Polynomial inequalities in regions with interior zero angles in the Bergman space
We investigate the order of growth of the moduli of arbitrary algebraic polynomials in the weighted Bergman space $A_p(G, h),\; p > 0$, in regions with interior zero angles at finitely many boundary points. We obtain estimations for algebraic polynomials in bounded regions with piecewise smoo...
Збережено в:
| Дата: | 2018 |
|---|---|
| Автори: | , , , , , |
| Формат: | Стаття |
| Мова: | Англійська |
| Опубліковано: |
Institute of Mathematics, NAS of Ukraine
2018
|
| Онлайн доступ: | https://umj.imath.kiev.ua/index.php/umj/article/view/1559 |
| Теги: |
Додати тег
Немає тегів, Будьте першим, хто поставить тег для цього запису!
|
| Назва журналу: | Ukrains’kyi Matematychnyi Zhurnal |
| Завантажити файл: |  |
Репозитарії
Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860507360764624896 |
|---|---|
| author | Abdullayev, F. G. Balci, S. Imash, kyzy M. Абдуллаєв, Ф. Г. Балчи, С. Імаш, кизи М. |
| author_facet | Abdullayev, F. G. Balci, S. Imash, kyzy M. Абдуллаєв, Ф. Г. Балчи, С. Імаш, кизи М. |
| author_sort | Abdullayev, F. G. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2019-12-05T09:18:29Z |
| description | We investigate the order of growth of the moduli of arbitrary algebraic polynomials in the weighted Bergman space
$A_p(G, h),\; p > 0$, in regions with interior zero angles at finitely many boundary points. We obtain estimations for algebraic
polynomials in bounded regions with piecewise smooth boundary. |
| first_indexed | 2026-03-24T02:08:05Z |
| format | Article |
| fulltext |
UDC 517.5
S. Balci, M. Imash kyzy (Kyrgyz-Turkish Manas Univ., Bishkek, Kyrgyzstan),
F. G. Abdullayev (Kyrgyz-Turkish Manas Univ., Bishkek, Kyrgyzstan and Mersin Univ., Turkey)
POLYNOMIAL INEQUALITIES IN REGIONS
WITH INTERIOR ZERO ANGLES IN THE BERGMAN SPACE*
ПОЛIНОМIАЛЬНI НЕРIВНОСТI В ОБЛАСТЯХ
IЗ ВНУТРIШНIМИ НУЛЬОВИМИ КУТАМИ У ПРОСТОРI БЕРГМАНА
We investigate the order of growth of the moduli of arbitrary algebraic polynomials in the weighted Bergman space
Ap(G, h), p > 0, in regions with interior zero angles at finitely many boundary points. We obtain estimations for algebraic
polynomials in bounded regions with piecewise smooth boundary.
Вивчається порядок зростання модулiв довiльних алгебраїчних полiномiв у ваговому просторi Бергмана Ap(G, h),
p > 0, в областях iз внутрiшнiми нульовими кутами у скiнченнiй кiлькостi точок. Отримано оцiнки для алгебраїчних
полiномiв в обмежених областях з кусково-гладкою межею.
1. Introduction and main results. Let G \subset \BbbC be a finite region, with 0 \in G, bounded by a Jordan
curve L := \partial G, \Omega := \mathrm{e}\mathrm{x}\mathrm{t}L := \BbbC \setminus G, where \BbbC := \BbbC \cup \{ \infty \} , \Delta := \{ w : | w| > 1\} and let \wp n denote
the class of arbitrary algebraic polynomials Pn(z) of degree at most n \in \BbbN . Let w = \Phi (z) be the
univalent conformal mapping of \Omega onto the \Delta with usual normalization, and \Psi := \Phi - 1. For t \geq 1,
z \in \BbbC , we us set
Lt := \{ z : | \Phi (z)| = t\} (L1 \equiv L), Gt := \mathrm{i}\mathrm{n}\mathrm{t}Lt, \Omega t := \mathrm{e}\mathrm{x}\mathrm{t}Lt.
Let \{ zj\} mj=1 be a fixed system of distinct points on curve L, located in the positive direction. For
some fixed R0, 1 < R0 < \infty , and z \in GR0 , consider a so-called generalized Jacobi weight function
h (z) being defined as follows:
h(z) := h0(z)
m\prod
j=1
| z - zj | \gamma j , z \in GR0 , (1.1)
where \gamma j > - 2, for all j = 1, 2, . . . ,m, and the function h0 is uniformly separated from zero in
GR0 , i.e., there exists a constant c0 := c0(GR0) > 0 such that, for all z \in GR0
h0(z) \geq c0 > 0.
For any p > 0 and for Jordan region G, lets define
\| Pn\| p := \| Pn\| Ap(h,G) :=
\left( \int \int
G
h(z) | Pn(z)| p d\sigma z
\right) 1/p
< \infty , 0 < p < \infty ,
\| Pn\| \infty := \| Pn\| A\infty (1,G) := \| Pn\| C(G) , p = \infty ,
(1.2)
where \sigma z is the two-dimensional Lebesgue measure. Clearly, \| \cdot \| Ap
is the quasinorm (i.e., a norm
for 1 \leq p \leq \infty and a p-norm for 0 < p < 1).
* This work is supported by Kyrgyz-Turkey Manas University (project No. 2016 FBE 13).
c\bigcirc S. BALCI, M. IMASH KYZY, F. G. ABDULLAYEV, 2018
318 ISSN 1027-3190. Укр. мат. журн., 2018, т. 70, № 3
POLYNOMIAL INEQUALITIES IN REGIONS WITH INTERIOR ZERO ANGLES IN THE BERGMAN SPACE 319
In this work, we study the following Nikol’skii-type inequality:
\| Pn\| \infty \leq c1\lambda n(G, h, p) \| Pn\| p , (1.3)
where c1 = c1(G, h, p) > 0 is a constant independent of n and Pn, and \lambda n(G, h, p) \rightarrow \infty , n \rightarrow \infty ,
depending on the geometrical properties of region G, weight function h and of p. The estimate of
(1.3)-type for some (G, p, h) was investigated in [21, p. 122 – 133], [15], [20] (Sect. 5.3), [2 – 8, 14,
23] (see also references therein).
Further, analogous of (1.3) for some regions and the weight function h(z) were obtained: in
[8] for p > 1 and for regions bounded by piecewise Dini-smooth boundary without cusps; in [9]
(h(z) \equiv 1) and [11] (h(z) \not = 1) for p > 0 and for regions bounded by quasiconformal curve; in [7]
for p > 1 and for regions bounded by piecewise smooth curve without cusps; in [10] for p > 0 and
for regions bounded by asymptotically conformal curve.
In this work, we investigate similar problems for z \in G in regions bounded by piecewise smooth
curve having interior zero angles and for weight function h (z) , defined in (1.1) and for p > 0.
Let us give some definitions and notations that will be used later in the text.
Following [18, p. 97, 22], the Jordan curve (or arc) L is called K -quasiconformal (K \geq 1), if
there is a K -quasiconformal mapping f of the region D \supset L such that f(L) is a circle (or line
segment).
Let S be rectifiable Jordan curve or arc and let z = z(s), s \in [0, | S| ] , | S| := \mathrm{m}\mathrm{e}\mathrm{s}S, be the
natural parametrization of S.
Definition 1.1. We say that a Jordan curve or arc S \in C\theta , if S has a continuous tangent
\theta (z) := \theta (z(s)) at every point z(s). We will write a region G \in C\theta , if \partial G \in C\theta .
According to [22], we have the following fact.
Corollary 1.1. If S \in C\theta , then S is (1 + \varepsilon )-quasiconformal for arbitrary small \varepsilon > 0.
According to the
”
three-point” criterion [12, p. 100], every piecewise smooth curve (without any
cusps) is quasiconformal.
Now we define a new class of regions with piecewise smooth boundary, where having exterior
corners and interior cusps simultaneously.
Throughout this paper, c, c0,c1, c2, . . . are positive and \varepsilon 0, \varepsilon 1, \varepsilon 2, . . . are sufficiently small positive
constants (generally, different in different relations), which depend on G in general. Also note that,
for any k \geq 0 and m > k, notation j = k,m denotes j = k, k + 1, . . . ,m.
