On the improvement of the rate of convergence of the generalized Bieberbach polynomials in domains with zero angles
Let ℂ be the complex plane, let C¯=C∪{∞}, let G ⊂ ℂ be a finite Jordan domain with 0 ∈ G; let L := ∂G; let Ω := C¯∖G¯, and let w = φ(z) be a conformal mapping of G onto a disk B(0, ρ₀) := {w:|w|<ρ₀} normalized by the conditions φ(z) = 0 and φ′(0)=1, where ρ₀ = ρ₀(0, G) is the conformal radius of...
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irk-123456789-1644212020-02-23T20:20:47Z On the improvement of the rate of convergence of the generalized Bieberbach polynomials in domains with zero angles Abdullayev, F.G. Özkartepe, N.P. Статті Let ℂ be the complex plane, let C¯=C∪{∞}, let G ⊂ ℂ be a finite Jordan domain with 0 ∈ G; let L := ∂G; let Ω := C¯∖G¯, and let w = φ(z) be a conformal mapping of G onto a disk B(0, ρ₀) := {w:|w|<ρ₀} normalized by the conditions φ(z) = 0 and φ′(0)=1, where ρ₀ = ρ₀(0, G) is the conformal radius of G with respect to 0. Let φp(z):=∫₀z[φ′(ζ)]2/pdζ and let π n,p (z) be the generalized Bieberbach polynomial of degree n for the pair (G, 0) that minimizes the integral ∬G∣∣φ′p(z)−P′n(z)∣∣pdσz in the class of all polynomials of degree deg Pn ≤ n such that Pn(0) = 0 and P′n(0)=1. We study the uniform convergence of the generalized Bieberbach polynomials π n,p (z) to φ p (z) on G¯ with interior and exterior zero angles determined depending on properties of boundary arcs and the degree of their tangency. In particular, for Bieberbach polynomials, we obtain improved estimates for the rate of convergence in these domains. Нехай C — комплексна площина, C¯=C⋃{∞}, G⊂C — скiнченна жорданова область iз 0∈G, L:=∂G, Ω:=C¯ G¯ i w=φ(z) — конформне вiдображення G на круг B(0,ρ):={w:|w|<ρ₀0}, нормоване умовами φ(0)==0,φ′(0)=1, де ρ₀=ρ₀(0,G) — конформний радiус G вiдносно 0. Покладемо φρ(z):=∫₀z[φ′(ζ)]2/pdζ. Нехай πn,p(z) — узагальнений полiном Бiбербаха степеня n для пари (G,0), що мiнiмiзує iнтеграл ∫∫G|φ′(z)−P′n(z)|pdσz у класi всiх полiномiв степеня degPn≤n таких, що Pn(0)=0, P′n(0)=1. Вивчається рiвномiрна збiжнiсть узагальнених полiномiв Бiбербаха πn,p(z) до φρ(z) у G¯ iз внутрiшнiми та зовнiшнiми нульовими кутами, що визначаються в залежностi вiд властивостей граничних дуг та степеня їхнього дотику. Зокрема, для полiномiв Бiбербаха отримано покращенi оцiнки швидкостi збiжностi у цих областях. 2012 Article On the improvement of the rate of convergence of the generalized Bieberbach polynomials in domains with zero angles / F.G. Abdullayev, N.P. Özkartepe // Український математичний журнал. — 2012. — Т. 64, № 5. — С. 582-596. — Бібліогр.: 27 назв. — англ. 1027-3190 http://dspace.nbuv.gov.ua/handle/123456789/164421 517.5 en Український математичний журнал Інститут математики НАН України |
| institution |
Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| collection |
DSpace DC |
| language |
English |
| topic |
Статті Статті |
| spellingShingle |
Статті Статті Abdullayev, F.G. Özkartepe, N.P. On the improvement of the rate of convergence of the generalized Bieberbach polynomials in domains with zero angles Український математичний журнал |
| description |
Let ℂ be the complex plane, let C¯=C∪{∞}, let G ⊂ ℂ be a finite Jordan domain with 0 ∈ G; let L := ∂G; let Ω := C¯∖G¯, and let w = φ(z) be a conformal mapping of G onto a disk B(0, ρ₀) := {w:|w|<ρ₀} normalized by the conditions φ(z) = 0 and φ′(0)=1, where ρ₀ = ρ₀(0, G) is the conformal radius of G with respect to 0. Let
φp(z):=∫₀z[φ′(ζ)]2/pdζ
and let π n,p (z) be the generalized Bieberbach polynomial of degree n for the pair (G, 0) that minimizes the integral
∬G∣∣φ′p(z)−P′n(z)∣∣pdσz
in the class of all polynomials of degree deg Pn ≤ n such that Pn(0) = 0 and P′n(0)=1. We study the uniform convergence of the generalized Bieberbach polynomials π n,p (z) to φ p (z) on G¯ with interior and exterior zero angles determined depending on properties of boundary arcs and the degree of their tangency. In particular, for Bieberbach polynomials, we obtain improved estimates for the rate of convergence in these domains. |
| format |
Article |
| author |
Abdullayev, F.G. Özkartepe, N.P. |
| author_facet |
Abdullayev, F.G. Özkartepe, N.P. |
| author_sort |
Abdullayev, F.G. |
| title |
On the improvement of the rate of convergence of the generalized Bieberbach polynomials in domains with zero angles |
| title_short |
On the improvement of the rate of convergence of the generalized Bieberbach polynomials in domains with zero angles |
| title_full |
On the improvement of the rate of convergence of the generalized Bieberbach polynomials in domains with zero angles |
| title_fullStr |
On the improvement of the rate of convergence of the generalized Bieberbach polynomials in domains with zero angles |
| title_full_unstemmed |
On the improvement of the rate of convergence of the generalized Bieberbach polynomials in domains with zero angles |
