On the Bernstein - Walsh-type lemmas in regions of the complex plane
Let $G \subset C$ be a finite region bounded by a Jordan curve $L := \partial G,\quad \Omega := \text{ext} \; \overline{G}$ (respect to $\overline{C}$), $\Delta := \{z : |z| > 1\}; \quad w = \Phi(z)$ be the univalent conformal mapping of $\Omega$ ont $\Phi$ normalized by $\Phi(\infty) = \inf...
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| Мова: | Англійська |
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Institute of Mathematics, NAS of Ukraine
2011
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Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860508670792564736 |
|---|---|
| author | Abdullayev, F. G. Aral, N. D. Абдуллаєв, Ф. Г. Арал, Н. Д. |
| author_facet | Abdullayev, F. G. Aral, N. D. Абдуллаєв, Ф. Г. Арал, Н. Д. |
| author_sort | Abdullayev, F. G. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2020-03-18T19:34:39Z |
| description | Let $G \subset C$ be a finite region bounded by a Jordan curve $L := \partial G,\quad \Omega := \text{ext} \; \overline{G}$ (respect to $\overline{C}$), $\Delta := \{z : |z| > 1\}; \quad w = \Phi(z)$ be the univalent conformal mapping of $\Omega$ ont $\Phi$ normalized by $\Phi(\infty) = \infty,\quad \Phi'(\infty) > 0$.
Let $A_p(G),\; p > 0$, denote the class of functions $f$ which are analytic in $G$ and satisfy the condition
$$||f||^p_{A_p(G)} := \int\int_G |f(z)|^p d \sigma_z < \infty,\quad (∗)$$
where $\sigma$ is a two-dimensional Lebesque measure.
Let $P_n(z)$ be arbitrary algebraic polynomial of degree at most $n$. The well-known Bernstein – Walsh
lemma says that
$$P_n(z)k ≤ |\Phi(z)|^{n+1} ||P_n||_{C(\overline{G})}, \; z \in \Omega. \quad (∗∗)$$
Firstly, we study the estimation problem (∗∗) for the norm (∗). Secondly, we continue studying the
estimation (∗∗) when we replace the norm $||P_n||_{C(\overline{G})}$ by $||P_n||_{A_2(G)}$
for some regions of complex plane. |
| first_indexed | 2026-03-24T02:28:54Z |
| format | Article |
| fulltext |
UDC 517.5
F. G. Abdullayev, N. D. Aral (Mersin Univ., Turkey)
ON THE BERNSTEIN – WALSH-TYPE LEMMAS
IN REGIONS OF THE COMPLEX PLANE
ПРО ЛЕМИ ТИПУ БЕРНШТЕЙНА – УОЛША
В ОБЛАСТЯХ КОМПЛЕКСНОЇ ПЛОЩИНИ
Let G ⊂ C be a finite region bounded by a Jordan curve L := ∂G, Ω := extG (respect to C), ∆ :=
:= {z : |z| > 1} ; w = Φ(z) be the univalent conformal mapping of Ω onto ∆ normalized by Φ(∞) =∞,
Φ
′
(∞) > 0.
Let Ap(G), p > 0, denote the class of functions f which are analytic in G and satisfy the condition
‖f‖p
Ap(G)
:=
∫∫
G
|f(z)|pdσz <∞, (∗)
where σ is a two-dimensional Lebesque measure.
Let Pn(z) be arbitrary algebraic polynomial of degree at most n. The well-known Bernstein – Walsh
lemma says that
‖Pn(z)‖ ≤ |Φ(z)|n+1‖Pn‖C(G), z ∈ Ω. (∗∗)
Firstly, we study the estimation problem (∗∗) for the norm (∗). Secondly, we continue studying the
estimation (∗∗) when we replace the norm ‖Pn‖C(G) by ‖Pn‖A2(G) for some regions of complex plane.
Припустимо, що G ⊂ C — скiнченна область, що обмежена кривою Жордана L := ∂G, Ω := extG
(вiдносно C),∆ := {z : |z| > 1} ;w = Φ(z) — унiвалентне конформне вiдображення Ω на ∆, нормоване
з використанням Φ(∞) =∞, Φ
′
(∞) > 0.
Нехай Ap(G), p > 0, позначає клас функцiй f, якi є аналiтичними в G i задовольняють умову
‖f‖p
Ap(G)
:=
∫∫
G
|f(z)|pdσz <∞, (∗)
де σ — двовимiрна мiра Лебега.
Припустимо, що Pn(z) — довiльний алгебраїчний полiном степеня не бiльше n. У вiдомiй лемi
Бернштейна – Уолша стверджується, що
‖Pn(z)‖ ≤ |Φ(z)|n+1‖Pn‖C(G), z ∈ Ω. (∗∗)
По-перше, розглянуто задачу оцiнювання (∗∗) для норми (∗). По-друге, продовжено дослiдження
оцiнювання (∗∗) у випадку, коли норма ‖Pn‖C(G) замiнюється нормою ‖Pn‖A2(G) для деяких областей
комплексної площини.