Definition 1.2 [7]. We say that a Jordan region G \in C\theta (\lambda 1, . . . , \lambda m), 0 < \lambda j \leq 2, j = 1,m,
if L = \partial G consists of the union of finite smooth arcs \{ Lj\} mj=1 , such that they have exterior (with
respect to G) angles \lambda j\pi , 0 < \lambda j \leq 2, at the corner points \{ zj\} mj=1 \in L, where two arcs meet.
Let m1 be the number of exterior angles, which are not cusps, and thus m - m1 is the number of
cusps. It is clear from Definition 1.2, that each region G \in C\theta (\lambda 1, . . . , \lambda m), 0 < \lambda j \leq 2, j = 1,m,
may have exterior nonzero \lambda j\pi , 0 < \lambda j < 2, angles at the points \{ zj\} m1
j=1 \in L, and interior zero
angles (\lambda j = 2) at the the points \{ zj\} mj=m1+1 \in L. If m1 = m = 0, then the region G doesn’t
have such angles, and in this case we will write G \in C\theta ; if m1 = m \geq 1, then G has only \lambda i\pi ,
0 < \lambda i < 2, i = 1,m1, exterior nonzero angles; if m1 = 0 and m \geq 1, then G has only interior
zero angles, and in this case we will write G \in C\theta (2, . . . , 2).
ISSN 1027-3190. Укр. мат. журн., 2018, т. 70, № 3
320 S. BALCI, M. IMASH KYZY, F. G. ABDULLAYEV
Throughout this work, we will assume that the points \{ zj\} mj=1 \in L defined in (1.1) and in
Definition 1.2 are identical and wj := \Phi (zj).
For the simplicity of exposition and in order to avoid cumbersome calculations, without loss of
generality, we will take m1 = 1, m = 2. Then, after this assumption, in the future we will have
region G \in C\theta (\lambda 1, 2), 0 < \lambda 1 < 2, such that at the point z1 \in L the region G have exterior nonzero
angle \lambda 1\pi , 0 < \lambda 1 < 2, and at the point z2 \in L-interior zero angle.
Now we can state our new results.
Theorem 1.1. Let p > 0, G \in C\theta (\lambda 1, \lambda 2) for some 0 < \lambda 1, \lambda 2 < 2, h(z) be defined as in
(1.1). Then, for any Pn \in \wp n, n \in \BbbN , \gamma j > - 2, j = 1, 2, and arbitrary small \varepsilon > 0, we have
\| Pn\| \infty \leq c1\mu n,1 \| Pn\| p , (1.4)
where c1 = c1(G, \gamma 1, \gamma 2, \lambda 1, \lambda 2, p, \varepsilon ) > 0 is the constant, independent of z and n, and
\mu n,1 :=
\left\{
n
(2+\widetilde \gamma )\cdot \widetilde \lambda
p , if (2 + \gamma ) \cdot \widetilde \lambda > 1,
(n \mathrm{l}\mathrm{n}n)1/p, if (2 + \gamma ) \cdot \widetilde \lambda = 1,
n1/p, if (2 + \gamma ) \cdot \widetilde \lambda < 1,
(1.5)
\gamma := \mathrm{m}\mathrm{a}\mathrm{x} \{ \gamma 1, \gamma 2\} , \widetilde \gamma j := \mathrm{m}\mathrm{a}\mathrm{x} \{ 0, \gamma j\} ; \widetilde \lambda := \mathrm{m}\mathrm{a}\mathrm{x}
\Bigl\{ \widetilde \lambda 1, \widetilde \lambda 2
\Bigr\}
, \widetilde \lambda j := \mathrm{m}\mathrm{a}\mathrm{x} \{ 1;\lambda j\} + \varepsilon .
Now, we assume that the curve L at both points have interior zero angles. In this case we obtain
the following theorem.
Theorem 1.2. Let p > 0, G \in C\theta (2, 2), h(z) be defined as in (1.1). Then, for any Pn \in \wp n,
n \in \BbbN , \gamma j > - 2, j = 1, 2, we have
\| Pn\| \infty \leq c2\mu n,2 \| Pn\| p , (1.6)
where c2 = c2(G, \gamma 1, \gamma 2, p) > 0 is the constant, independent of z and n, and \widetilde \gamma is defined as in (1.5)
and
\mu n,2 :=
\left\{
n
2(2+\widetilde \gamma )
p , if \gamma > - 3
2
,
(n \mathrm{l}\mathrm{n}n)1/p , if \gamma = - 3
2
,
n1/p, if \gamma < - 3
2
.
(1.7)
Now we will estimate of | Pn(z)| at the critical points zj , j = 1, 2.
Theorem 1.3. Let p > 0, G \in C\theta (\lambda 1, 2) for some 0 < \lambda 1 < 2, h(z) be defined as in (1.1).
Then, for any Pn \in \wp n, n \in \BbbN , \gamma j > - 2, j = 1, 2, and arbitrary small \varepsilon > 0, we obtain
| Pn(zj)| \leq c3\mu n,3 \| Pn\| p , (1.8)
where c3 = c3(G, \gamma 1, \gamma 2, \lambda 1, p, \varepsilon ) > 0 is the constant, independent of z and n;
ISSN 1027-3190. Укр. мат. журн., 2018, т. 70, № 3
POLYNOMIAL INEQUALITIES IN REGIONS WITH INTERIOR ZERO ANGLES IN THE BERGMAN SPACE 321
\mu n,3 :=
\left\{
n
(2+\gamma 1)\cdot \widetilde \lambda 1
p , if \gamma 1 >
1\widetilde \lambda 1
- 2,
(n \mathrm{l}\mathrm{n}n)1/p, if \gamma 1 =
1\widetilde \lambda 1
- 2,
n1/p, if \gamma 1 <
1\widetilde \lambda 1
- 2,
for j = 1, and
\mu n,3 :=
\left\{
n
2(2+\gamma 2)
p , if \gamma 2 > - 3
2
,
(n \mathrm{l}\mathrm{n}n)1/p , if \gamma 2 = - 3
2
,
n1/p, if \gamma 2 < - 3
2
,
for j = 2.
Combining Theorems 1.1 and 1.2 with the estimate for | Pn(z)| , z \in \Omega , in [25] (Corollaries 1.2
and 1.3), we can obtain estimation for | Pn(z)| in the whole complex plane.
For z \in \BbbC and M \subset \BbbC , we set that d(z,M) = \mathrm{d}\mathrm{i}\mathrm{s}\mathrm{t}(z,M) := \mathrm{i}\mathrm{n}\mathrm{f} \{ | z - \zeta | : \zeta \in M\} and
R := 1 +
\varepsilon 0
n
.
Corollary 1.2. Under the conditions of Theorem 1.1, the following is true:
| Pn(z)| \leq c4 \| Pn\| p
\left\{
\mu n,1, z \in GR,
| \Phi (z)| n+1
d2/p(z, LR)
\mu n,4, z \in \Omega R,
(1.9)
where c4 = c4(G, \gamma 1, \gamma 2, \lambda 1, p, \varepsilon ) > 0 is the constant, independent of z and n; \mu n,1 is defined as in
(1.5) and
\mu n,4 :=
\left\{
n
\widetilde \gamma \cdot \widetilde \lambda 1
p , if \gamma \cdot \widetilde \lambda 1 > 1,
(n \mathrm{l}\mathrm{n}n)1/p, if \gamma \cdot \widetilde \lambda 1 = 1,
n1/p, if \gamma \cdot \widetilde \lambda 1 < 1.
Corollary 1.3. Under the conditions of Theorem 1.2, the following is true:
| Pn(z)| \leq c3 \| Pn\| p
\left\{
\mu n,2, z \in GR,
| \Phi (z)| n+1
d2/p(z, LR)
\mu n,5, z \in \Omega R,
(1.10)
where c5 = c5(G, \gamma 1, \gamma 2, p) > 0 is the constant, independent of z and n; \mu n,2 is defined as in (1.7)
and
ISSN 1027-3190. Укр. мат. журн., 2018, т. 70, № 3
322 S. BALCI, M. IMASH KYZY, F. G. ABDULLAYEV
\mu n,5 :=
\left\{
n
2\widetilde \gamma
p , if \gamma >
1
2
,
(n \mathrm{l}\mathrm{n}n)1/p, if \gamma =
1
2
,
n1/p, if \gamma <
1
2
.
The sharpness of the estimations (1.4), (1.6), (1.8), (1.9) and (1.10), can be discussed by compa-
ring them with the following result.
