditions [5, 6, p. 26]:
|y(ζ)− z| � |ζ − z| , z ∈ L, ε < |ζ| < 1
ε
,∣∣yζ∣∣ � ∣∣yζ∣∣ � 1, ε < |ζ| < 1
ε
, (2.1)∣∣yζ∣∣ � |y(ζ)|2 , |ζ| < ε,
∣∣yζ∣∣ � |ζ|−2 , |ζ| > 1
ε
,
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ON THE BEHAVIOR OF THE ALGEBRAIC POLYNOMIAL IN UNBOUNDED REGIONS . . . 585
and for the Jacobian Jy = |yz|2 − |yz|2 of y(·) the relation
∣∣yζ∣∣2 ≤ 1
1− κ2
Jy is hold, where
κ =
K2 − 1
K2 + 1
.
Lemma 2.1 [1]. Let L be a K-quasiconformal curve, z1 ∈ L, z2, z3 ∈ Ω ∩ {z : |z − z1| �
� d(z1, Lr0)}; wj = Φ(zj), j = 1, 2, 3. Then
(a) the statements |z1 − z2| � |z1 − z3| and |w1 − w2| � |w1 − w3| are equivalent; so are
|z1 − z2| � |z1 − z3| and |w1 − w2| � |w1 − w3| ;
(b) if |z1 − z2| � |z1 − z3|, then∣∣∣∣w1 − w3
w1 − w2
∣∣∣∣ε � ∣∣∣∣z1 − z3
z1 − z2
∣∣∣∣ � ∣∣∣∣w1 − w3
w1 − w2
∣∣∣∣c ,
where ε < 1, c > 1, 0 < r0 < 1 are constants, depending on G.
The following lemma is a consequence of the results given in [7, p. 32 – 36; 11, p. 48].
Lemma 2.2. Assume that a Jordan region G has a piecewise Dini-smooth boundary L := ∂G.
Then
(i) for any w ∈ ∆j , and 0 < λj < 2, |Ψ(w)−Ψ(wj)| � |w − wj |λj , |Ψ′(w)| � |w − wj |λj−1 ;
(ii) for any w ∈ ∆\∆j , |Ψ(w)−Ψ(wj)| � |w − wj | , |Ψ′(w)| � 1.
Let {zj}mj=1 be a fixed system of distinct points on curve L located in the positive direction and
the weight function h(z) is defined as (1.4).
Lemma 2.3 [3]. Let L be a K-quasiconformal curve and h(z) be defined in (1.4). Then, for
arbitrary Pn(z) ∈ ℘n, any R > 1, and n = 1, 2, . . .
‖Pn‖Ap(h,GR) � R̃
n+
1
p ‖Pn‖Ap(h,G) , p > 0, (2.2)
where R̃ = 1 + c(R− 1) and c is independent from n and R.
Lemma 2.4. Let L be a K-quasiconformal curve, R = 1 +
c
n
. Then, for any fixed ε ∈ (0, 1)
there exists a level curve L1+ε(R−1) such that the following holds for any polynomial Pn(z) ∈ ℘n,
n ∈ N :
‖Pn‖Lp
(
h
Φ′ , L1+ε(R−1)
) � n1
p ‖Pn‖Ap(h, G) , p > 0. (2.3)
Proof. Without loss of generality, we can take ε =
1
2
. Then R1 := 1 +
R− 1
2
. We have
An :=
∫
LR1
h(z) |Pn(z)|p |dz|
|Φ′(z)|
=
=
∫
|w|=R1
m∏
j=1
|Ψ(w)−Ψ(wj)|γj
∣∣∣∣Pn(Ψ(w))(Ψ′(w))
2
p
∣∣∣∣p |dw| = ∫
|w|=R1
|fn,p(w)|p |dw| , (2.4)
where
fn,p(w) :=
m∏
j=1
(Ψ(w)−Ψ(wj))
γj
p Pn (Ψ(w))
(
Ψ′(w)
)2
p , w ∈ ∆.
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586 F. G. ABDULLAYEV, P. ÖZKARTEPE
We now split the circle |t| = R1 into n equal parts δn with mes δn =
2πR1
n
. Applying the mean-value
theorem to the integral An, we get
An =
∫
|t|=R1
|fn,p (w)|p |dw| =
n∑
k=1
∫
δk
|fn,p (w)|p |dw| =
=
n∑
k=1
∣∣fn,p (t′k)∣∣p mes δk, t
′
k ∈ δk.
On the other hand, by applying the mean-value estimation,∣∣fn,p (t′k)∣∣p ≤ 1
π
(∣∣t′k∣∣− 1
)2 ∫∫
|ξ−t′k|<|t′k|−1
|fn,p (ξ)|p dσξ,
we obtain
An �
n∑
k=1
mes δk
π
(∣∣t′k∣∣− 1
)2 ∫∫
|w−t′k|<|t′k|−1
|fn,p (w)|p dσw, t′k ∈ δk.