1. Introduction and main results. Let G ⊂ C be a finite region, with 0 ∈ G,
bounded by a Jordan curve L := ∂G, B := B(0, 1) := {z : |z| < 1} , ∆ := ∆(0, 1) :=
:= {w : |w| > 1} , Ω := extG (respect to C); w = Φ(z) be the univalent conformal
mapping of Ω onto the ∆ normalized by Φ(∞) =∞, Φ
′
(∞) > 0, and Ψ := Φ−1. Let
℘n denote the class of arbitrary algebraic polynomials Pn(z) of degree at most n.
Let σ be the two-dimensional Lebesque measure and let h (z) be a weight function
defined in G.
Let Ap(h,G), p > 0 denote the class of functions f which are analytic in G and
satisfy the condition
c© F. G. ABDULLAYEV, N. D. ARAL, 2011
ISSN 1027-3190. Укр. мат. журн., 2011, т. 63, № 3 291
292 F. G. ABDULLAYEV, N. D. ARAL
‖f‖Ap(h,G) :=
∫∫
G
h(z)|f(z)|pdσz
1/p
<∞
and Ap(1, G) ≡ Ap(G).
In case of when L is rectifiable, let Lp(L), p > 0, we denote the class of functions
f which are integrable on L and satisfy the condition
‖f‖Lp(L) :=
∫
L
|f(z)|p|dz|
1/p
<∞.
Well known Bernstein – Walsh lemma [1] says that
|Pn(z)| ≤ |Φ(z)|n+1‖Pn‖C(G), z ∈ Ω. (1.1)
For R > 1, let us set LR := {z : |Φ(z)| = R} , GR := intLR, ΩR := extLR. Then,
(1.1) can be written as following:
‖Pn‖C(GR) ≤ R
n+1‖Pn‖C(G). (1.2)
For R = 1 +
c1
n
, according to (1.2), we see that the C-norm of polynomials Pn(z)
in GR and G is identical, i.e., the norm ‖Pn‖C(G) increases with at most a constant.
Similar estimation to (1.2) in space Lp(L) was investigated in [2] and obtained as
following:
‖Pn‖Lp(LR) ≤ c1Rn+1/p‖Pn‖Lp(L), p > 0. (1.3)
Here and throughout this paper, c, c0,c1, c2, . . . are positive constants (in general,
different in different relations), which are depended on G in general.
Definition 1.1 [3, p. 97; 4]. The Jordan arc (or curve) L is calledK-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).
F (L) denotes 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 defines
KL := inf {K(f) : f ∈ F (L)} ,
where K(f) is the maximal dilatation of a such mapping f. L is a quasiconformal
curve, if KL <∞, and L is a K-quasiconformal curve, if KL ≤ K.
We well know that there exists quasiconformal curve which is not rectifiable [3,
p. 104].
Let L be a K-quasiconformal and y(·) be a regular quasiconformal reflection across
L (for detail see Section 2). For R > 1, let L∗ := y(LR), G∗ := intL∗, Ω∗ := extL∗;
w = ΦR(z) be the conformal mapping of Ω∗ onto the ∆ normalized by ΦR(∞) = ∞,
Φ
′
R(∞) > 0, and ΨR := Φ−1
R ; d(Γ, L) := inf {|ζ − z| : z ∈ Γ, ζ ∈ L} .
The Bernstein – Walsh-type estimation in the space Ap(h,G), p > 0 is contained in
[5]. In particular,
‖Pn‖Ap(GR)
≤ c2 R∗
n+1/p
‖Pn‖Ap(G)
, p > 0, (1.4)
ISSN 1027-3190. Укр. мат. журн., 2011, т. 63, № 3
ON THE BERNSTEIN – WALSH-TYPE LEMMAS IN REGIONS OF THE COMPLEX PLANE 293
where R∗ := 1 + c3(R − 1). Therefore, if we choose R = 1 +
c1
n
, then (1.4) can be
shown that Ap-norm of polynomials Pn(z) in GR and G is identical.
N. Stylianopoulos in [9] obtained the following result by changing the norm ‖Pn‖C(G)
in (1.1) with the norm ‖Pn‖A2(G).
Lemma 1.1 [9]. Assume that L is quasiconformal and rectifiable. Then, for any
Pn ∈ ℘n
|Pn(z)| ≤ c(L)
d(z, L)
√
n‖Pn‖A2(G)|Φ(z)|n+1, z ∈ Ω. (1.5)
In this work, we study the similar problems to (1.5) for domains withK-quasiconformal
(non-rectifiable!) boundary. Now, we give the main results.
Theorem 1.1. Assume that L is K-quasiconformal. Then, for any Pn ∈ ℘n and
R > 1 we have
|Pn(z)| ≤ c4
d(L,LR)
‖Pn‖A2(GR)|Φ(z)|n+1, z ∈ Ω. (1.6)
Theorem 1.2. Assume that L is K-quasiconformal. Then, for any Pn ∈ ℘n we
have
|Pn(z)| ≤ c5
d(L,L1+ c
n
)
‖Pn‖A2(G)|Φ(z)|n+1, z ∈ Ω. (1.7)
Remark 1.1. If z ∈ G1+c/n ∩ Ω for some c > 1, then (1.7) is better than (1.5).