Remark 1.1 ([9] (Theorem 1.15), [2]). (a) For any n \in \BbbN there exist polynomials Q\ast
n, T
\ast
n \in \wp n
such that for unit disk B and weight function h\ast (z) = | z - z1| 2 the following is true:
| Q\ast
n(z)| \geq c6n \| Q\ast
n\| A2(B) for all z \in B,
| T \ast
n(z1)| \geq c7n
2 \| T \ast
n\| A2(h\ast ,B .
(b) For any n \in \BbbN there exists a polynomial P \ast
n \in \wp n, region G\ast
1 \subset \BbbC , compact F \ast \Subset \Omega \setminus G\ast
1
and constant c8 = c8(G
\ast
1, F
\ast ) > 0 such that
| P \ast
n(z)| \geq c8
\surd
n
d(z, L)
\| P \ast
n\| A2(G\ast
1)
| \Phi (z)| n+1 for all z \in F \ast .
2. Some auxiliary results. Throughout this work, for the nonnegative functions a > 0 and
b > 0, we will use the notations a \preceq b (order inequality), if a \leq cb and a \asymp b are equivalent to
c1a \leq b \leq c2a for some constants c, c1, c2 (independent of a and b), respectively.
Lemma 2.1 [1]. Let L be a K -quasiconformal curve, z1 \in L, z2, z3 \in \Omega \cap \{ z : | z - z1| \preceq
\preceq d(z1, Lr0)\} ; wj = \Phi (zj), j = 1, 2, 3. Then:
(a) The statements | z1 - z2| \preceq | z1 - z3| and | w1 - w2| \preceq | w1 - w3| are equivalent. So, the
statements | z1 - z2| \asymp | z1 - z3| and | w1 - w2| \asymp | w1 - w3| also are equivalent.
(b) If | z1 - z2| \preceq | z1 - z3| , then\bigm| \bigm| \bigm| \bigm| w1 - w3
w1 - w2
\bigm| \bigm| \bigm| \bigm| K2
\preceq
\bigm| \bigm| \bigm| \bigm| z1 - z3
z1 - z2
\bigm| \bigm| \bigm| \bigm| \preceq \bigm| \bigm| \bigm| \bigm| w1 - w3
w1 - w2
\bigm| \bigm| \bigm| \bigm| K - 2
,
where 0 < r0 < 1 is constants, depending on G.
Corollary 2.1. Under the assumptions of Lemma 2.1, for z3 \in Lr0 ,
| w1 - w2| K
2
\preceq | z1 - z2| \preceq | w1 - w2| K
- 2
.
Corollary 2.2. If L \in C\theta , then, for all \varepsilon > 0,
| w1 - w2| 1+\varepsilon \preceq | z1 - z2| \preceq | w1 - w2| 1 - \varepsilon .
For 0 < \delta j < \delta 0 :=
1
4
\mathrm{m}\mathrm{i}\mathrm{n} \{ | z1 - z2| \} , we put \Omega (zj , \delta j) := \Omega \cap \{ z : | z - zj | \leq \delta j\} , \delta :=
:= \mathrm{m}\mathrm{i}\mathrm{n}1\leq j\leq m \delta j , \Omega (\delta ) :=
\bigcup 2
j=1
\Omega (zj , \delta ), \widehat \Omega := \Omega \setminus \Omega (\delta ). Additionally, let \Delta j := \Phi (\Omega (zj , \delta )),
\Delta (\delta ) :=
\bigcup m
j=1
\Phi (\Omega (zj , \delta )), \widehat \Delta (\delta ) := \Delta \setminus \Delta (\delta ).
ISSN 1027-3190. Укр. мат. журн., 2018, т. 70, № 3
POLYNOMIAL INEQUALITIES IN REGIONS WITH INTERIOR ZERO ANGLES IN THE BERGMAN SPACE 323
The following lemma is a consequence of the results given in [17, 19, 27], and of estimate for
the | \Psi \prime | (see, for example, [13], Theorem 2.8) for 0 < \lambda j < 2, j = 1,m:\bigm| \bigm| \Psi \prime (\tau )
\bigm| \bigm| \asymp d(\Psi (\tau ) , L)
| \tau | - 1
. (2.1)
Lemma 2.2 [27]. Let G \in C\theta (\lambda 1, . . . , \lambda m), 0 < \lambda j < 2, j = 1,m. Then, for all \varepsilon > 0:
i) for any w \in \Delta j , | w - wj | \lambda j+\varepsilon \preceq | \Psi (w) - \Psi (wj)| \preceq | w - wj | \lambda j - \varepsilon , | w - wj | \lambda j - 1+\varepsilon \preceq
\preceq | \Psi \prime
(w)| \preceq | w - wj | \lambda j - 1 - \varepsilon ,
ii) for any w \in \Delta \setminus \Delta j , (| w| - 1)1+\varepsilon \preceq d(\Psi (w), L)| \preceq (| w| - 1)1 - \varepsilon , (| w| - 1)\varepsilon \preceq | \Psi \prime (w)| \preceq
\preceq (| w| - 1) - \varepsilon .
Let \{ zj\} mj=1 be a fixed system of the points on L and the weight function h (z) be defined as
in (1.1).
Lemma 2.3 [5]. Let L be a K -quasiconformal curve, h(z) is defined in (1.1). Then, for arbi-
trary Pn(z) \in \wp n, any R > 1 and n = 1, 2, . . . , we have
\| Pn\| Ap(h,GR) \preceq \widetilde Rn+ 1
p \| Pn\| Ap(h,G) , p > 0, (2.2)
where \widetilde R = 1 + c(R - 1) and c is independent of n and R.
Lemma 2.4. Let G \in C\theta (\lambda 1, . . . , \lambda m), 0 < \lambda j \leq 2, j = 1,m. Then, for arbitrary Pn(z) \in \wp n
and any p > 0, we obtain
\| Pn\| Ap(h,G1+c/n)
\preceq \| Pn\| Ap(h,G) . (2.3)
Proof. For 0 < \lambda j < 2, j = 1,m, this follows from Lemma 2.4 and Corollary 1.1 and from the
fact, that according to the
”
three-point” criterion [18, p. 100], any piecewise smooth curve without
cusps is a quasiconformal. If \lambda j = 2, for all j = 1,m, then the region G have exterior 2\pi angles
(i.e., interior cusps) at the every point zj , j = 1,m. Then in the neighborhood of the this points the
region G have a boundary with outside wedge. Therefore, as well known from theory of conformal
mappings, the distance from the corner point to the level curve LR is less than of such distance from
the other points. Furthermore, the area between boundary L and level curve LR in the neighborhood
of the such corners will be smaller than such in the case of without angles.
3. Proof of theorems. 3.1. Proof of Theorems 1.1 and 1.2. Suppose that G \in C\theta (\lambda 1; 2), for
some 0 < \lambda 1 < 2 and h(z) be defined as in (1.1). Let \{ \xi j\} , 1 \leq j \leq m \leq n, be the zeros (if any
exist) of Pn(z) lying on \Omega . Lets define the function Blaschke with respect to the zeros \{ \xi j\} of the
polynomial Pn(z) :
\widetilde Bj(z) :=
\Phi (z) - \Phi (\xi j)
1 - \Phi (\xi j)\Phi (z)
, z \in \Omega , (3.1)
and let
Bm(z) :=
m\prod
j=1
\widetilde Bj(z), z \in \Omega . (3.2)
It is easy that the
Bm(\xi j) = 0, | Bm(z)| \equiv 1, z \in L; | Bm(z)| < 1, z \in \Omega . (3.3)
ISSN 1027-3190. Укр. мат. журн., 2018, т. 70, № 3
324 S. BALCI, M. IMASH KYZY, F. G. ABDULLAYEV
Then, for each \varepsilon 1, 0 < \varepsilon 1 < 1, there exists circle
\Bigl\{
w : | w| = R1 := 1 + \varepsilon 2, 0 < \varepsilon 2 <
\varepsilon 1
n
\Bigr\}
such that
for any j = 1, 2, the following is holds:\bigm| \bigm| \bigm| \widetilde Bj(\zeta )
\bigm| \bigm| \bigm| > 1 - \varepsilon 2, \zeta \in LR1 .