Taking into account that discs with origin at the points t′k at most two may be crossing, we have
An �
mes δ1(∣∣t′1∣∣− 1
)2 ∫∫
1<|w|<R
|fn,p (w)|p dσw � n ·
∫∫
1<|w|<R
|fn,p (w)|p dσw =
= n ·
∫∫
1<|w|<R
m∏
j=1
|Ψ(w)−Ψ(wj)|γj |Pn (Ψ(w))|p
∣∣Ψ′(w)
∣∣2 dσw ≤
≤ n ·
∫∫
1<|w|<R
m∏
j=1
|Ψ(w)−Ψ(wj)|γj |Pn (Ψ(w))|p
∣∣Ψ′(w)
∣∣2 dσw �
� n ·
∫∫
GR\G
h(z) |Pn (z)|p dσz.
According to (2.3), for An, we get
An � n ·
∫∫
GR\G
h(z) |Pn (z)|p dσz � n · ‖Pn‖pAp(h,G) . (2.5)
Combining (2.4), (2.5), we prove estimate (2.3).
Lemma 2.4 is proved.
3. Proofs. Proof of Theorem 1.1. Let for z ∈ Ω:
Tn(z) :=
Pn(z)
Φn+1(z)
. (3.1)
For any R > 1 and R1 := 1 +
R− 1
2
, the Cauchy integral representation for the region ΩR1 gives
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ON THE BEHAVIOR OF THE ALGEBRAIC POLYNOMIAL IN UNBOUNDED REGIONS . . . 587
Tn(z) = − 1
2πi
∫
LR1
Tn (ζ)
dζ
ζ − z
, z ∈ ΩR1 .
Since |Φ(ζ)| > 1, for ζ ∈ LR1 , then we have
|Pn(z)| = |Φ(z)|n+1
2π
∫
LR1
|Pn (ζ)| |dζ|
|ζ − z|
≤ |Φ(z)|n+1
2πd(z, LR1)
∫
LR1
|Pn (ζ)| |dζ| . (3.2)
Let’s
An :=
∫
LR1
|Pn (ζ)| |dζ| =
m∑
i=1
∫
Li
R1
|Pn (ζ)| |dζ| =
=
m∑
i=1
∫
F i
R1
|Pn (Ψ(τ))|
∣∣Ψ′(τ)
∣∣ |dτ | , (3.3)
where F iR1
:= Φ(LiR1
) = ∆′i ∩ {τ : |τ | = R1} , i = 1,m. Replacing the variable τ = Φ(ζ) and
multiplying the numerator and denominator of the by multiplier
∏m
j=1
|Ψ(τ)−Ψ(wj)|
γj
p
∣∣Ψ′(τ)
∣∣2p ,
after then, according to the H’́older inequality, we obtain
An =
m∑
i=1
∫
F i
R1
∏m
j=1
|Ψ(τ)−Ψ(wj)|
γj
p
∣∣∣∣Pn (Ψ(τ))
(
Ψ′(τ)
)2
p
∣∣∣∣ ∣∣Ψ′(τ)
∣∣1−2
p
∏m
j=1
|Ψ(τ)−Ψ(wj)|
γj
p
|dτ | ≤
≤
m∑
i=1
∫
F i
R1
m∏
j=1
|Ψ(τ)−Ψ(wj)|γj |Pn (Ψ(τ))|p
∣∣Ψ′(τ)
∣∣2 |dτ |
1
p
×
×
∫
F i
R1
|Ψ′(τ)|1−
2
p∏m
j=1
|Ψ(τ)−Ψ(wj)|
γj
p
q
|dτ |
1
q
≤
≤
m∑
i=1
Ain, (3.4)
where
Ain :=
∫
F i
R1
|fn,p(τ)|p |dτ |
1
p
∫
F i
R1
|Ψ′(τ)|2−q∏m
j=1
|Ψ(τ)−Ψ(wj)|γj(q−1)
|dτ |
1
q
=: J in,1 · J in,2,
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588 F. G. ABDULLAYEV, P. ÖZKARTEPE
fn,p(τ) :=
m∏
j=1
(Ψ(τ)−Ψ(wj))
γj
p Pn (Ψ(τ))
(
Ψ′(τ)
)2
p , |τ | = R1.