| title_sort |
on the improvement of the rate of convergence of the generalized bieberbach polynomials in domains with zero angles |
| publisher |
Інститут математики НАН України |
| publishDate |
2012 |
| topic_facet |
Статті |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/164421 |
| citation_txt |
On the improvement of the rate of convergence of the generalized Bieberbach polynomials in domains with zero angles / F.G. Abdullayev, N.P. Özkartepe // Український математичний журнал. — 2012. — Т. 64, № 5. — С. 582-596. — Бібліогр.: 27 назв. — англ. |
| series |
Український математичний журнал |
| work_keys_str_mv |
AT abdullayevfg ontheimprovementoftherateofconvergenceofthegeneralizedbieberbachpolynomialsindomainswithzeroangles AT ozkartepenp ontheimprovementoftherateofconvergenceofthegeneralizedbieberbachpolynomialsindomainswithzeroangles |
| first_indexed |
2025-07-14T16:59:07Z |
| last_indexed |
2025-07-14T16:59:07Z |
| _version_ |
1837642387410649088 |
| fulltext |
UDC 517.5
F. G. Abdullayev, N. P. Özkartepe (Mersin Univ., Turkey)
ON THE IMPROVEMENT OF THE RATE OF CONVERGENCE
OF THE GENERALIZED BIEBERBACH POLYNOMIALS
IN DOMAINS WITH ZERO ANGLES
ПРО ПОКРАЩЕННЯ ШВИДКОСТI ЗБIЖНОСТI
УЗАГАЛЬНЕНИХ ПОЛIНОМIВ БIБЕРБАХА
В ОБЛАСТЯХ З НУЛЬОВИМИ КУТАМИ
Let C be the complex plane, let C = C ∪ {∞} , let G ⊂ C be a finite Jordan domain with 0 ∈ G, let L := ∂G, let Ω :=
:= C\G, and let w = ϕ(z) be the conformal mapping of G onto a disk B(0, ρ0) := {w : |w| < ρ0} normalized by ϕ(0) =
= 0, ϕ′(0) = 1, where ρ0 = ρ0 (0, G) is the conformal radius of G with respect to 0. Let ϕp(z) :=
∫ z
0
[
ϕ′(ζ)
]2/p
dζ
and let πn,p(z) be the generalized Bieberbach polynomial of degree n for the pair (G, 0) that minimizes the integral∫∫
G
∣∣ϕ′p(z)− P ′n(z)
∣∣p dσz in the class of all polynomials of degree degPn ≤ n such that Pn(0) = 0 and P ′n(0) = 1.
We study the uniform convergence of the generalized Bieberbach polynomials πn,p(z) to ϕp(z) on G with interior and
exterior zero angles determined depending on properties of boundary arcs and the degree of their tangency. In particular,
for Bieberbach polynomials, we obtain better estimates for the rate of convergence in these domains.
Нехай C — комплексна площина, C = C ∪ {∞} , G ⊂ C — скiнченна жорданова область iз 0 ∈ G, L :=
:= ∂G, Ω := C\G i w = ϕ(z) — конформне вiдображення G на круг B(0, ρ0) := {w : |w| < ρ0}, нормоване
умовами ϕ(0) = 0, ϕ′(0) = 1, де ρ0 = ρ0 (0, G) — конформний радiус G вiдносно 0. Покладемо ϕp(z) :=
:=
∫ z
0
[ϕ′(ζ)]
2/p
dζ. Нехай πn,p(z) — узагальнений полiном Бiбербаха степеня n для пари (G, 0), що мiнiмiзує
iнтеграл
∫∫
G
∣∣ϕ′p(z)− P ′n(z)
∣∣p dσz у класi всiх полiномiв степеня degPn ≤ n таких, що Pn(0) = 0, P ′n(0) = 1.
Вивчається рiвномiрна збiжнiсть узагальнених полiномiв Бiбербаха πn,p(z) до ϕp(z) у G iз внутрiшнiми та
зовнiшнiми нульовими кутами, що визначаються в залежностi вiд властивостей граничних дуг та степеня їхнього
дотику. Зокрема, для полiномiв Бiбербаха отримано покращенi оцiнки швидкостi збiжностi у цих областях.
1. Introduction and main result. Let C be the complex plane, let C = C ∪ {∞}, let G ⊂ C be a
finite Jordan domain with 0 ∈ G, let L := ∂G, let Ω := C\G, and let w = ϕ(z) be the conformal
mapping of G onto a disk B(0, ρ0) := {w : |w| < ρ0} normalized by the conditions ϕ(0) = 0 and
ϕ′(0) = 1, where ρ0 = ρ0 (0, G) is the conformal radius of G with respect to 0.
For p > 0, let A1
p(G) denote the set of functions f(z) analytic in G, normalized by the conditions
f(0) = 0 and f ′(0) = 1, and such that
‖f‖p := ‖f‖A1
p(G) :=
∫∫
G
∣∣f ′(z)∣∣p dσz
1/p
<∞,
where σ denotes the two-dimensional Lebesgue measure.
Consider the extremal problem {
‖f‖p , f ∈ A
1
p(G)
}
→ inf . (1.1)
It is well known [22, p. 426] that the function
c© F. G. ABDULLAYEV, N. P. ÖZKARTEPE, 2012
582 ISSN 1027-3190. Укр. мат. журн., 2012, т. 64, № 5
ON THE IMPROVEMENT OF THE RATE OF CONVERGENCE . . . 583
ϕp(z) :=
z∫
0
[
ϕ′(ζ)
]2/p
dζ, z ∈ G, (1.2)
is the unique solution of the extremal problem (1.1). This function is well known in the geometric
theory of functions and is of great interest (see, e.g., [21]).
Let ℘n denote the class of all polynomials Pn(z), degPn(z) ≤ n, satisfying the conditions
Pn(0) = 0 and P ′n(0) = 1. For each p > 0, we consider the following extremal problem:{
‖ϕp − Pn‖p , Pn ∈ ℘n
}
→ inf . (1.3)
Using a method similar to that given in [13, p. 137], one can see that, for any p > 0, there
exists a polynomial P ∗n,p(z) that realizes a minimum of the integral ‖ϕp − Pn‖p in the class ℘n,
and for p > 1 this polynomial is uniquely determined [13, p. 142]. We call this polynomial the n-th
generalized Bieberbach polynomial for the pair (G, 0) and denote it by πn,p(z). In the case p = 2,
the polynomial πn,2(z) coincides with the Bieberbach polynomial for the pair (G, 0) (see, e.g., [14]).