So, from (3.2), we get
| Bm(\zeta )| > (1 - \varepsilon 2)
m \succeq 1, \zeta \in LR1 . (3.4)
For any p > 0 and z \in \Omega let us set
Qn,p (z) :=
\biggl[
Pn (z)
Bm(z)\Phi n+1(z)
\biggr] p/2
. (3.5)
The function Qn,p (z) is analytic in \Omega , continuous on \Omega , Qn,p (\infty ) = 0 and does not have zeros in
\Omega . We take an arbitrary continuous branch of the Qn,p (z) and for this branch, we maintain the same
designation. According to Cauchy integral representation for the unbounded region \Omega , we have
Qn,p (z) = - 1
2\pi i
\int
LR1
Qn,p (\zeta )
d\zeta
\zeta - z
, z \in \Omega R1 . (3.6)
According to (3.1) – (3.5), we get
| Pn (z)| p/2 =
\bigm| \bigm| Bm(z)\Phi n+1(z)
\bigm| \bigm| p/2
2\pi d(z, LR1)
\int
LR1
\bigm| \bigm| \bigm| \bigm| Pn (\zeta )
Bm(\zeta )\Phi n+1(\zeta )
\bigm| \bigm| \bigm| \bigm| p/2 | d\zeta | \preceq
\preceq
\bigm| \bigm| \Phi n+1(z)
\bigm| \bigm| p/2 \int
LR1
| Pn (\zeta )| p/2
| d\zeta |
| \zeta - z|
. (3.7)
Multiplying the numerator and the denominator of the last integrand by h1/2(\zeta ), replacing the
variable w = \Phi (z) and applying the Hölder inequality, we obtain\left( \int
LR1
| Pn (\zeta )|
p
2 | d\zeta |
\right)
2
\leq
\int
| t| =R1
h(\Psi (t)) | Pn (\Psi (t))| p
\bigm| \bigm| \Psi \prime (t)
\bigm| \bigm| 2 | dt| \int
| t| =R1
| dt|
h(\Psi (t)) | \Psi (t) - \Psi (w)| 2
\leq
\leq
\int
| t| =R1
h(\Psi (t)) | Pn (\Psi (t))| p
\bigm| \bigm| \Psi \prime (t)
\bigm| \bigm| 2 | dt| \int
| t| =R1
| dt|
h(\Psi (t)) | \Psi (t) - \Psi (w)| 2
=
=
\int
| t| =R1
| fn,p(t)| p | dt|
\int
| t| =R1
| dt|
h(\Psi (t)) | \Psi (t) - \Psi (w)| 2
=: AnDn(w), (3.8)
where fn,p(t) := h1/p(\Psi (t))Pn(\Psi (t))(\Psi \prime (t))2/p, | t| = R1.
To estimate integral An, we separate the circle | t| = R1 to n equal parts \delta n with \mathrm{m}\mathrm{e}\mathrm{s} \delta n =
2\pi R1
n
and by applying the mean value theorem, we get
ISSN 1027-3190. Укр. мат. журн., 2018, т. 70, № 3
POLYNOMIAL INEQUALITIES IN REGIONS WITH INTERIOR ZERO ANGLES IN THE BERGMAN SPACE 325
An : =
\int
| t| =R1
| fn,p (t)| p | dt| =
=
n\sum
k=1
\int
\delta k
| fn,p (t)| p | dt| =
n\sum
k=1
\bigm| \bigm| fn,p\bigl( t\prime k\bigr) \bigm| \bigm| p\mathrm{m}\mathrm{e}\mathrm{s} \delta k, t\prime k \in \delta k.
On the other hand, by applying mean value estimation\bigm| \bigm| fn,p \bigl( t\prime k\bigr) \bigm| \bigm| p \leq 1
\pi
\bigl( \bigm| \bigm| t\prime k\bigm| \bigm| - 1
\bigr) 2 \int \int
| \xi - t\prime k| <| t\prime k| - 1
| fn,p (\xi )| p d\sigma \xi ,
we obtain
(An)
2 \preceq
n\sum
k=1
\mathrm{m}\mathrm{e}\mathrm{s} \delta k
\pi
\bigl( \bigm| \bigm| t\prime k\bigm| \bigm| - 1
\bigr) 2 \int \int
| \xi - t\prime k| <| t\prime k| - 1
| fn,p (\xi )| p d\sigma \xi , t\prime k \in \delta k.
Taking into account that at most two of the discs with center t\prime k are intersecting, we have
An \preceq \mathrm{m}\mathrm{e}\mathrm{s} \delta 1
(| t\prime 1| - 1)2
\int \int
1<| \xi | <R
| fn,p (\xi )| p d\sigma \xi \preceq n
\int \int
1<| \xi | <R
| fn,p (\xi )| p d\sigma \xi .
According to Lemma 2.4, for An we get
An \preceq n
\int \int
GR\setminus G
h(\zeta ) | Pn(\zeta )| p d\sigma \zeta \preceq n \| Pn\| pp . (3.9)
To estimate the integral Dn(w), denote by wj := \Phi (zj), \varphi j := \mathrm{a}\mathrm{r}\mathrm{g}wj , for any fixed \rho > 1, we
introduce
\Delta 1(\rho ) :=
\biggl\{
t = rei\theta : r > \rho ,
\varphi 0 + \varphi 1
2
\leq \theta <
\varphi 1 + \varphi 2
2
\biggr\}
,
\Delta 2(\rho ) :=
\biggl\{
t = rei\theta : r > \rho ,
\varphi 1 + \varphi 2
2
\leq \theta <
\varphi 1 + \varphi 0
2
\biggr\}
,
(3.10)
\Delta j := \Delta j(1), \Omega j := \Psi (\Delta j), \Omega j
\rho := \Psi (\Delta j(\rho )),
Lj := L \cap \Omega
j
, Lj
\rho := L\rho \cap \Omega
j
\rho , j = 1, 2; L = L1 \cup L1, L\rho = L1
\rho \cup L2
\rho .
Under these notations, from (3.8) for the Dn(w), we get
Dn(w) =
\int
| t| =R1
| dt|
h(\Psi (t)) | \Psi (t) - \Psi (w)| 2
\preceq
\preceq
2\sum
j=1
\int
\Phi (Lj
R1
)
| dt| \prod 2
j=1
| \Psi (t) - \Psi (wj)| \gamma j | \Psi (t) - \Psi (w)| 2
\asymp
ISSN 1027-3190. Укр. мат. журн., 2018, т. 70, № 3
326 S. BALCI, M. IMASH KYZY, F. G. ABDULLAYEV
\asymp
2\sum
j=1
\int
\Phi (Lj
R1
)
| dt|
| \Psi (t) - \Psi (wj)| \gamma j | \Psi (t) - \Psi (w)| 2
=:
2\sum
j=1
Dn,j(w), (3.11)
since the points \{ zj\} mj=1 \in L are distinct. So, we need to evaluate the Dn,j(w). For this, we take
z \in LR and introduce the notations:
\Phi (LR1) = \Phi
\left( 2\bigcup
j=1
Lj
R1
\right) =
2\bigcup
j=1
\Phi (Lj
R1
) =
2\bigcup
j=1
3\bigcup
i=1
Kj
i (R1), (3.12)
where
Kj
1(R1) :=
\Bigl\{
t \in \Phi (Lj
R1
) : | t - wj | <
c1
n
\Bigr\}
,
Kj
2(R1) :=
\Bigl\{
t \in \Phi (Lj
R1
) :
c1
n
\leq | t - wj | < c2
\Bigr\}
,
Kj
3(R1) :=
\Bigl\{
t \in \Phi (Lj
R1
) : c2 \leq | t - wj | < c3 < \mathrm{d}\mathrm{i}\mathrm{a}\mathrm{m} G
\Bigr\}
, j = 1, 2.
Analogously,
\Phi (LR) = \Phi
\left( 2\bigcup
j=1
Lj
R
\right) =
2\bigcup
j=1
\Phi (Lj
R) =
2\bigcup
j=1
3\bigcup
i=1
Kj
i (R),
where
Kj
1(R) :=
\biggl\{
\tau \in \Phi (Lj
R) : | \tau - wj | <
2c1
n
\biggr\}
,
Kj
2(R) :=
\biggl\{
\tau \in \Phi (Lj
R) :
2c1
n
\leq | \tau - wj | < c2
\biggr\}
,
Kj
3(R) :=
\Bigl\{
\tau \in \Phi (Lj
R) : c2 \leq | \tau - wj | < c3 < \mathrm{d}\mathrm{i}\mathrm{a}\mathrm{m} G
\Bigr\}
, j = 1, 2.