Applying to Lemma 2.4, we get
J in,1 � n
1
p ‖Pn‖Ap(h, G) , i = 1,m. (3.5)
For the estimation the integral J in,2
(
J in,2
)q
:=
∫
F i
R1
|Ψ′(τ)|2−q∏m
j=1
|Ψ(τ)−Ψ(wj)|γj(q−1)
|dτ | �
∫
F i
R1
|Ψ′(τ)|2−q
|Ψ(τ)−Ψ(wi)|γi(q−1)
|dτ | , (3.6)
since the points wj := Φ(zj) are distinct.
For simplicity, we may take i = 1, J in,1 =: J1; J1
n,2 =: J2. Let us set: w1 := Φ(z1),
E11
R1
:=
{
τ : τ ∈ F 1
R1
, |τ − w1| < c1(R1 − 1)
}
,
E12
R1
:=
{
τ : τ ∈ F 1
R1
, c1(R1 − 1) ≤ |τ − w1| < c2
}
,
E13
R1
:=
{
τ : τ ∈ F 1
R1
, |τ − w1| ≥ c2
}
, j = 1, 2, 3,
F 1
R1
=
3⋃
k=1
E1k
R1
.
Taking into consideration these designations, (3.6) can be written as
J2 = J2(E11
R1
) + J2(E12
R1
) + J2(E13
R1
) =: J1
2 + J2
2 + J3
2 (3.7)
and, consequently,
A1
n =: J1 · (J1
2 + J2
2 + J3
2 ) =: A1
n,1 +A1
n,2 +A1
n,3, (3.8)
where
A1
n,k := n
1
p ‖Pn‖Ap(h,G)
∫
E1k
R1
|Ψ′(τ)|2−q
|Ψ(τ)−Ψ(w1)|γ1(q−1)
|dτ | , k = 1, 2, 3. (3.9)
Given the possible values q (q > 2 and q < 2), λ1 (0 < λ1 < 1 and 1 < λ1 < 2), and γ1
(−2 < γ1 < 0 and γ1 ≥ 0), we will consider separately the cases.
Case 1. Let 1 < q < 2 (p > 2). Then
(
J1
2
)q � ∫
E11
R1
|Ψ′(τ)|2−q
|Ψ(τ)−Ψ(w1)|γ1(q−1)
|dτ | .
1.1. Let 1 ≤ λ1 < 2.
1.1.1. If γ1 ≥ 0, applying Lemma 2.2 to (3.9), we get
(
J1
2
)q � ∫
E11
R1
|τ − w1|(λ1−1)(2−q)
|τ − w1|γ1λ1(q−1)
|dτ | �
(
1
n
)(λ1−1)(2−q) ∫
E11
R1
|dτ |
(|τ | − 1)γ1λ1(q−1)
�
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ON THE BEHAVIOR OF THE ALGEBRAIC POLYNOMIAL IN UNBOUNDED REGIONS . . . 589
� nγ1λ1(q−1)−(λ1−1)(2−q)−1, if γ1λ1(q − 1) > 1,
J1
2 � n
γ1λ1(q−1)−(λ1−1)(2−q)−1
q , if γ1λ1(q − 1) > 1,
(
J2
2
)q � ∫
E12
R1
|τ − w1|(λ1−1)(2−q)
|τ − w1|γ1λ1(q−1)
|dτ | �
(
1
n
)(λ1−1)(2−q) ∫
E12
R1
|dτ |
(|τ | − 1)γ1λ1(q−1)
�
� nγ1λ1(q−1)−(λ1−1)(2−q)−1, if γ1λ1(q − 1) > 1,
J2
2 � n
γ1λ1(q−1)−(λ1−1)(2−q)−1
q , if γ1λ1(q − 1) > 1.
In this case, from (3.7) and (3.8), we obtain
A1
n,1 � n
1
p+
γ1λ1(q−1)−(λ1−1)(2−q)−1
q ‖Pn‖Ap(h,G) =
= n
(2+γ1
p −1
)
λ1 ‖Pn‖Ap(h,G) , γ1λ1(q − 1) > 1, if γ1λ1(q − 1) > 1, (3.10)
A1
n,2 � n
(2+γ1
p −1
)
λ1 ‖Pn‖Ap(h,G) , if γ1λ1(q − 1) > 1. (3.11)
1.1.2. If γ1 < 0, analogously we have
(
J1
2
)q � ∫
E11
R1
|τ − w1|(λ1−1)(2−q)
|τ − w1|γ1λ1(q−1)
|dτ | �
∫
E11
R1
|τ − w1|(λ1−1)(2−q)+(−γ1)λ1(q−1) |dτ | �
�
(
1
n
)(λ1−1)(2−q)+(−γ1)λ1(q−1)
·mesE11
R1
,
J1
2 � n
γ1λ1(q−1)−(λ1−1)(2−q)−1
q ,
(
J2
2
)q � ∫
E12
R1
|τ − w1|(λ1−1)(2−q)
|τ − w1|γ1λ1(q−1)
|dτ | �
∫
E12
R1
|τ − w1|(λ1−1)(2−q)+(−γ1)λ1(q−1) |dτ | �
�
∫
E12
R1
|dτ | � 1.