Theorem 1.3. Assume that L is K-quasiconformal. Then, for any Pn ∈ ℘n we
have
|Pn(z)| ≤ c6
d(z, L)
nµ−µ
−1
‖Pn‖A2(G)|Φ1+1/n(z)|n+1, z ∈ Ω, (1.8)
where µ := min
{
2,K4
}
.
Remark 1.2. For K ≤ 4
√
1 +
√
17
4
and for z ∈ Ω such that far away from L, (1.8)
is better than (1.5).
Theorem 1.4. Assume that L is K-quasiconformal. Then, for any Pn ∈ ℘n we
have
|Pn(z)| ≤ c7nµ‖Pn‖A2(G)|Φ1+1/n(z)|n+1, z ∈ Ω, (1.9)
where µ := min
{
2,K4
}
.
Theorem 1.5. Assume that L is K-quasiconformal. Then, for any Pn ∈ ℘n we
have
|Pn(z)| ≤ c8
d(z, L)
nν−ν
−1
‖Pn‖A2(G)|Φ(z)|n+1, z ∈ Ω1+1/n, (1.10)
where ν := min
{
2,K2
}
.
Remark 1.3. For any K ≤ 1
2
√
1 +
√
17 and the points z ∈ Ω1+1/n, (1.10) is
better than (1.5).
Theorem 1.6. Assume that L is K-quasiconformal. Then, for any Pn ∈ ℘n we
have
|Pn(z)| ≤ c9nν‖Pn‖A2(G)|Φ(z)|n+1, z ∈ Ω1+1/n, (1.11)
where ν := min
{
2,K2
}
.
ISSN 1027-3190. Укр. мат. журн., 2011, т. 63, № 3
294 F. G. ABDULLAYEV, N. D. ARAL
2. Some auxiliary results. Let G ⊂ C be a finite region bounded by Jordan curve
L and let w = ϕ(z) be the univalent conformal mapping of G onto the B normalized
by ϕ(0) = 0, ϕ
′
(0) > 0 and ψ := ϕ−1.
The level curve (interior or exterior) can be defined for t > 0 as
Lt := {z : |ϕ(z)| = t, if t < 1, |Φ(z)| = t, if t > 1}, L1 ≡ L,
and let Gt := intLt, Ωt := extLt.
We note that, the region D in Definition 1.1 may be taken as D ⊂ C or D ≡ C. Case
D ≡ C gives the global definition of a K-quasiconformal arc or curve consequently. At
the same time, we can consider the domain D ⊃ L as the neighborhood of the curve
L. In this case, Definition 1.1 will be called local definition. This local definition has an
advantage in determining the coefficients of quasiconformality for some simple arcs or
curves.
Let us denote natural representation of L by z = z(s), s ∈ [0,mesL] .
Definition 2.1 [12]. We say that G ∈ Cθ if L := ∂G has a continuous tangent
θ(z) := θ(z(s)) for every points z(s).
According to [4], we have the following facts:
Corollary 2.1. If G ∈ Cθ, then K = 1 + ε, for all ε > 0.
Corollary 2.2. If L is an analytic curve or arc, then K = 1.
For a > 0 and b > 0, we shall use the notations “a ≺ b” (order inequality), if a ≤ cb
and “a � b” are equivalent to c1a ≤ b ≤ c2a for some constants c, c1, c2 (independent of
a and b) respectively. Throughout this paper ε, ε1, ε2, . . . are sufficiently small positive
constants (in general, different in different relations), which depend on G in general.
Let L be a K-quasiconformal curve and D = C. Then [8] there exists a quasicon-
formal reflection y(·) across L such that y(G) = Ω, y(Ω) = G and y(·) fixes the points
of L. The quasiconformal reflection y(·) is such that it satisfied the following condition
[8, 7; 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
,
and for the Jacobian Jy = |yz|2 − |yz|2 of y(·) the relation Jy � 1 is hold.
On the other hand, let L be a K-quasiconformal curve and D ⊂ C. Then the
region D in the Definition 1.1 can be chosen to be the region D := GR0
\Gr0 , for a
certain number 1 < R0 ≤ 2, depending on ϕ, Φ, f, and r0 = R−1
0 . In this case, it
is known that the function α(·) = f−1
{[
f(·)
]−1
}
is a K2-quasiconformal reflection
across L as shown in [11, p. 28] by analogously in [8, p. 75], that is, α(·) is a K2-
quasiconformal mapping leaving the points on L fixed and satisfying the conditions
α(GR̃\G) ⊂ G\Gr0 , α(G\Gr̃) ⊂ GR0
\G for some 1 < R̃ < R0, r0 < r̃ < 1.
Therefore, by means of the extension theorem of a quasiconformal mapping [3, p. 98],
without loss of generality, we may assume that
ISSN 1027-3190. Укр. мат. журн., 2011, т. 63, № 3
ON THE BERNSTEIN – WALSH-TYPE LEMMAS IN REGIONS OF THE COMPLEX PLANE 295
y(z) = α(z), z ∈ D.