Then, after these definitions, taking arbitrary fixed w = \Phi (z) \in \Phi (LR), the quantity Dn,j(w)
can be written as follows:
Dn,j(w) =
3\sum
i=1
\int
Kj
i (R1)
| dt|
| \Psi (t) - \Psi (wj)| \gamma j | \Psi (t) - \Psi (w)| 2
=:
3\sum
i=1
Di
n,j(w). (3.13)
The quantity Di
n,j(w) we will estimate for each i = 1, 2, 3 and j = 1, 2 separately, depending of
location of the w \in \Phi (LR). Let \varepsilon > 0 be an arbitrary small fixed number.
Case 1. Let w \in \Phi (L1
R).
According to the above notations, we will make evaluations for case w \in K1
i (R) for each
i = 1, 2, 3.
ISSN 1027-3190. Укр. мат. журн., 2018, т. 70, № 3
POLYNOMIAL INEQUALITIES IN REGIONS WITH INTERIOR ZERO ANGLES IN THE BERGMAN SPACE 327
1.1. Let w \in K1
1 (R). In this case, we will estimate the quantity
Dn,1(w) =
3\sum
i=1
\int
K1
i (R1)
| dt|
| \Psi (t) - \Psi (w1)| \gamma 1 | \Psi (t) - \Psi (w)| 2
=:
3\sum
i=1
Di
n,1(w) (3.14)
for \gamma 1 \geq 0 and \gamma 1 < 0 separately.
For each i = 1, 2, 3 and j = 1, 2 we put Kj
i,1(R1) :=
\Bigl\{
t \in \Phi (Lj
R1
) : | t - wj | \geq | t - w|
\Bigr\}
,
Kj
i,2(R1) := Kj
i (R1) \setminus Kj
i,1(R1).
1.1.1. If \gamma 1 \geq 0, then
D1
n,1(w) =
\int
K1
1 (R1)
| dt|
| \Psi (t) - \Psi (w1)| \gamma 1 | \Psi (t) - \Psi (w)| 2
=
=
\int
K1
1,1(R1)
| dt|
| \Psi (t) - \Psi (w)| 2+\gamma 1
+
\int
K1
1,2(R1)
| dt|
| \Psi (t) - \Psi (w1)| 2+\gamma 1
=:
=: D1,1
n,1(w) +D1,2
n,1(w) (3.15)
and so Lemma 2.2 yields
D1,1
n,1(w) \preceq
\int
K1
1,1(R1)
| dt|
| t - w| (2+\gamma 1)(\lambda 1+\varepsilon )
\preceq
\left\{
n(2+\gamma 1)\lambda 1 - 1+\varepsilon , if (2 + \gamma 1)\lambda 1 > 1 - \varepsilon ,
\mathrm{l}\mathrm{n}n, if (2 + \gamma 1)\lambda 1 > 1 - \varepsilon ,
1, if (2 + \gamma 1)\lambda 1 < 1 - \varepsilon ,
(3.16)
and
D1,2
n,1(w) \preceq
\int
K1
1,2(R1)
| dt|
| t - w1| (2+\gamma 1)(\lambda 1+\varepsilon )
\preceq
\left\{
n(2+\gamma 1)\lambda 1 - 1+\varepsilon , if (2 + \gamma 1)\lambda 1 > 1 - \varepsilon ,
\mathrm{l}\mathrm{n}n, if (2 + \gamma 1)\lambda 1 = 1 - \varepsilon ,
1, if (2 + \gamma 1)\lambda 1 < 1 - \varepsilon .
(3.17)
If \gamma 1 < 0, then
D1
n,1(w) =
\int
K1
1 (R1)
| \Psi (t) - \Psi (w1)| ( - \gamma 1) | dt|
| \Psi (t) - \Psi (w)| 2
\preceq
\preceq
\int
K1
1 (R1)
| t - w1| ( - \gamma 1)(\lambda 1 - \varepsilon ) | dt|
| t - w| 2(\lambda 1+\varepsilon )
\preceq
\biggl(
1
n
\biggr) ( - \gamma 1)(\lambda 1 - \varepsilon ) \int
K1
1 (R1)
| dt|
| t - w| 2(\lambda 1+\varepsilon )
\preceq
\preceq
\biggl(
1
n
\biggr) ( - \gamma 1)(\lambda 1 - \varepsilon )
\left\{
n2(\lambda 1+\varepsilon ) - 1, if 2\lambda 1 > 1 - \varepsilon ,
\mathrm{l}\mathrm{n}n, if 2\lambda 1 > 1 - \varepsilon ,
1, if 2\lambda 1 < 1 - \varepsilon ,
\preceq
ISSN 1027-3190. Укр. мат. журн., 2018, т. 70, № 3
328 S. BALCI, M. IMASH KYZY, F. G. ABDULLAYEV
\preceq
\left\{
n(2+\gamma 1)\lambda 1 - 1+\varepsilon , if \lambda 1 >
1
2
- \varepsilon ,
1, if \lambda 1 \leq
1
2
- \varepsilon .
(3.18)
1.1.2. If \gamma 1 \geq 0, then
D2
n,1(w) =
\int
K1
2 (R1)
| dt|
| \Psi (t) - \Psi (w1)| \gamma 1 | \Psi (t) - \Psi (w)| 2
=
=
\int
K1
2,1(R1)
| dt|
| \Psi (t) - \Psi (w)| 2+\gamma 1
+
\int
K1
2,2(R1)
| dt|
| \Psi (t) - \Psi (w1)| 2+\gamma 1
=:
=: D2,1
n,1(w) +D2,2
n,1(w) (3.19)
and, so from Lemma 2.2, we get
D2,1
n,1(w) \preceq
\int
K1
2,1(R1)
| dt|
| t - w| (2+\gamma 1)(\lambda 1+\varepsilon )
\preceq n(2+\gamma 1)\lambda 1+\varepsilon \mathrm{m}\mathrm{e}\mathrm{s}K1
2,1(R1) \preceq n(2+\gamma 1)\lambda 1 - 1+\varepsilon (3.20)
and
D2,2
n,1(w) \preceq
\int
K1
2,2(R1)
| dt|
| t - w1| (2+\gamma 1)(\lambda 1+\varepsilon )
\preceq
\left\{
n(2+\gamma 1)\lambda 1 - 1+\varepsilon , if (2 + \gamma 1)\lambda 1 > 1 - \varepsilon ,
\mathrm{l}\mathrm{n}n, if (2 + \gamma 1)\lambda 1 = 1 - \varepsilon ,
1, if (2 + \gamma 1)\lambda 1 < 1 - \varepsilon .
(3.21)
Therefore, from (3.19) – (3.21) for \gamma 1 \geq 0, we have
D2
n,1(w) \preceq
\left\{
n(2+\gamma 1)\lambda 1 - 1+\varepsilon , if (2 + \gamma 1)\lambda 1 > 1 - \varepsilon ,
\mathrm{l}\mathrm{n}n, if (2 + \gamma 1)\lambda 1 = 1 - \varepsilon ,
1, if (2 + \gamma 1)\lambda 1 < 1 - \varepsilon .