Also
A1
n,1 � n
1
p+
γ1λ1(q−1)−(λ1−1)(2−q)−1
q ‖Pn‖Ap(h,G) = n
(2+γ1
p −1
)
λ1 ‖Pn‖Ap(h,G) ,
A1
n,2 � n
1
p ‖Pn‖Ap(h,G) .
(3.12)
1.2. Let 0 < λ1 < 1.
1.2.1. If γ1 ≥ 0, applying Lemma 2.2 to (3.9), we get
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590 F. G. ABDULLAYEV, P. ÖZKARTEPE
(
J1
2
)q � ∫
E11
R1
|τ − w1|(λ1−1)(2−q)
|τ − w1|γ1λ1(q−1)
|dτ | �
∫
E11
R1
|dτ |
(|τ | − 1)γ1λ1(q−1)+(1−λ1)(2−q) �
� nγ1λ1(q−1)+(1−λ1)(2−q)−1, if γ1λ1(q − 1) + (1− λ1)(2− q) > 1,
J1
2 � n
γ1λ1(q−1)+(1−λ1)(2−q)−1
q , if γ1λ1(q − 1) + (1− λ1)(2− q) > 1,
(
J2
2
)q � ∫
E12
R1
|τ − w1|(λ1−1)(2−q)
|τ − w1|γ1λ1(q−1)
|dτ | �
∫
E12
R1
|dτ |
(|τ | − 1)γ1λ1(q−1)+(1−λ1)(2−q) �
� nγ1λ1(q−1)+(1−λ1)(2−q)−1, if γ1λ1(q − 1) + (1− λ1)(2− q) > 1,
J2
2 � n
γ1λ1(q−1)+(1−λ1)(2−q)−1
q , if γ1λ1(q − 1) + (1− λ1)(2− q) > 1.
In this case, from (3.8) for A1
n,1 and A1
n,2, we obtain
A1
n,1 � n
1
p+
γ1λ1(q−1)+(1−λ1)(2−q)−1
q ‖Pn‖Ap(h,G) =
= n
γ1λ1
p −
(
1−2
p
)
λ1 ‖Pn‖Ap(h,G) , if γ1λ1(q − 1) + (1− λ1)(2− q) > 1, (3.13)
A1
n,2 � n
1
p+
γ1λ1(q−1)+(1−λ1)(2−q)−1
q ‖Pn‖Ap(h,G) =
= n
γ1λ1
p −
(
1−2
p
)
λ1 ‖Pn‖Ap(h,G) , if γ1λ1(q − 1) + (1− λ1)(2− q) > 1. (3.14)
1.2.2. If γ1 < 0, analogously we have
(
J1
2
)q � ∫
E11
R1
|τ − w1|(−γ1)λ1(q−1)
|τ − w1|(1−λ1)(2−q) |dτ | �
�
(
1
n
)(−γ1)λ1(q−1) ∫
E11
R1
|dτ |
|τ − w1|(1−λ1)(2−q) �
(
1
n
)(−γ1)λ1(q−1)
� 1,
(
J2
2
)q � ∫
E12
R1
|τ − w1|(−γ1)λ1(q−1)
|τ − w1|(1−λ1)(2−q) |dτ | �
∫
E12
R1
|dτ |
|τ − w1|(1−λ1)(2−q) � 1.
Also
A1
n,1 � n
1
p ‖Pn‖Ap(h,G) ,
A1
n,2 � n
1
p ‖Pn‖Ap(h,G) .
(3.15)
Case 2. Let q > 2(p < 2). Then, 2− q < 0 and, so
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ON THE BEHAVIOR OF THE ALGEBRAIC POLYNOMIAL IN UNBOUNDED REGIONS . . . 591
(
J1
2
)q � ∫
E11
R1
|dτ |
|Ψ′(τ)|q−2 |ζ − z1|γ1(q−1)
. (3.16)