Then, taking y(z) that satisfies (2.1) and denoting the restriction of y(z) in D as α∗(·),
we see that the following conditions are also satisfied for α∗(·):
|z1 − α∗(z)| � |z1 − z|, z1 ∈ L, ε < |z| < 1
ε
,
|α∗z| � |α∗z| � 1, ε < |z| < 1
ε
, (2.2)
|α∗z| � |α∗(z)|2, |z| < ε, |α∗z| � |z|−2, |z| > 1
ε
,
and for the Jacobian Jα∗ = |α∗z|2 − |α∗z|2 of α∗(·) the relation Jα∗ � 1 is hold.
For simplicity of notation, we denote the α∗(·) also as α(·). Throughout this paper
we assume that D ⊂ C.
For R > 1, we denote L∗ := y(LR), G∗ := intL∗, Ω∗ := extL∗; w = ΦR(z) be
the conformal mapping of Ω∗ onto the ∆ normalized by ΦR(∞) = ∞, Φ
′
R(∞) > 0;
ΨR := Φ−1
R . For t > 1, let L∗t := {z : |ΦR(z)| = t} , G∗t := intL∗t , Ω∗t := extL∗t .
According to [10], for all z ∈ L∗ and t ∈ L such that |z − t| = d(z, L) we have
d(z, L) � d(t, LR) � d(z, L∗R).
|ΦR(z)| ≤ |ΦR(t)| ≤ 1 + c(R− 1).
(2.3)
Lemma 2.1 [11]. 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
∣∣∣∣K−2
≺
∣∣∣∣z1 − z3
z1 − z2
∣∣∣∣ ≺ ∣∣∣∣w1 − w3
w1 − w2
∣∣∣∣K2
,
where 0 < r0 < 1 a constant, depending on G.
In particular, for arbitrary z1 ∈ L, 1 < R < R0 and fixed z3 ∈ LR0
we have
(R− 1)K
2
≺ d(z1, LR) ≺ (R− 1)1/K2
. (2.4)
Remark 2.1. The left part of (2.4) for arbitrary continuum can be replaced by (see,
for instance, [7])
(R− 1)2 ≺ d(z1, LR). (2.5)
Let {zj}mj=1 be a fixed system of the points on L and the weight function h (z) is
defined as the following:
h (z) = h0 (z)
m∏
j=1
|z − zj |γj , (2.6)
where γj > −2 for j = 1,m and h0 (z) is uniformly separated from zero in G:
h0 (z) ≥ c0 > 0 ∀z ∈ G.
ISSN 1027-3190. Укр. мат. журн., 2011, т. 63, № 3
296 F. G. ABDULLAYEV, N. D. ARAL
Lemma 2.2 [6]. Let L be a K-quasiconformal curve; h(z) is defined in (2.6). Then,
for arbitrary Pn(z) ∈ ℘n, any R > 1 and n = 1, 2, . . . , we have
‖Pn‖Ap(h,G1+c(R−1)) ≤ c1R
n+1/p‖Pn‖Ap(h,G), p > 0, (2.7)
where c, c1 are independent of n and R.
3. Proof of theorems. 3.1. Proof of Theorem 1.1. First of all, we note the estimation
|Pn(z)| ≺ 1
d(z, LR)
‖Pn‖A2(GR), z ∈ G. (3.1)
Since L is a K-quasiconformal, we conclude that any LR, R > 1, is also quasiconfor-
mal. Therefore, we can construct a c (K)-quasiconformal reflection yR (z) , yR (0) =∞,
across LR such that yR (GR) = ΩR, yR (ΩR) = GR and yR(·) fixes the points of LR
that satisfies conditions (2.1) described for yR (z) . By using this constructed yR (z) , we
can write the following integral representations for Pn(z) [7, p. 105]:
Pn (z) = − 1
π
∫∫
GR
Pn (ζ) yR,ζ
(yR (ζ)− z)2 dσζ , z ∈ GR. (3.2)
For ε > 0, let us set Uε(z) := {ζ : |ζ − z| < ε} and without loss of generality we may
take Uε := Uε(0) ⊂ G∗. For arbitrary fixed point z ∈ L we have
|Pn (z) | ≤ 1
π
∫∫
Uε
|Pn (ζ) ||yR,ζ̄ |
|yR (ζ)− z|2
dσζ +
1
π
∫∫
GR\Uε
|Pn (ζ) ||yR,ζ̄ |
|yR (ζ)− z|2
dσζ =: J1 + J2. (3.3)
To estimate the integral J1, applying the Hölder inequality we get
J2
1 ≤
∫∫
Uε
|Pn (ζ) |2dσζ
∫∫
Uε
|yR,ζ̄ |2
|yR (ζ)− z|4
dσζ ≺ ‖Pn‖2A2(G)
∫∫
Uε
|yR,ζ̄ |2
|yR (ζ)− z|4
dσζ .