If G is a Carathéodory domain, then ‖ϕp − πn,p‖p → 0 as n → ∞ [27, p. 63], and so the
sequence {πn,p(z)}∞n=0 converges uniformly to ϕp(z) on compact subsets of G. Our purpose is
to extend the uniform convergence of the sequence {πn,p(z)}∞n=0 to ϕp(z) on G. Moreover, we
investigate the estimate
‖ϕp − πn,p‖C(G)
:= max
{
|ϕp(z)− πn,p(z)| , z ∈ G
}
≤ const ·εn,p, (1.4)
where εn,p = εn,p(ϕ,G)→ 0, n→∞, and its dependence on the geometric properties of G.
For p = 2, estimate (1.4) was studied in [18, 20, 26] in the case where L satisfies certain
smoothness conditions and in [2, 5, 8 – 10, 14, 15, 24], etc., in the case where L has some zero or
nonzero angles.
In the case p 6= 2, the existence of a sequence {εn,p} → 0, n→∞, that satisfies (1.4) for some
domains with quasiconformal and piecewise-smooth (without cusps) boundary was investigated in
[17, 3, 6], etc. It is well known that quasiconformal curves have many properties, but they do not have
zero angles. Similar problems for domains of the class PQ (K,α, β) with piecewise-quasiconformal
boundaries having interior and exterior zero angles with “power tangency” (of the type cx1+α and
cx1+β for some α > 0 and β > 0) were investigated in [4].
Prior to introducing the class PQ (K,α, β) , we give several definitions.
Definition 1.1 [19, p. 97; 23]. A Jordan curve L is called K-quasiconformal (K ≥ 1) if there
is a K-quasiconformal mapping f of a domain D ⊃ L such that f(L) is a circle.
Let F (L) denote the set of all sense-preserving plane homeomorphisms f of domains D ⊃ L
such that f(L) is a circle and let
K (L) = inf {K(f) : f ∈ F (L)} ,
where K(f) is the maximal dilatation of a mapping f of this type. The curve L is K-quasiconformal
if and only if K (L) <∞. If L is K-quasiconformal, then K (L) ≤ K.
In this paper, we consider the case D ≡ C, i.e., we use the global definition of K-quasiconformal
curve.
ISSN 1027-3190. Укр. мат. журн., 2012, т. 64, № 5
584 F. G. ABDULLAYEV, N. P. ÖZKARTEPE
Definition 1.2. A Jordan arc ` is called K-quasiconformal if ` is a part of some closed K-
quasiconformal curve.
We can now define the class PQ (K,α, β) .
Note that, throughout this paper, c, c1, c2, . . . are positive constants and ε, ε1, ε2, . . . are suffi-
ciently small positive constants that depend, in general, on G.
Definition 1.3 [2]. We say that G ∈ PQ (K,α, β) , K > 1, α > 0, β > 0, if L := ∂G is
the union of a finite number of K-quasiconformal arcs (K = max1≤j≤m {Kj}) connected at the
points z0, z1, . . . , zm and such that L is a locally K-quasiconformal curve at z0 and the following
conditions are satisfied in the local coordinate system (x, y) with origin at zj , 1 ≤ j ≤ m :
a) for 1 ≤ j ≤ p, one has{
z = x+ iy : |z| ≤ ε1, c1x
1+α ≤ y ≤ c2x
1+α
}
⊂ Ω,
{z = x+ iy : |z| ≤ ε1, |y| ≥ ε2x} ⊂ G;
b) for p+ 1 ≤ j ≤ m, one has{
z = x+ iy : |z| ≤ ε3, c3x
1+β ≤ y ≤ c4x
1+β
}
⊂ G,
{z = x+ iy : |z| ≤ ε3, |y| ≥ ε4x} ⊂ Ω.
Here, −∞ < c1 < c2 <∞, −∞ < c3 < c4 <∞ and εi > 0, i = 1, 4, are some constants.
It is clear from Definition 1.3 that each domain G ∈ PQ (K,α, β) may have p exterior and
m − p interior zero angles. If a domain G does not have exterior (p = 0) (interior (p = m)) zero
angles, then we write G ∈ PQ (K, 0, β) (G ∈ PQ (K,α, 0)). If a domain G does not have these
angles (α = β = 0), then G is bounded by a K-quasiconformal curve. Further, PQ (K,α1, β) ⊂
⊂ PQ (K,α2, β) (PQ (K,α, β1) ⊂ PQ (K,α, β2)) for α2 > α1 (β2 > β1) and every fixed β > 0
and K > 1 (α > 0 and K > 1).
In this paper, we study the convergence of generalized Bieberbach polynomials in the closed
domains G ∈ PQ (K,α, β) and estimate an upper bound εn,p = εn,p(ϕ,G) → 0, n → ∞, and its
dependence on the geometric properties of G.
Prior to giving the main results, we introduce the following notation:
p0 := min
{
p− 1;
2
2 + p
}
, p1 :=
√
17− 1
2
, p̃2 :=
√
20K4 + 4K2 + 1− 2K2 − 1
2K2
,
p̃3 :=
√
33K4 + 2K2 + 1−K2 − 1
2K2
,
and
β̃2(p,K) :=
:=
√
(8K2+10+2pK2−p)2−16(K2+1) [4(K2 + 1)−2p(K2 + 1)−2]−8K2+10−2pK2 − p
8(K2+1)
.
ISSN 1027-3190. Укр. мат. журн., 2012, т. 64, № 5
ON THE IMPROVEMENT OF THE RATE OF CONVERGENCE . . . 585
Theorem 1.1. Suppose that p > 2, G ∈ PQ (K,α, β) for some K > 1, α <
2
p
, and 0 < β <
< min
{
p
2
− 1;
2
p+ 2
}
. Then, for any n ≥ 2, one has
‖ϕp − πn,p‖C(G) ≤ c1
(
1
lnn
)2−αp
2αp
.
Theorem 1.2. Suppose that 2 < p < 2
√
2 and G ∈ PQ
(
K,α,
p
2
− 1
)
for some K > 1,
α <
2
3p
. Then, for any n ≥ 2, one has
‖ϕp − πn,p‖C(G) ≤ c2
(
1
lnn
)2−3αp
2αp
.