2.1. Let 1 ≤ λ1 < 2.
2.1.1. If γ1 ≥ 0, applying Lemma 2.2 to (3.16), we obtain
(
J1
2
)q � ∫
E11
R1
|dτ |
|τ − w1|(λ1−1)(q−2) |τ − w1|γ1λ1(q−1)
�
∫
E11
R1
|dτ |
(|τ | − 1)γ1λ1(q−1)+(λ1−1)(q−2)
�
� nγ1λ1(q−1)+(λ1−1)(q−2)−1, if γ1λ1(q − 1) + (λ1 − 1)(q − 2) > 1,
J1
2 � n
γ1λ1(q−1)+(λ1−1)(q−2)−1
q , if γ1λ1(q − 1) + (λ1 − 1)(q − 2) > 1,(
J2
2
)q � ∫
E12
R1
|dτ |
|τ − w1|(λ1−1)(q−2) |τ − w1|γ1λ1(q−1)
�
∫
E12
R1
|dτ |
(|τ | − 1)γ1λ1(q−1)+(λ1−1)(q−2)
�
� nγ1λ1(q−1)+(λ1−1)(q−2)−1, if γ1λ1(q − 1) + (λ1 − 1)(q − 2) > 1,
J2
2 � n
γ1λ1(q−1)+(λ1−1)(q−2)−1
q , if γ1λ1(q − 1) + (λ1 − 1)(q − 2) > 1.
In this case, from (3.8), we have
A1
n,1 � n
1
p+
γ1λ1(q−1)+(λ1−1)(q−2)−1
q ‖Pn‖Ap(h,G) =
= n
γ1λ1
p +
(
2
p−1
)
λ1 ‖Pn‖Ap(h,G) , if γ1λ1(q − 1) + (λ1 − 1)(q − 2) > 1, (3.17)
A1
n,2 � n
1
p+
γ1λ1(q−1)+(λ1−1)(q−2)−1
q ‖Pn‖Ap(h,G) =
= n
γ1λ1
p +
(
2
p−1
)
λ1 ‖Pn‖Ap(h,G) , if γ1λ1(q − 1) + (λ1 − 1)(q − 2) > 1. (3.18)
2.1.2. If γ1 < 0, analogously we have
(
J1
2
)q � ∫
E11
R1
|τ − w1|(−γ1)λ1(q−1)
|τ − w1|(λ1−1)(q−2)
|dτ | �
(
1
n
)(−γ1)λ1(q−1) ∫
E11
R1
|dτ |
|τ − w1|(λ1−1)(q−2)
�
� n(λ1−1)(q−2)+γ1λ1(q−1)−1, if (λ1 − 1)(q − 2) > 1,
J1
2 � n
(λ1−1)(q−2)+γ1λ1(q−1)−1
q , if (λ1 − 1)(q − 2) > 1,
(
J2
2
)q � ∫
E12
R1
|τ − w1|(−γ1)λ1(q−1)
|τ − w1|(λ1−1)(q−2)
|dτ | �
∫
E12
R1
|dτ |
|τ − w1|(λ1−1)(q−2)
�
� n(λ1−1)(q−2)−1, if (λ1 − 1)(q − 2) > 1,
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592 F. G. ABDULLAYEV, P. ÖZKARTEPE
J2
2 � n
(λ1−1)(q−2)−1
q , if (λ1 − 1)(q − 2) > 1.
So,
A1
n,1 � n
1
p+
(λ1−1)(q−2)+γ1λ1(q−1)−1
q ‖Pn‖Ap(h,G) =
= n
(
2
p−1
)
λ1+
γ1λ1
p ‖Pn‖Ap(h,G) , if (λ1 − 1)(q − 2) > 1, (3.19)
A1
n,2 � n
1
p+
(λ1−1)(q−2)−1
q ‖Pn‖Ap(h,G) = n
(
2
p−1
)
λ1 ‖Pn‖Ap(h,G) , if (λ1 − 1)(q − 2) > 1.
(3.20)
2.2. Let 0 < λ1 < 1.
2.2.1. If γ1 ≥ 0, applying Lemma 2.2 to (3.16) we obtain
(
J1
2
)q � ∫
E11
R1
|τ − w1|(1−λ1)(q−2) |dτ |
|τ − w1|γ1λ1(q−1)
�
(
1
n
)(1−λ1)(q−2) ∫
E11
R1
|dτ |
|τ − w1|γ1λ1(q−1)
�
� n−(1−λ1)(q−2)+γ1λ1(q−1)−1, if γ1λ1(q − 1) > 1,
J1
2 � n
−(1−λ1)(q−2)+γ1λ1(q−1)−1
q , if γ1λ1(q − 1) > 1,
(
J2
2
)q � ∫
E12
R1
|τ − w1|(1−λ1)(q−2)
|τ − w1|γ1λ1(q−1)
|dτ | �
∫
E12
R1
|dτ |
|τ − w1|γ1λ1(q−1)
�
� nγ1λ1(q−1)−1, if γ1λ1(q − 1) > 1,
J2
2 � n
γ1λ1(q−1)−1
q , if γ1λ1(q − 1) > 1.