According to (2.1), |yR,ζ | � |yR(ζ)|2, for all ζ ∈ Uε, because of |ζ − z| ≥ ε, |yR(ζ)−
− z| � |yR(ζ)| for z ∈ L and ζ ∈ Uε. On the other hand, if Jy,R := |yR,ζ |2 − |yR,ζ |2 is
Jacobian of the reflection yR(ζ), we can obtain
|Jy,R | � |yR,ζ |
2
as in [12]. Then, we can find
J2
1 ≺ ‖Pn‖2A2(G)
∫∫
yR(Uε)
|yR,ζ̄ |2
|Jy,R||ζ − z|4
dσζ ≺
≺ ‖Pn‖2A2(G)
∫∫
|ζ−z|≥c1
dσζ
|ζ − z|4
≺ ‖Pn‖2A2(G). (3.4)
For the J2, we get
J2
2 =
∫∫
GR\Uε
|yR,ζ̄ |2dσζ
|yR (ζ)− z|4
∫∫
GR\Uε
|Pn (ζ) |2dσζ =: J21J22. (3.5)
ISSN 1027-3190. Укр. мат. журн., 2011, т. 63, № 3
ON THE BERNSTEIN – WALSH-TYPE LEMMAS IN REGIONS OF THE COMPLEX PLANE 297
For the integral J21, we get
J21 ≺
∫∫
y(GR\Uε)
dσζ
|ζ − z|4
≤
∫∫
|ζ−z|≥d(z,LR)
dσζ
|ζ − z|4
≺ d−2 (z, LR) (3.6)
and for J22 :
J22 =
∫∫
GR\Uε
|Pn (ζ) |2dσζ ≤
∫∫
GR
|Pn (ζ) |2dσζ = ‖Pn‖2A2(GR).
Then,
J2
2 = J21J22 ≺ d−2 (z, LR) ‖Pn‖2A2(GR). (3.7)
Combining (3.3), (3.4), (3.5) and (3.7), we prove the estimation (3.1). To complete the
proof of Theorem 1.1, according the maximum modulus principle, for any z ∈ Ω we
have ∣∣∣∣ Pn(z)
Φn+1(z)
∣∣∣∣ ≤ max
z∈Ω
∣∣∣∣ Pn(z)
Φn+1(z)
∣∣∣∣ = max
z∈L
|Pn(z)| ≺ 1
d(z, LR)
‖Pn‖A2(GR),
or
|Pn(z)| ≺ 1
d(L,LR)
‖Pn‖A2(GR)|Φ(z)|n+1.
Taking R = 1 + 1/n, according to Lemma 2.2, we obtain the proof of Theorem 1.2.
3.2. Proof of Theorem 1.3. For the arbitrary fixed R > 1, let us set L∗ := y(LR).
According to (2.3), the number ε1 (consuquently ρ1 := 1 + ε1(R − 1)) can be chosen
such that G
∗
ρ1 ⊆ G. Let R1 := 1 +
ρ1 − 1
2
.
For z ∈ Ω and w = ΦR(z) let us get
hR (w) :=
Pn (ΨR (w))
wn+1
.
Cauchy integral representation for unbounded region gives
hR (w) = − 1
2πi
∫
|t|=R1
hR (t)
dt
t− w
.
For all |t| = R1 > 1, |t|n+1 = Rn+1
1 > 1, then
An := |Pn (ΨR (w)) | ≤ |w|n+1 1
2π
∫
|t|=R1
|Pn (ΨR (t)) | |dt|
|t− w|
. (3.8)
Applying the Hölder inequality, we get
An ≺ |w|n+1
∫
|t|=R1
|Pn (ΨR (t)) Ψ′R (t) |2|dt|
1/2
×
×
∫
|t|=R1
1
|Ψ′R (t) |2|t− w|2
|dt|
1/2
=: |w|n+1(A1
nB
1
n)1/2. (3.9)
ISSN 1027-3190. Укр. мат. журн., 2011, т. 63, № 3
298 F. G. ABDULLAYEV, N. D. ARAL
Let us set
fn (t) := Pn (ΨR (t)) Ψ′R (t) .
Now, we separate the circle |t| = R1 to n equal partδn with mes δn =
2πR1
n
and by
applying the mean value theorem to the integral A1
n we get
A1
n =
n∑
k=1
∫
δk
|fn(t)|2|dt| =
n∑
k=1
∣∣∣fn(t
′
k)
∣∣∣2 mes δk, t′k ∈ δk.
On the other hand, by applying mean value estimation,∣∣∣fn (t′k)∣∣∣2 ≤ 1
π
(
|t′k| − 1
)2 ∫∫
|ξ−t′k|<|t
′
k|−1
|fn (ξ) |2dσξ,
we obtain
A1
n ≺
n∑
k=1
mes δk
π
(
|t′k| − 1
)2 ∫∫
|ξ−t′k|<|t
′
k|−1
|fn (ξ) |2dσξ, t′k ∈ δk.
Taking into account that discs with origin at the points t′k at most two may be crossing,
we have
A1
n ≺
mes δ1(
|t′1| − 1
)2 ∫∫
1<|ξ|<ρ1
|fn (ξ) |2dσξ ≺ n
∫∫
1<|ξ|<ρ1
|fn (ξ) |2dσξ.