(3.22)
According to well known inequality
(a+ b)\varepsilon \leq c(\varepsilon )(a\varepsilon + b\varepsilon ), a, b > 0, \varepsilon > 0, (3.23)
and using estimations
| t - w1| \leq | t - w| + | w - w1| \preceq | t - w| + 1
n
and consequently,
| t - w1| ( - \gamma 1)(\lambda 1 - \varepsilon ) \preceq | t - w| ( - \gamma 1)(\lambda 1 - \varepsilon ) +
\biggl(
1
n
\biggr) ( - \gamma 1)(\lambda 1 - \varepsilon )
,
for \gamma 1 < 0, from (3.14), we get
ISSN 1027-3190. Укр. мат. журн., 2018, т. 70, № 3
POLYNOMIAL INEQUALITIES IN REGIONS WITH INTERIOR ZERO ANGLES IN THE BERGMAN SPACE 329
D2
n,1(w) =
\int
K1
2 (R1)
| \Psi (t) - \Psi (w1)| ( - \gamma 1) | dt|
| \Psi (t) - \Psi (w)| 2
\preceq
\preceq
\int
K1
2 (R1)
| t - w1| ( - \gamma 1)(\lambda 1 - \varepsilon ) | dt|
| t - w| 2(\lambda 1+\varepsilon )
\preceq
\preceq n
\gamma 1(\lambda 1 - \varepsilon )
\int
K1
2 (R1)
| dt|
| t - w| 2(\lambda 1+\varepsilon )
+
\int
K1
2 (R1)
| dt|
| t - w| (2+\gamma 1)(\lambda 1+\varepsilon )
\preceq n
(2+\gamma 1)\lambda 1 - 1+\varepsilon
. (3.24)
1.1.3. If \gamma 1 \geq 0, then Lemma 2.2 implies
D3
n,1(w) =
\int
K1
3 (R1)
| dt|
| \Psi (t) - \Psi (w1)| \gamma 1 | \Psi (t) - \Psi (w)| 2
\preceq
\preceq c - \gamma 1
2
\int
K1
3 (R1)
| dt|
| t - w| 2\lambda 1+\varepsilon
\preceq n
2\lambda 1 - 1+\varepsilon
, (3.25)
and for \gamma 1 < 0, also Lemma 2.4 yields
D3
n,1(w) \preceq c - \gamma 1
3
\int
K1
3 (R1)
| dt|
| t - w| 2\lambda 1+\varepsilon
\preceq n
2\lambda 1 - 1+\varepsilon
. (3.26)
1.2. Let w \in K1
2 (R).
1.2.1. For any \gamma 1 > - 2
D1
n,1(w) =
\int
K1
1,1(R1)
| dt|
| \Psi (t) - \Psi (w)| 2+\gamma 1
+
\int
K1
1,2(R1)
| dt|
| \Psi (t) - \Psi (w1)| 2+\gamma 1
=:
=: D1,1
n,1(w) +D1,2
n,1(w), (3.27)
and so, according to Lemmas 2.1 and 2.2, we obtain
D1,1
n,1(w) \preceq
\int
K1
1,1(R1)
| dt|
| t - w| (2+\gamma 1)(\lambda 1+\varepsilon )
\preceq
c\int
1/n
ds
s(2+\gamma 1)(\lambda 1+\varepsilon )
\preceq
\preceq
\left\{
n(2+\gamma 1)\lambda 1 - 1+\varepsilon , if (2 + \gamma 1)\lambda 1 > 1 - \varepsilon ,
\mathrm{l}\mathrm{n}n, if (2 + \gamma 1)\lambda 1 = 1 - \varepsilon ,
1, if (2 + \gamma 1)\lambda 1 < 1 - \varepsilon ,
(3.28)
and
D1,2
n,1(w) \preceq
\int
K1
1,2(R1)
| dt|
| t - w1| (2+\gamma 1)(\lambda 1+\varepsilon )
\preceq n(2+\gamma 1)(\lambda 1+\varepsilon )\mathrm{m}\mathrm{e}\mathrm{s}K1
1,2(R1) \preceq n(2+\gamma 1)\lambda 1 - 1+\varepsilon . (3.29)
ISSN 1027-3190. Укр. мат. журн., 2018, т. 70, № 3
330 S. BALCI, M. IMASH KYZY, F. G. ABDULLAYEV
1.2.2. For any \gamma 1 > - 2, according to Lemmas 2.1 and 2.2, we have
D2
n,1(w) \preceq
\int
K1
2,1(R1)
| dt|
| \Psi (t) - \Psi (w)| 2+\gamma 1
+
\int
K1
2,2(R1)
| dt|
| \Psi (t) - \Psi (w1)| 2+\gamma 1
\preceq
\preceq
\int
K1
2,1(R1)
| dt|
| t - w| (2+\gamma 1)(\lambda 1+\varepsilon )
+
\int
K1
2,2(R1)
| dt|
| t - w| (2+\gamma 1)(\lambda 1+\varepsilon )
\preceq
\preceq
c1\int
1/n
ds
s(2+\gamma 1)(\lambda 1+\varepsilon )
+
c2\int
1/n
ds
s(2+\gamma 1)(\lambda 1+\varepsilon )
\preceq
\preceq
\left\{
n(2+\gamma 1)\lambda 1 - 1+\varepsilon , if (2 + \gamma 1)\lambda 1 > 1 - \varepsilon ,
\mathrm{l}\mathrm{n}n, if (2 + \gamma 1)\lambda 1 > 1 - \varepsilon ,
1, if (2 + \gamma 1)\lambda 1 < 1 - \varepsilon .
(3.30)
1.2.3. For any \gamma 1 > - 2, according to Lemmas 2.1 and 2.2, we get
D3
n,1(w) \preceq
\int
K1
3 (R1)
| dt|
| \Psi (t) - \Psi (w)| 2
\preceq
\int
K1
3 (R1)
| dt|
| t - w| 2(\lambda 1 +\varepsilon )
\preceq
\preceq
c3\int
1/n
ds
s2(\lambda 1+\varepsilon )
\preceq
\left\{
n2\lambda 1 - 1+\varepsilon , if 2\lambda 1 > 1 - \varepsilon ,
\mathrm{l}\mathrm{n}n, if 2\lambda 1 = 1 - \varepsilon ,
1, if 2\lambda 1 < 1 - \varepsilon .
1.3. Let w \in K1
3 (R).
1.3.1. If \gamma 1 \geq 0, from Lemmas 2.1 and 2.2, we have
D1
n,1(w) \preceq
\int
K1
1 (R1)
| dt|
| \Psi (t) - \Psi (w1)| \gamma 1
\preceq
\int
K1
1 (R1)
| dt|
| t - w1| \gamma 1(\lambda 1+\varepsilon )
\preceq
\preceq n\gamma 1(\lambda 1+\varepsilon )\mathrm{m}\mathrm{e}\mathrm{s}K1
1 (R1) \preceq n\gamma 1(\lambda 1+\varepsilon ) - 1 (3.31)
and, for \gamma 1 < 0,
D1
n,1(w) \preceq
\int
K1
1 (R1)
| t - w1| ( - \gamma 1)(\lambda 1 - \varepsilon ) | dt| \preceq
\biggl(
1
n
\biggr) ( - \gamma 1)(\lambda 1 - \varepsilon )
\mathrm{m}\mathrm{e}\mathrm{s}K1
1 (R1) \preceq
\preceq
\biggl(
1
n
\biggr) ( - \gamma 1)(\lambda 1 - \varepsilon )+1
\preceq 1. (3.32)
1.3.2. In this case for any \gamma 1 > - 2, according to Lemmas 2.1 and 2.2, we obtain
ISSN 1027-3190. Укр. мат. журн., 2018, т. 70, № 3
POLYNOMIAL INEQUALITIES IN REGIONS WITH INTERIOR ZERO ANGLES IN THE BERGMAN SPACE 331
D2
n,1(w) \preceq
\int
K1
2,1(R1)
| dt|
| \Psi (t) - \Psi (w)| 2+\gamma 1
+
\int
K1
2,2(R1)
| dt|
| \Psi (t) - \Psi (w1)| 2+\gamma 1
\preceq
\preceq
\int
K1
2,1(R1)
| dt|
| t - w| (2+\gamma 1)(\lambda 1+\varepsilon )
+
\int
K1
2,2(R1)
| dt|
| t - w| (2+\gamma 1)(\lambda 1+\varepsilon )
\preceq
\preceq
c1\int
1/n
ds
s(2+\gamma 1)(\lambda 1+\varepsilon )
+
c2\int
1/n
ds
s(2+\gamma 1)(\lambda 1+\varepsilon )
\preceq
\preceq
\left\{
n(2+\gamma 1)\lambda 1 - 1+\varepsilon , if (2 + \gamma 1)\lambda 1 > 1 - \varepsilon ,
\mathrm{l}\mathrm{n}n, if (2 + \gamma 1)\lambda 1 > 1 - \varepsilon ,
1, if (2 + \gamma 1)\lambda 1 < 1 - \varepsilon .