Corollary 1.1. Suppose that p = 2 and G ∈ PQ(K,α, 0) for some K > 1, 0 < α <
1
2
. Then,
for any n ≥ 2, one has
‖ϕ− πn‖C(G) ≤ c3
(
1
lnn
)1−2α
2α
. (1.5)
In approximation problems, it is well known that if a domain has an exterior zero angle, then the
rate of approximation is “slower” than in its absence. Therefore, the right-hand sides of estimates for
such rates usually involve quantities of the type
(
1
lnn
)λ
. In the presence of an exterior zero angle,
quantities of the type
(
1
lnn
)λ
, λ > 0, cannot be replaced by
(
1
n
)µ
for any µ > 0. Moreover,
since PQ(K,α1, β) ⊂ PQ(K,α2, β) for α2 > α1, we may claim that the rate of approximation
improves as the class PQ(K,α, β) becomes narrower with respect to α (for the same K and β). In
other words, as the exterior zero angle of a domain becomes “wider” (for the same K and β), the
degree of approximation improves.
In particular, for p = 2 and a domain G ∈ PQ(K,α, 0), a result corresponding to Corollary
1.1 was obtained by Andrievskii in [8] (see also [9] (Th. 2)), in which the right-hand side of (1.5)
contains the additional multiplier
√
ln lnn. However, Corollary 1.1 shows that this multiplier can be
omitted.
Now assume that there is no exterior zero angle (α = 0). Then, in theory, the rate of approx-
imation must increase. The theorems presented below confirm this: the degree of approximation is
measured along the scale of
(
1
n
)µ
but not the scale of
(
1
lnn
)λ
.
Theorem 1.3. Let p > 2 and G ∈ PQ (K, 0, β) for some K > 1 and 0 < β < min
{
p
2
−
− 1;
K2 − 1
1+pK2 + 3K2
}
. Then, for any n ≥ 2, one has
‖ϕp − πn,p‖C(G) ≤ c4n
−γ ,
for every γ such that 0 < γ <
1
pK2
.
ISSN 1027-3190. Укр. мат. журн., 2012, т. 64, № 5
586 F. G. ABDULLAYEV, N. P. ÖZKARTEPE
Theorem 1.4. Let 2 < p < p̃3 and G ∈ PQ
(
K, 0,
p
2
− 1
)
for some K > 1. Then, for any
n ≥ 2, one has
‖ϕp − πn,p‖C(G) ≤ c5
lnn
nγ
,
for every γ such that 0 < γ <
1
pK2
.
Theorem 1.5. Let 2 − 1
2K2
< p < p̃3 and G ∈ PQ (K, 0, β) for some K > 1 and
max
{p
2
− 1; 0
}
< β < min
{
K2 − 1
1+pK2 + 3K2
;
1
4K2
+
p
2
− 1
}
. Then, for any n ≥ 2, one has
‖ϕp − πn,p‖C(G) ≤ c6n
−γ ,
for every γ such that 0 < γ <
1− 2K2(2β + 2− p)
pK2
.
Corollary 1.2. Let p = 2 and G ∈ PQ (K, 0, β) for some K > 1 and 0 ≤ β < min
{
K2 − 1
1+5K2
;
1
4K2
}
. Then, for any n ≥ 2, one has
‖ϕ− πn‖C(G) ≤ c7n
−γ ,
for every γ such that 0 < γ <
1
2K2
− 2β.
Theorem 1.6. Let p > p̃3 and G ∈ PQ (K, 0, β) for some K > 1 and
K2 − 1
1+pK2 + 3K2
< β <
< min
{
p
2
− 1;
2
2 + p
}
. Then, for any n ≥ 2, one has
‖ϕp − πn,p‖C(G) ≤ c8n
−γ ,
for every γ such that 0 < γ <
2− (p+ 2)β
p(1 + β)(K2 + 1)
.
Theorem 1.7. Let p̃3 < p < 2
√
2 and G ∈ PQ
(
K, 0,
p
2
− 1
)
for some K > 1. Then, for any
n ≥ 2, one has
‖ϕp − πn,p‖C(G) ≤ c9
lnn
nγ
,
for every γ such that 0 < γ <
2− (p+ 2)β
p(1 + β)(K2 + 1)
.
Theorem 1.8. Let
3
2
< p < 2
√
2 and G ∈ PQ (K, 0, β) for some 1 < K < K̃1, where K̃1 :=
:= max
{
K :
K2 − 1
1+pK2 + 3K2
< β̃2(p,K)
}
, and max
{
p
2
− 1;
K2 − 1
1+pK2 + 3K2
}
< β < min
{
p0;
β̃2(p,K)
}
. Then, for any n ≥ 2, one has
‖ϕp − πn,p‖C(G) ≤ c10n
−γ ,
for every γ such that 0 < γ <
2− (p+ 2)β
p(1 + β)(K2 + 1)
− 2
p
(2β + 2− p).
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ON THE IMPROVEMENT OF THE RATE OF CONVERGENCE . . . 587
Remark 1.1. a) Theorems 1.1 – 1.8 are unimprovable, and, in some cases, they improve the
corresponding theorems of [4],
b) Theorems 1.1 – 1.8 extend the results of [3, 4, 6, 18, 20] to domains with zero angles.
c) Corollary 1.2 improves the corresponding theorems of [4].
d) Theorems 1.3 – 1.8 show that, as indicated above, the rate of approximation changes in this
case according to the state of β, i.e., since PQ(K,α, β1) ⊂ PQ(K,α, β2) for β2 > β1, we may
claim that the rate of approximation improves as the class PQ(K,α, β) becomes narrower with
respect to β (for the same K and α).
2. Some auxiliary facts. Throughout this paper, the notation “a ≺ b” means that a ≤ c1b for
a constant c1 that does not depend on a and b. The relation “a � b” indicates that c2b ≤ a ≤ c3b,
where c2 and c3 are independent of a and b.
Let G ⊂ C be a finite domain bounded by a Jordan curve L and let w = Φ(z) (w = ϕ̂ (z))
be a conformal mapping of Ω := extG (G) onto ∆ = {w : |w| > 1} (B(0, 1)) normalized by the
conditions Φ (∞) =∞ and Φ′ (∞) > 0
(
ϕ̂(0) = 0 and ϕ̂′(0) > 0
)
.
The (exterior or interior) level curve can be defined for t > 0 as follows:
Lt := {z : |ϕ̂ (z)| = t if t < 1; |Φ (z)| = t if t > 1} , L1 ≡ L.