According to (2.2), for A1
n, we get
A1
n ≺ n
∫∫
G∗ρ1
\G∗
|Pn (z) |2dσz ≺ n‖Pn‖2A2(G). (3.10)
To estimate the integral B1
n, taking into account the estimation for the Ψ′R (see, for
instance, [7], Theorem 2.8) and Lemma 2.1 written for ΦR(z), from (2.4) and (2.5) we
get
B1
n ≺
∫
|t|=R1
(|t| − 1)
2
d2(ΨR (t) , L∗)
|dt|
|t− w|2
=
=
∫
|t|=R1
(|t| − 1)
2
d2(ΨR (t) , L∗)
|dt|
|t− w|2−
2
µ |t− w|
2
µ
≺
≺ 1
d2(z, L∗R1
)
∫
|t|=R1
(|t| − 1)
2
(|t| − 1)
2µ
|dt|
|t− w|2−
2
µ
=
=
1
d2(z, L∗R1
)
∫
|t|=R1
1
(|t| − 1)
2(µ−1)
|dt|
|t− w|2(1− 1
µ )
≺
ISSN 1027-3190. Укр. мат. журн., 2011, т. 63, № 3
ON THE BERNSTEIN – WALSH-TYPE LEMMAS IN REGIONS OF THE COMPLEX PLANE 299
≺ 1
d2(z, L∗R1
)
n2(µ−µ−1)−1 ≺ 1
d2(z, L)
n2(µ−µ−1)−1, (3.11)
where µ := min
{
2,K4
}
. Relations (3.8), (3.9), (3.10), and (3.11) yield
|Pn(z)| ≺ |w|n+1
√
n‖Pn‖A2(G)
1
d(z, L)
n(µ−µ−1)−1/2 =
=
n(µ−µ−1)
d(z, L)
|ΦR(z)|n+1‖Pn‖A2(G), z ∈ Ω.
Theorem 1.3 is proved.
3.3. Proof of Theorem 1.4. Proof of the Theorem 1.4 will be similar to proof of
Theorem 1.3. The term in (3.11) will be treated as the following:
B1
n ≺
∫
|t|=R1
(|t| − 1)
2
d2(ΨR (t) , L∗)
|dt|
|t− w|2
≺
≺
∫
|t|=R1
1
(|t| − 1)
2µ−2
|dt|
|t− w|2
≺
∫
|t|=R1
|dt|
(|t| − 1)
2µ ≺ n
2µ−1. (3.12)
And, consequently,
|Pn(z)| ≺ |w|n+1
√
n‖Pn‖A2(G)n
µ−1/2 = nµ‖Pn‖A2(G)|ΦR(z)|n+1, z ∈ Ω.
3.4. Proof of Theorem 1.5. Let R > 1 be arbitrary fixed and let R1 := 1 +
R− 1
2
.
For z ∈ ΩR and w = Φ(z) let us get
h (w) :=
Pn (Ψ (w))
wn+1
.
Cauchy integral representation for unbounded region gives
h (w) = − 1
2πi
∫
|t|=R1
h (t)
dt
t− w
.
Following the method used in proof of Theorem 1.3, similar terms are treated as
below:
Ãn := |Pn (Ψ (w)) | ≤ |w|n+1 1
2π
∫
|t|=R1
|Pn (Ψ (t)) | |dt|
|t− w|
≺
≺ |w|n+1
∫
|t|=R1
|Pn (Ψ (t)) Ψ′ (t) |2|dt|
1/2
×
×
∫
|t|=R1
1
|Ψ′ (t) |2|t− w|2
|dt|
1/2
=: |w|n+1(Ã1
nB̃
1
n)1/2. (3.13)
ISSN 1027-3190. Укр. мат. журн., 2011, т. 63, № 3
300 F. G. ABDULLAYEV, N. D. ARAL
Let us set
f̃n (t) := Pn (Ψ (t)) Ψ′ (t) .
Now, we separate the circle |t| = R1 to n equal part δn with mes δn =
2πR1
n
and by
applying the mean value theorem to the integral Ã1
n we get
Ã1
n =
n∑
k=1
∫
δk
|f̃n (t) |2|dt| =
n∑
k=1
|f̃n
(
t
′
k
)
|2mes δk, t′k ∈ δk.
On the other hand, applying mean value estimation, we obtain
Ã1
n ≺
n∑
k=1
mes δk
π
(
|t′k| − 1
)2 ∫∫
|ξ−t′k|<|t
′
k|−1
|f̃n (ξ) |2dσξ, t′k ∈ δk,
Ã1
n ≺
mes δ1(
|t′1| − 1
)2 ∫∫
1<|ξ|<R
|f̃n (ξ) |2dσξ ≺ n
∫∫
1<|ξ|<R
|f̃n (ξ) |2dσξ.
According to (2.2), for Ã1
n, we get
Ã1
n ≺ n
∫∫
GR\G
|Pn (z) |2dσz ≺ n‖Pn‖2A2(GR). (3.14)
To estimate the integral B̃1
n, taking into account that the estimation for the Ψ′ (see, for
instance, [7], Theorem 2.8) and Lemma 2.1, we get
B̃1
n ≺
∫
|t|=R1
(|t| − 1)
2
d2(Ψ (t) , L)
|dt|
|t− w|2
=
=
∫
|t|=R1
(|t| − 1)
2
d2(Ψ (t) , L)
|dt|
|t− w|2−2/ν |t− w|2/ν
≺
≺ 1
d2(z, LR1
)
∫
|t|=R1
(|t| − 1)
2
(|t| − 1)
2ν
|dt|
|t− w|2−2/ν
=
=
1
d2(z, LR1)
∫
|t|=R1
1
(|t| − 1)
2(ν−1)
|dt|
|t− w|2(1−1/ν)
≺
≺ 1
d2(z, LR1
)
n2(ν−ν−1)−1, (3.15)
where ν := min
{
2,K2
}
.