In this case, from (3.8), we have
A1
n,1 � n
1
p+
−(1−λ1)(q−2)+γ1λ1(q−1)−1
q ‖Pn‖Ap(h,G) =
= n
(2+γ1
p −1
)
λ1 ‖Pn‖Ap(h,G) , if γ1λ1(q − 1) > 1, (3.21)
A1
n,2 � n
1
p+
γ1λ1(q−1)−1
q ‖Pn‖Ap(h,G) = n
(
2
p−1
)
+
γ1λ1
p ‖Pn‖Ap(h,G) , if γ1λ1(q − 1) > 1.
(3.22)
2.2.2. If γ1 < 0, analogously we have
(
J1
2
)q � ∫
E11
R1
|τ − w1|(1−λ1)(q−2) |dτ |
|τ − w1|γ1λ1(q−1)
�
∫
E11
R1
|τ − w1|(1−λ1)(q−2)+(−γ1)λ1(q−1) |dτ | �
�
(
1
n
)(1−λ1)(q−2)+(−γ1)λ1(q−1)
·mesE1
R1
,
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ON THE BEHAVIOR OF THE ALGEBRAIC POLYNOMIAL IN UNBOUNDED REGIONS . . . 593
J2
2 � n
−(1−λ1)(q−2)+γ1λ1(q−1)−1
q � 1,
(
J2
2
)q � ∫
E12
R1
|τ − w1|(1−λ1)(q−2) |dτ |
|τ − w1|γ1λ1(q−1)
�
∫
E12
R1
|τ − w1|(1−λ1)(q−2)+(−γ1)λ1(q−1) |dτ | �
∫
E12
R1
|dτ | � 1,
(3.23)
A1
n,1 � n
1
p ‖Pn‖Ap(h,G) ,
A1
n,2 � n
1
p ‖Pn‖Ap(h,G) .
To estimate A1
n,3, note that for each ζ ∈ E3
R1
|ζ − z1| � 1, and so
J3
2 � 1,
A1
n,3 � n
1
p ‖Pn‖Ap(h,G) .
(3.24)
Therefore, combining the (3.8) – (3.24), for An, we get
A1
n =
3∑
k=1
A1
n,k � ‖Pn‖Ap(h,G)×
×
n
(2+γ1
p −1
)
λ1 + n
(2+γ1
p −1
)
λ1 + n
1
p , p > 2, λ1 ≥ 1, γ1λ1(q − 1) > 1,
n
γ1λ1
p −
(
1−2
p
)
λ1 + n
γ1λ1
p −
(
1−2
p
)
λ1 + n
1
p , p > 2, λ1 < 1, γ1λ1(q − 1) > 1,
n
γ1λ1
p +
(
2
p−1
)
λ1 + n
γ1λ1
p +
(
2
p−1
)
λ1 + n
1
p , p < 2, λ1 ≥ 1, γ1λ1(q − 1)+
+(λ1 − 1)(q − 2) > 1,
n
(2+γ1
p −1
)
λ1 + n
(
2
p−1
)
+
γ1λ1
p + n
1
p , p < 2, λ1 < 1, γ1λ1(q − 1) > 1,
if γ1 ≥ 0, and
A1
n =
3∑
k=1
A1
n,k � ‖Pn‖Ap(h,G)×
×
n
(2+γ1
p −1
)
λ1 + n
1
p + n
1
p , p > 2, λ1 ≥ 1, γ1 < 0,
n
1
p + n
1
p + n
1
p , p > 2, λ1 < 1, γ1 < 0,
n
(
2
p−1
)
λ1+
γ1λ1
p + n
(
2
p−1
)
λ1 + n
1
p , p < 2, λ1 ≥ 1, (λ1 − 1)(q − 2) > 1,
n
(
2
p−1
)
+
γ1λ1
p + n
1
p + n
1
p , p < 2, λ1 < 1,
if γ1 < 0.