Denote Gt := intLt, Ωt := extLt, and d(z, L) := inf {|ς − z| : ς ∈ L} .
Let L be a K-quasiconformal curve. Then there exists a quasiconformal reflection y(·) across L
such that y(G) = Ω, y(Ω) = G, and y(·) fixes the points of L. By using the results of [7, p. 76] (see
also [14], Lemma 1), we can find a C(K)-quasiconformal reflection α(·) across L such that
|z1 − α (z)| � |z1 − z| , z1 ∈ L, ε < |z| < 1
ε
,
|αz| � |αz| � 1, ε < |z| < 1
ε
,
|αz| � |αz|2 , |z| < ε, |αz| � |z|−2 , |z| > 1
ε
, (2.1)
∣∣α(z)− z′
∣∣ � ∣∣z − z′∣∣, z′ ∈ L;
Jα := |αz|2 − |αz|2 : Jα � 1
in a certain neighborhood of L.
Lemma 2.1 [1]. Suppose that L is aK-quasiconformal curve, z1 ∈ L, z2, z3 ∈ G∩{z : |z − z1|≤
≤ c1d(z1, LR0)}, wj = ϕ̂(zj) (z2, z3 ∈ Ω ∩ {z : |z − z1| ≤ c2d(z1, Lr0)}, and wj = Φ(zj)), j =
= 1, 2, 3. Then the following assertions are true:
1) the statements |z1 − z2| ≺ |z1 − z3| and |w1 − w2| ≺ |w1 − w3| are equivalent, and so are
|z1 − z2| � |z1 − z3| and |w1 − w2| � |w1 − w3|;
2) if |z1 − z2| ≺ |z1 − z3|, then∣∣∣∣w1 − w3
w1 − w2
∣∣∣∣K−2
≺
∣∣∣∣z1 − z3
z1 − z2
∣∣∣∣ ≺ ∣∣∣∣w1 − w3
w1 − w2
∣∣∣∣K2
.
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588 F. G. ABDULLAYEV, N. P. ÖZKARTEPE
Lemma 2.2. Suppose that L is a K-quasiconformal curve, z1 ∈ L, and z2 ∈ G, w2 = ϕ̂(z2)
(z2 ∈ Ω, w2 = Φ(z2)). Then
|w1 − w2|
2K2
K2+1 ≺ |z1 − z2| ≺ |w1 − w2|
2
K2+1 . (2.2)
Proof. Let L be a K-quasiconformal curve. Then there exists a K2-quasiconformal reflection
y(·) across L. Therefore, L is a k-quasicircle with k =
K2 − 1
K2 + 1
. According to [21, p. 287] and the
estimate for Ψ
′
[11] (Theorem 2.8), we have
|w1 − w2|1+k ≺ |Ψ(w1)−Ψ(w2)| ≺ |w1 − w2|1−k. (2.3)
Lemma 2.3 [3, 12]. Let L be a K-quasiconformal curve. Then, for every z ∈ L and z0 ∈ G,
there exists an arc `(z, z0) in G that joins z and z0 and possesses the following properties:
i) d(ζ, L) � |ζ − z| for every ζ ∈ `(z, z0);
ii) for every ζ1, ζ2 ∈ `(z, z0), if ˜̀(ζ1, ζ2) is a subarc of `(z, z0), then mes ˜̀(ζ1, ζ2) ≺ |ζ1 − ζ2|.
Let Gε :=
{
z : z ∈ G ∩D, d(z, L) <ε <
1
2
d(∂D,L)
}
.
Lemma 2.4 [9]. Let L be a K-quasiconformal curve. Then, for every rectifiable arc ` ⊂ Gε,
one has mes ` � mesα(`).
Lemma 2.5. Let L be a K-quasiconformal curve. Then
a) mesGε � mesα(Gε);
b) mes ϕ̂(Gε) ≺ εδ, δ =
K2 + 1
2K2 .
Proof. a) Let Jα(z) := |αz(z)|2− |αz(z)|2 be a Jacobian of an antiquasiconformal mapping α(·)
across L. Then, according to (2.1), we obtain
mesα(Gε) =
∫∫
Gε
(−Jα(z))dσz �
∫∫
Gε
dσz � mesGε.
b) It is obvious that
mes ϕ̂(Gε) ≤ sup
z∈Gε
π(1− |ϕ̂(z)|2) ≺ sup
z∈Gε
(1− |ϕ̂(z)|). (2.4)
According to (2.2) we get
d(z, L) � (1− |ϕ̂(z)|)
2K2
K2+1 . (2.5)
Using (2.4), (2.5), we complete the proof.
Lemma 2.6. Let L be a K-quasiconformal curve. Then, for every u, 0 < u < R0−1, one has
mesα(G1+u\G) ≺ u
1
K2 .
Proof. The required statement follows from Lemma 2.5 and [16].
3. Some properties of domains G ∈ PQ(K,α, β). Suppose that a domain G ∈ PQ(K,α, β)
is given. Then, for simplicity but without loss of generality, we can assume that α > 0, β > 0, p = 1,
m = 2, z1 = 1, z2 = −1, (−1, 1) ⊂ G, the local coordinate axes in Definition 1.3 are parallel to OX
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ON THE IMPROVEMENT OF THE RATE OF CONVERGENCE . . . 589
and OY in the coordinate system, L1 := {z : z ∈ L, Im z ≥ 0}, and L2 := {z : z ∈ L, Im z ≤ 0}.
Then z0 is taken as an arbitrary point on L2 (or on L1, depending on the chosen direction).
Recall that the domain G ∈ PQ(K,α, β) has interior and exterior zero angles in the nearest
neighborhoods of each of the points z1 = 1 and z2 = −1 respectively. Therefore, following the
argument presented in [9], we can say that, for the domain G ∈ PQ(K,α, β), the function w = Φ(z)
(w = ϕ̂(z)) satisfies the conditions given in Lemma 2.2 in the nearest neighborhood of the point
z2 = −1 (z1 = 1). Thus, using Lemma 2.2, we can easily get
d(z, L) ≺ (|ϕ̂(z)| − 1)
2
K2+1 , |z − 1| ≺ |ϕ̂(z)−ϕ̂(1)|
2
K2+1
∀z ∈M1 := {z ∈ G : |z + 1| > ε1},
d(z, L) ≺ (|Φ(z)| − 1)
2
K2+1 ; |z + 1| ≺ |Φ(z)− Φ(−1)|
2
K2+1
∀z ∈M2 := {z ∈ Ω: |z − 1| > ε2}.