Let us denote ζ = Ψ(τ) ∈ L, ζ1 = Ψ(τ1) ∈ LR1
such that d(z, L) = |z −
− ζ|, d(z, LR1) = |z − ζ1|, and denote this image from τ = Φ(ζ), τ1 = Φ(ζ1). Also
we denote points |τ∗| = 1, |w − τ∗| = |w| − 1, |τ∗1 | = R1, |w − τ∗1 | = |w| − R1.
According to R1 := 1 +
R− 1
2
we have |w − τ1| � |w − τ∗1 | � |w − τ∗| � |w − τ |.
Then, by Lemma 2.1 we get d(z, LR1
) � d(z, L). Therefore, we obtain
ISSN 1027-3190. Укр. мат. журн., 2011, т. 63, № 3
ON THE BERNSTEIN – WALSH-TYPE LEMMAS IN REGIONS OF THE COMPLEX PLANE 301
B̃1
n ≺
1
d2(z, L)
n2(ν−ν−1)−1. (3.16)
Relations (3.13), (3.14), (3.16) and Lemma 2.2 yield
|Pn(z)| ≺ |w|n+1
√
n‖Pn‖A2(GR)
1
d(z, L)
n(ν−ν−1)−1/2 =
=
n(ν−ν−1)
d(z, L)
|Φ(z)|n+1‖Pn‖A2(G), z ∈ Ω1+1/n.
Theorem 1.5 is proved.
3.5. Proof of Theorem 1.6. Analogous to proof of the Theorem 1.4, the proof of the
Theorem 1.6 is identical to proof of the proof Theorem 1.5. In this case, the following
the method used in the proof of Theorem 1.5 and (3.15) will be treated as the following:
B1
n ≺
∫
|t|=R1
(|t| − 1)
2
d2(Ψ (t) , L)
|dt|
|t− w|2
≺
≺
∫
|t|=R1
1
(|t| − 1)
2ν−2
|dt|
|t− w|2
=
∫
|t|=R1
|dt|
(|t| − 1)
2ν ≺ n
2ν−1.
And, consequently,
|Pn(z)| ≺ |w|n+1
√
n‖Pn‖A2(GR)n
ν−1/2 ≺ nν‖Pn‖A2(G)|Φ(z)|n+1, z ∈ Ω1+1/n.
Theorem 1.6 is proved.
We note that the Theorems 1.2 – 1.6 are sharp. This can be clearly seen by the
example Pn(z) =
∑n
j=0
(j + 1)zj , G = B. In this case,
‖Pn‖C(G) =
(n+ 1)(n+ 2)
2
, ‖Pn‖A2(G) =
√
π(n+ 1)(n+ 2)
2
.
Then, for all z ∈ L1+1/n such that |Pn(z)| = ‖Pn‖C(GR), we have
|Pn(z)| ≥ ‖Pn‖C(G) ≥
1√
2π
n‖Pn‖A2(G) =
=
1√
2π
d(L,L1+1/n)
d(L,L1+1/n)
n‖Pn‖A2(G)
|Φ(z)|n+1
|Φ(z)|n+1
≥
≥ c10
1
d(L,L1+1/n)
‖Pn‖A2(G)|Φ(z)|n+1.
1. Walsh J. L. Interpolation and approximation by rational functions // Complex Domain. – Amer. Math.
Soc., 1960.
2. Hille E., Szegö G., Tamarkin J. D. On some generalization of a theorem of A. Markoff // Duke Math. J.
– 1937. – 3. – P. 729 – 739.
3. Lehto O., Virtanen K. I. Quasiconformal mapping in the plane. – Berlin: Springer, 1973.
4. Rickman S. Characterisation of quasiconformal arcs // Ann. Acad. Sci. Fenn. Ser. A. Math. – 1966. –
395. – 30 p.
ISSN 1027-3190. Укр. мат. журн., 2011, т. 63, № 3
302 F. G. ABDULLAYEV, N. D. ARAL
5. Abdullayev F. G. On the some properties of the orthogonal polynomials over the region of the complex
plane (Pt III) // Ukr. Math. J. – 2001. – 53, № 12. – P. 1934 – 1948.
6. Abdullayev F. G. The properties of the orthogonal polynomials with weight having singulerity on the
boundary contour // J. Comp. Anal. and Appl. – 2004. – 6, № 1. – P. 43 – 59.
7. Andrievskii V. V., Belyi V. I., Dzyadyk V. K. Conformal invariants in constructive theory of functions of
complex plane. – Atlanta: World Federation Publ. Comp., 1995.