(3.33)
1.3.3. Analogously, for any \gamma 1 > - 2,
D3
n,1(w) \preceq
\int
K1
3 (R1)
| dt|
| \Psi (t) - \Psi (w)| 2
\preceq
\int
K1
3 (R1)
| dt|
| t - w| 2(\lambda 1+\varepsilon )
\preceq n2\lambda 1 - 1+\varepsilon . (3.34)
Combining estimates (3.14) – (3.34), for w \in \Phi (LR), we have
Dn,1 \preceq
\left\{
n(2+\widetilde \gamma 1)\lambda 1 - 1+\varepsilon , if (2 + \gamma 1) \widetilde \lambda 1 > 1 - \varepsilon ,
\mathrm{l}\mathrm{n}n, if (2 + \gamma 1) \widetilde \lambda 1 > 1 - \varepsilon ,
1, if (2 + \gamma 1) \widetilde \lambda 1 < 1 - \varepsilon ,
(3.35)
where \widetilde \gamma 1 := \mathrm{m}\mathrm{a}\mathrm{x} \{ 0; \gamma 1\} , \widetilde \lambda 1 := \mathrm{m}\mathrm{a}\mathrm{x} \{ 1;\lambda 1\} .
Case 2. Let w \in \Phi (L2
R).
Analogously to the case 1, in this case we will obtain estimates for w \in K2
1 (R), w \in K2
2 (R)
and w \in K2
3 (R).
2.1. Let w \in K2
1 (R) \cup K2
2 (R). We will estimate the quantity
Dn,2(w) =
3\sum
i=1
\int
K2
i (R1)
| dt|
| \Psi (t) - \Psi (w2)| \gamma 2 | \Psi (t) - \Psi (w)| 2
=:
3\sum
i=1
Di
n,2(w) (3.36)
for \gamma 1 \geq 0 and \gamma 1 < 0 separately.
According to the estimation [24, p. 181] for arbitrary continuum with simple connected comple-
mentary, the following holds:
| \Psi (t) - \Psi (w2)| \succeq | t - w2| 2 . (3.37)
We will use this fact in evaluations in this section instead of Lemma 2.2.
ISSN 1027-3190. Укр. мат. журн., 2018, т. 70, № 3
332 S. BALCI, M. IMASH KYZY, F. G. ABDULLAYEV
2.1.1. For each i = 1, 2, we obtain
2\sum
i=1
Di
n,2(w) =
2\sum
i=1
\int
K2
i (R1)
| dt|
| \Psi (t) - \Psi (w2)| \gamma 2 | \Psi (t) - \Psi (w)| 2
\preceq
\preceq
\left( \int
K2
1,1(R1)
+
\int
K2
2,1(R1)
\right) | dt|
| \Psi (t) - \Psi (w)| 2+\gamma 2
+
\left( \int
K2
1,2(R1)
+
\int
K2
2,2(R1)
\right) | dt|
| \Psi (t) - \Psi (w2)| 2+\gamma 2
\preceq
\preceq
\left( \int
K2
1,1(R1)
+
\int
K2
2,1(R1)
\right) | dt|
| t - w| 2(2+\gamma 2)
\preceq n2(2+\gamma 2) - 1, (3.38)
if \gamma 2 \geq 0, and
2\sum
i=1
Di
n,2(w) =
2\sum
i=1
\int
K2
i (R1)
| \Psi (t) - \Psi (w2)| ( - \gamma 2) | dt|
| \Psi (t) - \Psi (w)| 2
\preceq n3, (3.39)
if \gamma 2 < 0.
2.1.2. For i = 3 we get
D3
n,2(w) =
\int
K2
3 (R1)
| dt|
| \Psi (t) - \Psi (w2)| \gamma 2 | \Psi (t) - \Psi (w)| 2
\preceq
\preceq c - \gamma 2
2
\int
K2
3 (R1)
| dt|
| \Psi (t) - \Psi (w)| 2
\preceq
\int
K2
3 (R1)
| dt|
| t - w| 2+\varepsilon \preceq n1+\varepsilon , (3.40)
if \gamma 2 \geq 0, and
D3
n,2(w) \preceq n1+\varepsilon , (3.41)
if \gamma 2 < 0.
2.2. Let w \in K2
3 (R). For each \gamma 2 > - 2, analogously to subcase 2.1.1, we obtain
2\sum
i=1
Di
n,2(w) =
2\sum
i=1
\int
K2
i (R1)
| dt|
| \Psi (t) - \Psi (w2)| \gamma 2 | \Psi (t) - \Psi (w)| 2
\preceq
\preceq
\left( \int
K2
1,1(R1)
+
\int
K2
2,1(R1)
\right) | dt|
| \Psi (t) - \Psi (w)| 2+\gamma 2
+
\left( \int
K2
1,2(R1)
+
\int
K2
2,2(R1)
\right) | dt|
| \Psi (t) - \Psi (w2)| 2+\gamma 2
\preceq
\preceq
\left( \int
K2
1,1(R1)
+
\int
K2
2,1(R1)
\right) | dt|
| t - w| 2(2+\gamma 2)
+
\left( \int
K2
1,2(R1)
+
\int
K2
2,2(R1)
\right) | dt|
| t - w2| 2(2+\gamma 2)
\preceq
ISSN 1027-3190. Укр. мат. журн., 2018, т. 70, № 3
POLYNOMIAL INEQUALITIES IN REGIONS WITH INTERIOR ZERO ANGLES IN THE BERGMAN SPACE 333
\preceq
\left\{
n2(2+\gamma 2) - 1, if 2 (2 + \gamma 2) > 1,
\mathrm{l}\mathrm{n}n, if 2 (2 + \gamma 2) = 1,
1, if 2 (2 + \gamma 2) < 1.
(3.42)
2.2.2. For each \gamma 2 > - 2, we have
D3
n,2(w) =
\int
K2
3 (R1)
| dt|
| \Psi (t) - \Psi (w2)| \gamma 2 | \Psi (t) - \Psi (w)| 2
\preceq
\preceq
\int
K2
3 (R1)
| dt|
| \Psi (t) - \Psi (w)| 2
\preceq
\int
K2
3 (R1)
| dt|
| t - w| 2+\varepsilon \preceq n1+\varepsilon . (3.43)
Combining (3.36) – (3.43), we obtain
Dn,2(w) \preceq
\left\{
n2(2+\widetilde \gamma 2) - 1, if 2 (2 + \gamma 2) > 1,
\mathrm{l}\mathrm{n}n, if 2 (2 + \gamma 2) = 1,
1, if 2 (2 + \gamma 2) < 1,
(3.44)
where \widetilde \gamma 2 := \mathrm{m}\mathrm{a}\mathrm{x} \{ 0; \gamma 2\} .
Therefore, comparing relations (3.11), (3.13), (3.35) and (3.44), we get
Dn(w) \preceq
\left\{
n(2+\widetilde \gamma 1)\widetilde \lambda 1 - 1+\varepsilon , if (2 + \gamma 1) \widetilde \lambda 1 > 1 - \varepsilon ,
\mathrm{l}\mathrm{n}n, if (2 + \gamma 1) \widetilde \lambda 1 > 1 - \varepsilon ,
1, if (2 + \gamma 1) \widetilde \lambda 1 < 1 - \varepsilon ,
+
\left\{
n2(2+\widetilde \gamma 2) - 1, if 2 (2 + \gamma 2) > 1,
\mathrm{l}\mathrm{n}n, if 2 (2 + \gamma 2) = 1,
1, if 2 (2 + \gamma 2) < 1,
and consequently, from (3.7), (3.8) and (3.9), we completed the proof of Theorems 1.1 and 1.2 for
any z \in LR. So, it also true for z \in G, and we completed the proofs.
3.2. Proof of Theorem 1.3. Suppose that G \in C\theta (\lambda 1; 2), for some 0 < \lambda 1 < 2; h(z) be defined
as in (1.1). For each R > 1, let w = \varphi R(z) denotes be a univalent conformal mapping GR onto the
B, normalized by \varphi R(0) = 0, \varphi \prime
R(0) > 0, and let \{ \zeta j\} , 1 \leq j \leq m \leq n, be a zeros of Pn(z) (if
any exist) lying on GR. Let
bm,R(z) :=
m\prod
j=1
\widetilde bj,R(z) =:
m\prod
j=1
\varphi R(z) - \varphi R(\zeta j)
1 - \varphi R(\zeta j)\varphi R(z)
, (3.45)
denotes a Blaschke function with respect to zeros \{ \zeta j\} , 1 \leq j \leq m \leq n, of Pn(z) [26]. Clearly,
| bm,R(z)| \equiv 1, z \in LR, and | bm,R(z)| < 1, z \in GR. (3.46)
For any p > 0 and z \in GR, let us set
Tn.p (z) :=
\biggl[
Pn (z)
bm,R(z)
\biggr] p/2
. (3.47)
ISSN 1027-3190. Укр. мат. журн., 2018, т. 70, № 3
334 S. BALCI, M. IMASH KYZY, F. G. ABDULLAYEV
The function Tn,p (z) is analytic in GR, continuous on GR and does not have zeros in GR. We take
an arbitrary continuous branch of the Tn,p (z) and for this branch we maintain the same designation.