(3.1)
On the other hand, if G ∈ PQ (K,α, β) , then, for points z ∈ Ω \M2 and z ∈ G \M1, using
the properties of the functions w = Φ(z) and w = ϕ̂(z) in the nearest neighborhoods of the points
z1 = 1 and z2 = −1, respectively, we obtain (see [9])
|z − 1| ≺ [− ln |Φ(z)− Φ(1)|]−α
−1
, |z + 1| ≺ [− ln |ϕ̂(z)−ϕ̂(−1)|]−β
−1
. (3.2)
Lemma 3.1 [4]. Let G be Jordan domain such that, for every z ∈ L, there exists an arc γ(z, 0)
in G that joins 0 and z and possesses the following properties:
i) mes γ(ζ1, ζ2) ≺ |ζ1 − ζ2| for every ζ1 , ζ2 ∈ γ(z, 0);
ii) there exists a monotonically increasing function f(t) such that d(ζ, L) � f(|ζ − z|) for every
ζ ∈ γ(z, 0).
Then, for all polynomials Pn(z), degPn ≤ n, Pn(0) = 0, one has
‖Pn‖C(G) ≺
c∫
εn−2
f−2/p(t)dt
‖Pn‖p, p > 0.
Corollary 3.1. Let G ∈ PQ(K,α, β) for some K > 1, α ≥ 0, and β > 0. Then
‖Pn‖C(G) ≺ An‖Pn‖p, (3.3)
where
An =
n
2
p (2β+2−p)
, β >
p
2
− 1,
lnn, β =
p
2
− 1,
c, β <
p
2
− 1.
(3.4)
Remark 3.1. If p = 2, then An =
√
lnn [9].
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590 F. G. ABDULLAYEV, N. P. ÖZKARTEPE
Let G be an arbitrary Jordan domain and let γ ∈ Ω be a rectifiable arc, except for one endpoint
z0 ∈ L, that satisfies the following conditions:
i) mes γ(ζ1, ζ2) ≺ |ζ1 − ζ2| for all ζ1, ζ2 ∈ γ;
ii) there exists a monotonically increasing function g(t) such that d(ζ, L) � g (|ζ − z0|) for all
ζ ∈ γ.
Lemma 3.2 [4]. Suppose that a measurable function f(z) is given on the arc γ and there exists
a monotonically increasing function ν(t), ν(0) = 0, such that |f(ζ)| ≺ ν(|ζ − z0|) for all ζ ∈ γ.
Then the function
Fγ(z) =
∫
γ
f(ζ)
ζ − z
dζ, z /∈ γ,
satisfies the following relation:
‖Fγ‖2p ≺ `
4(1−p)
p
`∫
0
ν(t)dt
2
+
+
`
2(2−p)
p
c`∫
0
ν2(t)
[
1
t
+
h0,1(t)
t2
+ h2,1(t)
]
dt, 1 < p < 2,
c`∫
0
ν2(t)
[
t
4−3p
p +
1
t2
h
2
p
0, p
2
(t) + h
2
p
p, p
2
(t)
]
dt, p ≥ 2,
where
hλ,µ(t) :=
t∫
0
r1−λdr
gµ(r)
.
Corollary 3.2. Let G ∈ PQ(K,α, 0) for some K > 1 and α > 0 and let ν(t) = t
1− 1
p . Then,
for any p > 1, one has
‖Fγ‖2p ≺ `
2−αp
2p , α < min
{
2
(
1− 1
p
)
;
2
p
}
.
Corollary 3.3. Let G ∈ PQ(K, 0, β) for some K > 1 and β > 0 and let ν(t) = t
1− 1+β
p . Then,
for any p > 1, one has
‖Fγ‖2p ≺ `
2−(2+p)β
2p , β <
2
p+ 2
.
We now give conditions under which the function ϕp admits a continuous extension to G.
Lemma 3.3 [4]. Let p > 1 and G ∈ PQ(K,α, β) for some K > 1, α ≥ 0, and β < p − 1.
Then the function ϕp(z) can be extended to G by continuity.
Corollary 3.4. Let p > 1 and G ∈ PQ(K,α, β). Then, for all z ∈ L and ζ ∈ G, one has
|ϕp(z)− ϕp(ζ)| ≺ |z − ζ|1−(1+β) 1
p .
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ON THE IMPROVEMENT OF THE RATE OF CONVERGENCE . . . 591
4. Polynomials approximation in the Ap-norm. Let a domain G ∈ PQ(K,α, β), α > 0,
β > 0, be given. For simplicity but without loss of generality, we can take the domain G as at the
beginning of Sec. 3.
Each Lj , i, j = 1, 2, is a Kj-quasiconformal arc. Let αj(·) be a quasiconformal reflection across
Lj . We also set
γ1
1 :=
{
z = x+ iy : y =
2c1 + c2
3
(x− 1)1+α
}
,
γ2
1 :=
{
z = x+ iy : y =
c1 + 2c2
3
(x− 1)1+α
}
,
γ1
2 := αj
{
z = x+ iy : y =
2c3 + c4
3
(x+ 1)1+β
}
,
γ2
2 := αj
{
z = x+ iy : y =
c3 + 2c4
3
(x+ 1)1+β
}
,
where the constants cj , j = 1, 4, are taken from the definition of the class PQ(K,α, β). It is easy to
check that mes γij(ζ1, ζ2) ≺ |ζ1 − ζ2| for all ζ1, ζ2 ∈ γij , i, j = 1, 2, from Lemma 2.4.
Let 0 < ε < 1 be sufficiently small and let R := 1 + cnε−1. We choose points zij , i, j = 1, 2,
so that they are the intersections of LR and γji and the first points in L̃1
R := {z : z ∈ LR, Im z ≥ 0}
or L̃2
R := LR \ L̃1
R (according to the motion on LR). These points divide LR into four parts:
L1
R := L1
R(z1
1 , z
1
2) (connecting the points z1
1 and z1
2), L2
R := L2
R(z2
2 , z
2
1), L3
R := L3
R(z2
1 , z
1
1), and
L4
R := L4
R(z1
2 , z
1
2). We have LR :=
⋃4
j=1
LjR, γ
j
i (R) = γji ∩ intLR, ΓjR := γj1(R) ∪ γj2(R) ∪ LjR,
and Uj := int
(
ΓjR ∪ Lj
)
, i, j = 1, 2.