8. Ahlfors L. V. Lectures on quasiconformal mappings. – Prinston, NJ: Van Nostrand, 1966.
9. Stylianopoulos N. Fine asymptotics for Bergman orthogonal polynomials over domains with corners //
CMFT. – 2009. – Ankara, 2009.
10. Andrievskii V. V. Constructive characterization of the harmonic functions in domains with quasiconformal
boundary // Quasiconformal Continuation and Approximation by Function in the Set of the Complex
Plane. – Kiev, 1985 [in Russian].
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angles // Acta Math. hung. – 1997. – 77, № 3. – P. 223 – 246.
Received 27.10.10
ISSN 1027-3190. Укр. мат. журн., 2011, т. 63, № 3
|
| id | umjimathkievua-article-2717 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T02:28:54Z |
| publishDate | 2011 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/e3/93dc2c3512d1bd0568284da65fa73ce3.pdf |
| spelling | umjimathkievua-article-27172020-03-18T19:34:39Z On the Bernstein - Walsh-type lemmas in regions of the complex plane Про леми типу Бернштейна – Уолша в областях комплексної площини Abdullayev, F. G. Aral, N. D. Абдуллаєв, Ф. Г. Арал, Н. Д. Let $G \subset C$ be a finite region bounded by a Jordan curve $L := \partial G,\quad \Omega := \text{ext} \; \overline{G}$ (respect to $\overline{C}$), $\Delta := \{z : |z| > 1\}; \quad w = \Phi(z)$ be the univalent conformal mapping of $\Omega$ ont $\Phi$ normalized by $\Phi(\infty) = \infty,\quad \Phi'(\infty) > 0$. Let $A_p(G),\; p > 0$, denote the class of functions $f$ which are analytic in $G$ and satisfy the condition $$||f||^p_{A_p(G)} := \int\int_G |f(z)|^p d \sigma_z < \infty,\quad (∗)$$ where $\sigma$ is a two-dimensional Lebesque measure. Let $P_n(z)$ be arbitrary algebraic polynomial of degree at most $n$. The well-known Bernstein – Walsh lemma says that $$P_n(z)k ≤ |\Phi(z)|^{n+1} ||P_n||_{C(\overline{G})}, \; z \in \Omega. \quad (∗∗)$$ Firstly, we study the estimation problem (∗∗) for the norm (∗). Secondly, we continue studying the estimation (∗∗) when we replace the norm $||P_n||_{C(\overline{G})}$ by $||P_n||_{A_2(G)}$ for some regions of complex plane. Припустимо, що $G \subset C$ — скiнченна область, що обмежена кривою Жордана $L := \partial G,\quad \Omega := \text{ext} \; \overline{G}$ (вiдносно $\overline{C}$), $\Delta := \{z : |z| > 1\}; \quad w = \Phi(z)$ — унiвалентне конформне вiдображення $\Omega$ на $\Phi$, нормоване з використанням $\Phi(\infty) = \infty,\quad \Phi'(\infty) > 0$. Нехай $A_p(G),\; p > 0$, позначає клас функцiй $f$, якi є аналiтичними в $G$ i задовольняють умову $$||f||^p_{A_p(G)} := \int\int_G |f(z)|^p d \sigma_z < \infty,\quad (∗)$$ де $\sigma$ — двовимiрна мiра Лебега. Припустимо, що $P_n(z)$ — довiльний алгебраїчний полiном степеня не бiльше $n$. У вiдомiй лемi Бернштейна – Уолша стверджується, що $$P_n(z)k ≤ |\Phi(z)|^{n+1} ||P_n||_{C(\overline{G})}, \; z \in \Omega. \quad (∗∗)$$ По-перше, розглянуто задачу оцiнювання (∗∗) для норми (∗). По-друге, продовжено дослiдження оцiнювання (∗∗) у випадку, коли норма $||P_n||_{C(\overline{G})}$ замiнюється нормою $||P_n||_{A_2(G)}$ для деяких областей комплексної площини. Institute of Mathematics, NAS of Ukraine 2011-03-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/2717 Ukrains’kyi Matematychnyi Zhurnal; Vol. 63 No. 3 (2011); 291-302 Український математичний журнал; Том 63 № 3 (2011); 291-302 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/2717/2190 https://umj.imath.kiev.ua/index.php/umj/article/view/2717/2191 Copyright (c) 2011 Abdullayev F. G.; Aral N. D. |
| spellingShingle | Abdullayev, F. G. Aral, N. D. Абдуллаєв, Ф. Г. Арал, Н. Д. On the Bernstein - Walsh-type lemmas in regions of the complex plane |
| title | On the Bernstein - Walsh-type lemmas in regions of the complex plane |
| title_alt | Про леми типу Бернштейна – Уолша в областях комплексної площини |
| title_full | On the Bernstein - Walsh-type lemmas in regions of the complex plane |
| title_fullStr | On the Bernstein - Walsh-type lemmas in regions of the complex plane |
| title_full_unstemmed | On the Bernstein - Walsh-type lemmas in regions of the complex plane |
| title_short | On the Bernstein - Walsh-type lemmas in regions of the complex plane |
| title_sort | on the bernstein - walsh-type lemmas in regions of the complex plane |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/2717 |
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