Hence,
ISSN 1027-3190. Укр. мат. журн., 2014, т. 66, № 5
594 F. G. ABDULLAYEV, P. ÖZKARTEPE
A1
n � ‖Pn‖Ap(h,G)
n
1
p , p > 2, λ1 ≥ 1, 0 ≤ γ1λ1 < 1 + λ1(p− 2),
n
γ1λ1
p −
(
1−2
p
)
λ1 , p > 2, λ1 ≥ 1, γ1λ1 ≥ 1 + λ1(p− 2),
n
1
p , p > 2, λ1 < 1, 0 ≤ γ1λ1 < 1 + λ1(p− 2),
n
γ1λ1
p −
(
1−2
p
)
λ1 , p > 2, λ1 < 1, γ1λ1 ≥ 1 + λ1(p− 2),
n
1
p , p < 2, λ1 > 1, 0 ≤ γ1λ1 < 1− λ1(2− p),
n
γ1λ1
p +
(
2
p−1
)
λ1 , p < 2, λ1 > 1, γ1λ1 ≥ 1− λ1(2− p),
n
1
p , p < 2, λ1 < 1, 0 ≤ γ1λ1 < p− 1,
n
γ1λ1
p +
(
2
p−1
)
, p < 2, λ1 < 1, γ1λ1 ≥ p− 1,
(3.25)
if γ1 ≥ 0, and
An � ‖Pn‖Ap(h,G)
n
1
p , p > 2, λ1 ≥ 1, γ1 < 0,
n
1
p , p > 2, λ1 < 1, γ1 < 0,
n
1
p , p < 2, λ1 ≥ 1, γ1 < 0,
n
1
p , p < 2, λ1 < 1, γ1 < 0,
(3.26)
if γ1 < 0. Therefore, taking into account also the case p = 2, and summing over all j = 1,m, from
(3.8) and (3.9) we get
An ≤
m∑
j=1
Ajn � ‖Pn‖Ap(h,G)×
×
n
1
p , p ≥ 2, 0 < λj < 2, −2 < γj <
1
λj
+ (p− 2),
m∑
j=1
n
γjλj
p −
(
1−2
p
)
λj , p ≥ 2, 0 < λj < 2, γj ≥
1
λj
+ (p− 2),
n
1
p , p < 2, 1 ≤ λj < 2, −2 < γj <
1
λj
− (2− p),
m∑
j=1
n
γjλj
p +
(
2
p−1
)
λj , p < 2, 1 ≤ λj < 2, γj ≥
1
λj
− (2− p),
n
1
p , p < 2, 0 < λj < 1, −2 < γj <
p− 1
λj
,
m∑
j=1
n
γjλj
p +
(
2
p−1
)
, p < 2, 0 < λj < 1, γj ≥
p− 1
λj
.
Also
An � ‖Pn‖Ap(h,G)×
ISSN 1027-3190. Укр. мат. журн., 2014, т. 66, № 5
ON THE BEHAVIOR OF THE ALGEBRAIC POLYNOMIAL IN UNBOUNDED REGIONS . . . 595
×
n
1
p , p ≥ 2, 0 < λj < 2, −2 < γj <
1
λ1
+ (p− 2) for all j = 1,m;
n
γ∗λ∗
p −
(
1−2
p
)
λ∗
, p ≥ 2, 0 < λj < 2, γj ≥
1
λm
+ (p− 2) for all j = 1,m;
n
γ∗kλ
∗
k
p −
(
1−2
p
)
λk∗ , p ≥ 2, 0 < λj < 2, µk ≤ γj < µk+1 for all k = 1,m− 1
and j = 1,m;
n
1
p , p < 2, 1 ≤ λj < 2, −2 < γj <
1
λ1
− (2− p) for all j = 1,m;
n
γ∗λ∗
p +
(
2
p−1
)
λ∗
, p < 2, 1 ≤ λj < 2, γj ≥
1
λm
− (2− p) for all j = 1,m;
n
γ∗kλ
∗
k
p +
(
2
p−1
)
λ∗k , p < 2, 1 ≤ λj < 2, ηk ≤ γj < ηk+1 for all k = 1,m− 1
and j = 1,m;
n
1
p , p < 2, 0 < λj < 1, −2 < γj <
p− 1
λ1
for all j = 1,m;
n
γ∗λ∗
p +
(
2
p−1
)
, p < 2, 0 < λj < 1, γj ≥
p− 1
λm
for all j = 1,m;
n
γ∗kλ
∗
k
p +
(
2
p−1
)
, p < 2, 0 < λj < 1, ωk ≤ γj < ωk+1 for all k = 1,m− 1
and j = 1,m.
Combining the formulas (3.2), (3.3), (3.5), (3.8), (3.25) and (3.26) we complete the proof of
Theorem 1.1.
Proof of Corollary 1.2. Let p = 2, m = 1. First of all we note that, from (3.5), (3.8) and (3.9),
for γ1 = 0 we can obtain easily
An � n
1
2 ‖Pn‖A2(G) . (3.27)
1. Let γ1 > 0. Then
(
J1
2
)2 � ∫
E1
R1
|dτ |
|τ − w1|γ1λ1
� nγ1λ1−1, γ1λ1 > 1, 0 < λ1 < 2,
(
J2
2
)2 � ∫
E2
R1
|dτ |
|τ − w1|γ1λ1
� nγ1λ1−1, γ1λ1 > 1, 0 < λ1 < 2.