We extend the function ϕp to U1 ∪ U2 as follows:
ϕ̃p(z) :=
{
ϕp(z), z ∈ G,
(ϕp ◦ αj)(z), z ∈ Uj .
(4.1)
Then
ϕ̃p,z(z) =
{
0, z ∈ G,
(ϕ
′
p ◦ αj)(z)αj,z, z ∈ Uj .
(4.2)
Using the Cauchy – Pompeiu formula [19, p. 148], we get
ϕp(z) =
1
2πi
∫
Γ1
R∪Γ2
R
ϕ̃p(ζ)
ζ − z
dζ − 1
π
∫∫
U1∪U2
ϕ̃p,ζ(ζ)
ζ − z
dσζ , z ∈ G.
Then, using the notation introduced above, we obtain
ϕp(z) =
1
2πi
∫
LR
fp(ζ)
ζ − z
dζ +
2∑
i,j=1
1
2πi
∫
γji (R)
ϕ̃p(ζ)− ϕp
(
(−1)i
)
ζ − z
dζ − 1
π
∫∫
U1∪U2
ϕ̃p,ζ(ζ)
ζ − z
dσζ , (4.3)
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592 F. G. ABDULLAYEV, N. P. ÖZKARTEPE
where
fp(ζ) :=
ϕ̃p(ζ), ζ ∈ L1
R ∪ L2
R,
ϕ̃p(1), ζ ∈ L3
R,
ϕ̃p(−1), ζ ∈ L4
R.
Lemma 4.1. Let p > 1 andG ∈ PQ(K,α, β) for someK > 1 and 0 < α < min
{
2
(
1− 1
p
)
;
2
p
}
, 0 ≤ β < p0. Then, for any n ≥ 2, one has
∥∥∥ϕp − πn,p∥∥∥p ≺
(
1
lnn
)2−αp
2αp
. (4.4)
Lemma 4.2. Let p > 1 and G ∈ PQ(K, 0, β) for some K > 1 and 0 < β < min
{
p− 1;
K2 − 1
1 + pK2 + 3K2
}
. Then, for any n ≥ 2 and arbitrary small ε > 0, one has
∥∥∥ϕp − πn,p∥∥∥p ≺
(
1
n
) 1−ε
pK2
. (4.5)
Lemma 4.3. Let p > p̃2 and G ∈ PQ(K, 0, β) for some K > 1 and
K2 − 1
1 + pK2 + 3K2
< β <
< p0. Then, for any n ≥ 2 and arbitrary small ε > 0, one has
‖ϕp−πn,p‖p ≺
(
1
n
) 2−(p+2)β−ε
p(1+β)(K2+1)
.
Proof. The proofs of Lemmas 4.1 – 4.3 an similar, and we present them together. Since the first
term in (4.3) is analytic in G, there is a polynomial Pn(z) of degree not higher than n [24, p. 142]
such that ∣∣∣∣∣∣∣
1
2πi
∫
LR
fp(ζ)
(ζ − z)2
dζ − P ′n(z)
∣∣∣∣∣∣∣ ≺
1
n
, z ∈ G. (4.6)
Hence, using (4.3), we get ∥∥ϕ′p − P ′n∥∥p ≺
≺ 1
n
+
2∑
i,j=1
∥∥∥∥∥∥∥∥
∫
γji (R)
ϕ̃p(ζ)− ϕp
(
(−1)i
)
ζ − z
dζ
∥∥∥∥∥∥∥∥
p
+
∥∥∥∥∥∥
∫∫
U1∪U2
ϕ̃,ζp(ζ)
ζ − z
dσζ
∥∥∥∥∥∥
p
=:
1
n
+
5∑
k=1
Jk. (4.7)
For all p > 1 and β < p− 1, we have
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ON THE IMPROVEMENT OF THE RATE OF CONVERGENCE . . . 593
|ϕ̃p(ζ)− ϕp(−1)| = |ϕp(αj(ζ))− ϕp(−1)| ≺ |ζ + 1|1−
1+β
p , ζ ∈ γ1
j (R), (4.8)
|ϕ̃p(ζ)− ϕp(1)| = |ϕp(αj(ζ))− ϕp(1)| ≺ |ζ − 1|1−
1
p , ζ ∈ γ2
j (R), (4.9)
by virtue of relation (2.1) and Corollary 3.4. Therefore, for every α < min
{
2
(
1− 1
p
)
;
2
p
}
and
β < p0, we obtain ∥∥∥∥∥∥∥
∫
γ1i (R)
ϕ̃p(ζ)− ϕp(−1)
ζ − z
dζ
∥∥∥∥∥∥∥
p
≺ `
2−(2+p)β
2p
i,2 , (4.10)
∥∥∥∥∥∥∥
∫
γ2i (R)
ϕ̃p(ζ)− ϕp(1)
ζ − z
dζ
∥∥∥∥∥∥∥
p
≺ `
2−αp
2p
i,1 (4.11)
by virtue of Corollaries 3.2 and 3.3 and the fact that `i,j = mes γji (R), i, j = 1, 2. On the other
hand, according to [21] (Lemma 9), we have
d(zj , L
j) ≺
(
1
n
) 2−ε
K2+1
.
Then, using (2.1), (3.1), and (3.2), we get
`i,j ≺
∣∣zij − (−1)i
∣∣ ≺
d(zi2, L
i)
1
1+β ≺
(
1
n
) 2−ε
(1+β)(K2+1)
∀ε > 0, i = 1, 2,
d(zi1, L
i) ≺ (lnn)−α
−1
, i = 1, 2.
Thus, it follows from (4.10) and (4.11) that∥∥∥∥∥∥∥
∫
γ1i (R)
ϕ̃p(ζ)− ϕp(−1)
ζ − z
dζ
∥∥∥∥∥∥∥
p
≺
(
1
n
) 2−(p+2)β−ε
p(1+β)(K2+1)
, β < min
{
p− 1;
2
p+ 2
}
, (4.12)
∥∥∥∥∥∥∥
∫
γ2i (R)
ϕ̃p(ζ)− ϕp(1)
ζ − z
dζ
∥∥∥∥∥∥∥
p
≺
(
1
lnn
)2−αp
2αp
, 0 < α < min
{
2
(
1− 1
p
)
;
2
p
}
. (4.13)
Since the Hilbert transformation
(Tf)(z) := − 1
π
∫ ∫
f(ζ)
(ζ − z)2
dσζ
is a bounded linear operator from Lp into itself for p > 1, we have∫∫
U1∪U2
∣∣∣ϕ̃p,,ζ(ζ)
∣∣∣pdσζ � ∫∫
U1∪U2
∣∣ϕ′(αj(ζ))
∣∣2dσζ ≺
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594 F. G. ABDULLAYEV, N. P. ÖZKARTEPE
≺
2∑
j=1
∫∫
α(Uj)
∣∣ϕ′(ζ)
∣∣2dσζ ≺ 2∑
j=1
mes ϕ (αj(Uj)).
According to (4.2) and (2.1), the Calderon – Zygmund inequality [7, p. 89] yields
J5 ≺
2∑
j=1
mesϕ (αj(Uj))
1
p
. (4.14)
For sufficiently large c and small ε0 <
1
2
, we set
V j
1 :=
{
ζ : ζ ∈ αj(Uj), |ζ − 1| ≤ c(lnn)−α
−1
}
,
V j
2 := αj(Uj)\V j
1 , j = 1, 2, α > 0,
Uε0 := {ζ : |ζ + 1| ≤ ε0}; Ṽ 1
j := Uj ∩ Uε0 , j = 1, 2, α = 0.
Then, by virtue of Lemma 2.6, we obtain
mesϕ(V
j)
1 ≺ (lnn)−α
−1
,
mesϕ(αj(Ṽ
1
j ) ≺ n
ε−2
K2+1
δ
= n
ε−1
K2 ,
mesϕ(αj(Uj\Ṽ 1
j )) ≺ n
ε−1
K2 ∀ε > 0,
and
J5 ≺
(
1
lnn
) 1
αp
, α > 0,
(
1
n
)1− ε
pK2
, ∀ε > 0, α = 0.
(4.15)
Using (4.8), (4.9), (4.12), (4.13), and (4.15), we get
‖ϕp − Pn‖p ≺
1
n
+
(
1
lnn
)2−αp
2αp
(
1
n
) 2−(p+2)β−ε
p(1+β)(K2+1)
+
+
(
1
lnn
) 1
αp
, 0 < α < min
{
2
(
1− 1
p
)
;
2
p
}
, β ≥ 0,
(
1
n
) 1−ε
pK2
∀ε > 0, α = 0, β > 0.
(4.16)
Case 1. Let p > 1, 0 < α < min
{
2
(
1− 1
p
)
;
2
p
}
, β ≥ 0. Then
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ON THE IMPROVEMENT OF THE RATE OF CONVERGENCE . . . 595
‖ϕp−Pn‖p ≺
1
n
+
(
1
lnn
)2−αp
2αp
+
(
1
lnn
) 1
αp
≺
(
1
lnn
)2−αp
2αp
. (4.17)
Case 2. Let p > 1, α = 0, β > 0. Then
‖ϕp−Pn‖p ≺
1
n
+
(
1
n
) 2−(p+2)β−ε
p(1+β)(K2+1)
+
(
1
n
) 1−ε
pK2
≺
≺
(
1
n
) 1−ε
pK2
, β < min
{
p0,
K2 − 1
1 + pK2 + 3K2
}
,
(
1
n
) 2−(p+2)β−ε
p(1+β)(K2+1)
,
K2 − 1
1 + pK2 + 3K2
≤ β < p0,
(4.18)
for any p > 1 and arbitrary small ε > 0.
Case 3. Let p > 1, α = 0, 0 < β < min
{
p0,
K2 − 1
1 + pK2 + 3K2
}
. Then
‖ϕp−Pn‖p ≺
(
1
n
) 1−ε
pK2
. (4.19)
Case 4. It is clear that
min
{
p− 1;
2
p+ 2
}
=
p− 1 if p < p1 :=
√
17− 1
2
,
2
p+ 2
if p ≥ p1.
Let p > p̃2, K > 1, α = 0,
K2 − 1
1 + pK2 + 3K2
< β < p0. Then
‖ϕp−Pn‖p ≺
(
1
n
) 2−(p+2)β−ε
p(1+β)(K2+1)
, (4.20)
for arbitrary small ε > 0. If P̃n(z) := Pn(z)−Pn(0)+z[1−P ′n(0)], then it is easy to see that relations
(4.17) – (4.20) are also satisfied for P̃n(z), P̃n(0) = 0, and P̃ ′n(0) = 1. Thus, we can complete the
proof of Lemmas 4.1 – 4.3 considering the extremal properties of πn,p(z).
5. Proof of Theorems 1.1 – 1.8. We use the known method given in [3, 4, 9].
Lemma 5.1. Suppose thatG is a Jordan domain such that, for {αn} ↓, {βn}↑, {γn := αnβn} ↓,
and n→∞, under the condition
‖ϕp − πn,p‖p ≺ αn, n = 2, 3, . . . ,
one has
‖Pn‖C(G)
≺ βn
∥∥P ′n∥∥p, n = 1, 2, . . . ,
for all polynomials Pn(z) of degree not higher than n with Pn(0) = 0. Also assume that there exists
a sequence of indices {nk}∞k=1 such that βnk+1
≤ cβnk and γnk+1
≤ εγnk for all k = 1, 2, . . . and
some c ≥ 1 and 0 < ε < 1. Then
‖ϕp − πn,p‖C(G) ≺ γn.
ISSN 1027-3190. Укр. мат. журн., 2012, т. 64, № 5
596 F. G. ABDULLAYEV, N. P. ÖZKARTEPE
The proof of this lemma is similar to that of [9] (Lemma 15). Therefore, by taking αn from
Lemmas 4.1 – 4.3 and βn from Corollary 3.1 and combining the results for G ∈ PQ(K,α, β) in the
case α = 0 or β = 0, we prove Theorems 1.1 – 1.8.
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Received 02.12.11
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