(3.28)
2. Let now −2 < γ1 < 0. Then
(
J1
2
)2 � ∫
E1
R1
|dτ |
|τ − w1|γ1λ1
� 1, 0 < λ1 < 2,
(
J2
2
)2 � ∫
E1
R1
|dτ |
|τ − w1|γ1λ1
� 1, 0 < λ1 < 2,
(3.29)
ISSN 1027-3190. Укр. мат. журн., 2014, т. 66, № 5
596 F. G. ABDULLAYEV, P. ÖZKARTEPE
J2
2 � 1, 0 < λ1 < 2. (3.30)
On the other hand, in case of all γ1 > −2 and 0 < λ1 < 2 we have
J3
2 � 1. (3.31)
Therefore, combining (3.2) – (3.5), (3.8), (3.27) – (3.31) , we get
An � ‖Pn‖A2(h,G)
{
n
γ1λ1
2 + n
1
2 , γ1λ1 > 1, �
� ‖Pn‖A2(h,G)
n
1
2 , −2 < γ1 <
1
λ1
, 0 < λ1 < 2,
n
γ1λ1
2 , γ1 ≥
1
λ1
, 0 < λ1 < 2.
Corollary 1.2 is proved.
Proof of Remark 1.2. Let the region G bounded by Dini-smooth curve L = ∂G. According to
the “three-point” criterion [10, p.100], the curve L is quasiconformal. Let {Kn(z)}, degKn = n,
denote of the system of Bergman polynomials for region G, i.e., system of polynomials {Kn(z)} ,
Kn(z) := αnz
n + αn−1z
n−1 + . . .+ α0, αn > 0 and satisfying the conditions∫∫
G
Kn(z)Km(z)dσz = δn,m,
where δn,m is the Kronecker symbol. According to [2], for arbitrary quasidiscs, we have
Kn(z) = αnρ
n+1Φn(z)Φ′(z)An(z), z ∈ F b Ω,
where √
n+ 1
π
≤ αnρn+1 ≤ c1
√
n+ 1
π
,
for some c1 = c1(G) > 1 and
c2 ≤ |An(z)| ≤ 1 +
c3√
|Φ(z)| − 1
,
for some ci = ci(G) > 0, i = 2, 3. Therefore, since ‖Kn‖A2(G) = 1, we have
|Kn(z)| ≥ c2
√
n+ 1
π
|Φ(z)|n |Φ(z)| − 1
d(z, L)
≥
≥ c3
√
n
d(z, L)
|Φ(z)|n+1
(
1− 1
|Φ(z)|
)
≥
≥ c4
√
n
d(z, L)
|Φ(z)|n+1 ‖Kn‖A2(G) .
Remark 1.2 is proved.
1. Abdullayev F. G., Andrievskii V. V. On the orthogonal polynomials in the domains with K-quasiconformal boundary
// Izv. Akad. Nauk Azerb. SSR. Ser. FTM. – 1983. – 1. – S. 3 – 7 (in Russian).
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ON THE BEHAVIOR OF THE ALGEBRAIC POLYNOMIAL IN UNBOUNDED REGIONS . . . 597
2. Abdullayev F. G. Dissertation (Ph. D.). – Donetsk, 1986. – 120 p.
3. Abdullayev F. G. On the some properties of the orthogonal polynomials over the region of the complex plane (Part III)
// Ukr. Math. J. – 2001. – 53, № 12. – P. 1934 – 1948.
4. Abdullayev F. G., Uğur D. On the orthogonal polynomials with weight having singularity on the boundary of regions
of the complex plane // Bull. Belg. Math. Soc. – 2009. – 16, № 2. – P. 235 – 250.
5. Ahlfors L. Lectures on quasiconformal mappings. – Princeton, NJ: Van Nostrand, 1966.
6. Andrievskii V. V., Belyi V. I., Dzyadyk V. K. Conformal invariants in constructive theory of functions of complex
plane. – Atlanta: World Federation Publ. Co., 1995.
7. Andrievskii V. V., Blatt H. P. Discrepancy of signed measures and polynomial approximation. – New York Inc.:
Springer, 2010.
8. Goluzin G. M. Func. of comp. var. Geom. Theory. – M.; L.: Gostekhizdat, 1952 (in Russian).
9. Hille E., Szegö G., Tamarkin J. D. On some generalization of a theorem of A. Markoff // Duke Math. J. – 1937. –
3. – P. 729 – 739.
10. Lehto O., Virtanen K. I. Quasiconformal mapping in the plane. – Berlin: Springer, 1973.
11. Pommerenke Ch. Boundary behavior of conformal maps. – Berlin: Springer, 1992.
12. Rickman S. Characterisation of quasiconformal arcs // Ann. Acad. Sci. Fenn. Ser. A. Math. – 1966. – 395. – 30 p.
13. Stylianopoulos N. Strong asymptotics for Bergman polynomials over domains with corners and applications // Const.
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14. Walsh J. L. Interpolation and approximation by rational functions in the complex domain. – Amer. Math. Soc., 1960.
Received 08.01.13
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