Then, the Cauchy integral representation for the Tn,p (z) at the z = zj , j = 1, 2, gives
Tn,p (z1) =
1
2\pi i
\int
LR
Tn,p (\zeta )
d\zeta
\zeta - z1
.
Then, according to (3.46), we obtain
| Pn (zj)|
p/2
\leq
| bm,R(z1)| p/2
2\pi
\int
LR
\bigm| \bigm| \bigm| \bigm| Pn (\zeta )
bm,R(\zeta )
\bigm| \bigm| \bigm| \bigm| p/2 | d\zeta |
| \zeta - zj |
\preceq
\preceq
\int
LR
| Pn (\zeta )|
p/2 | d\zeta |
| \zeta - zj |
. (3.48)
Multiplying the numerator and the denominator of the last integrand by h1/2(\zeta ), replacing the vari-
able w = \Phi (z) and applying the Hölder inequality, we get\left( \int
LR
| Pn (\zeta )|
p
2
| d\zeta |
| \zeta - zj |
\right)
2
\leq
\leq
\int
| t| =R
h(\Psi (t)) | Pn (\Psi (t))| p
\bigm| \bigm| \Psi \prime (t)
\bigm| \bigm| 2 | dt| \int
| t| =R
| dt|
h(\Psi (t)) | \Psi (t) - \Psi (wj)| 2
=
=
\int
| t| =R
| fn,p(t)| p | dt|
\int
| t| =R
| dt|
h(\Psi (t)) | \Psi (t) - \Psi (wj)| 2
, (3.49)
where fn,p(t) has been defined as in (3.8). Since R > 1 is arbitrary, then (3.49) holds also for
R = R1 := 1 +
\varepsilon 1
n
, 0 < \varepsilon 1 < 1. So, we have
\left( \int
LR1
| Pn (\zeta )|
p
2
| d\zeta |
| \zeta - zj |
\right)
2
\leq
\leq
\left( \int
| t| =R1
| fn,p(t)| p | dt|
\right)
\left( \int
| t| =R1
| dt|
h(\Psi (t)) | \Psi (t) - \Psi (wj)| 2
\right) =:
=: AnDn(wj), (3.50)
and, An and Dn(wj) has been defined as in (3.8) for R = R1. Therefore, from (3.48) and (3.50),
we obtain
| Pn (z1)| \preceq AnDn(wj), (3.51)
ISSN 1027-3190. Укр. мат. журн., 2018, т. 70, № 3
POLYNOMIAL INEQUALITIES IN REGIONS WITH INTERIOR ZERO ANGLES IN THE BERGMAN SPACE 335
where, according to (3.9), the estimate
An \preceq n \| Pn\| pp
is satisfied. For the estimate of the quantity Dn(wj) we use the notations at the estimation of the
Dn(w) as in (3.11) – (3.13). Therefore, under these notations, for the Dn(wj), we get
Dn(wj) \preceq
2\sum
j=1
\int
\Phi (Lj
R1
)
| dt|
| \Psi (t) - \Psi (wj)| 2+\gamma j
\preceq
\preceq
2\sum
j=1
3\sum
i=1
\int
Kj
i (LR1
)
| dt|
| \Psi (t) - \Psi (wj)| 2+\gamma j
=:
2\sum
j=1
3\sum
i=1
Di
n,j(wj), (3.52)
since the points \{ zj\} mj=1 \in L are distinct. So, we need to evaluate the Di
n,j(wj) for each j = 1, 2
and i = 1, 2, 3.
Case 1. j = 1:
D1
n,1(w1) +D2
n,1(w1) =
\int
K1
1 (LR1
)\cup K2
1 (LR1
)
| dt|
| \Psi (t) - \Psi (w1)| 2+\gamma 1
\preceq
\preceq
\int
K1
1 (LR1
)\cup K2
1 (LR1
)
| dt|
| t - w1| (2+\gamma 1)(\lambda 1+\varepsilon )
\preceq
\left\{
n(2+\gamma 1)\lambda 1 - 1+\varepsilon , if (2 + \gamma 1)\lambda 1 > 1 - \varepsilon ,
\mathrm{l}\mathrm{n}n, if (2 + \gamma 1)\lambda 1 = 1 - \varepsilon ,
1, if (2 + \gamma 1)\lambda 1 < 1 - \varepsilon ,
(3.53)
and
D3
n,1(w1) =
\int
K3
1 (LR1
)
| dt|
| \Psi (t) - \Psi (w1)| 2+\gamma 1
\preceq 1
c2+\gamma 1
2
\int
K3
1 (LR1
)
| dt| \preceq 1. (3.54)
Case 2. j = 2:
D1
n,2(w2) +D2
n,2(w2) =
\int
K1
2 (LR1
)\cup K2
2 (LR1
)
| dt|
| \Psi (t) - \Psi (w2)| 2+\gamma 2
\preceq
\preceq
\int
K1
2 (LR1
)\cup K2
2 (LR1
)
| dt|
| t - w2| 2(2+\gamma 2)
\preceq
\left\{
n2(2+\gamma 2) - 1, if 2 (2 + \gamma 2) > 1,
\mathrm{l}\mathrm{n}n, if 2 (2 + \gamma 2) = 1,
1, if 2 (2 + \gamma 2) < 1,
(3.55)
and
D3
n,2(w2) =
\int
K3
2 (LR1
)
| dt|
| \Psi (t) - \Psi (w2)| 2+\gamma 2
\preceq 1
c2+\gamma 2
2
\int
K3
2 (LR1
)
| dt| \preceq 1. (3.56)
Combining relations (3.51) – (3.56), we complete the proof.
ISSN 1027-3190. Укр. мат. журн., 2018, т. 70, № 3
336 S. BALCI, M. IMASH KYZY, F. G. ABDULLAYEV
References
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. – P. 3 – 7 (in Russian).
2. Abdullayev F. G. On the some properties of the orthogonal polynomials over the region of the complex plane
(Part I) // Ukr. Math. J. – 2000. – 52, № 12. – P. 1807 – 1821.
3. Abdullayev F. G. On the some properties of the orthogonal polynomials over the region of the complex plane
(Part II) // Ukr. Math. J. – 2001. – 53, № 1. – P. 1 – 14.
4. 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.
5. Abdullayev F. G. The properties of the orthogonal polynomials with weight having singulerity on the boundary
contour // J. Comput. Anal. and Appl. – 2004. – 6, № 1. – P. 43 – 59.
6. Abdullayev F. G., Deger U. 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.
7. Abdullayev F. G., Gün C. D. On the behavior of the algebraic polynomials in regions with piecewise smooth boundary
without cusps // Ann. Polon. Math. – 2014. – 111. – P. 39 – 58.
8. Abdullayev F. G., Özkartepe N. P. On the behavıor of the algebraıc polynomıal in unbounded regıons wıth pıecewıse
dını-smooth boundary // Ukr. Math. J. – 2014. – 66, № 5. – P. 579 – 597.
9. Abdullayev F. G., Özkartepe N. P. Uniform and pointwise Bernstein – Walsh-type inequalities on a quasidisk in the
complex plane // Bull. Belg. Math. Soc. – 2016. – 23, № 2. – P. 285 – 310.
10. Abdullayev F. G., Tunç T. Uniform and pointwise polynomial inequalities in regions with asymptotically conformal
curve on weighted Bergman space // Lobachevcki J. Math. – 2017. – 38, № 2. – P. 193 – 205.
11. Abdullayev F. G., Tunç T., Abdullayev G. A. Polynomial inequalities in quasidisks on weighted Bergman space // Ukr.
Mat. Zh. – 2017. – 69, № 5. – P. 582 – 598.
12. Ahlfors L. Lectures on quasiconformal mappings. – Princeton, NJ: Van Nostrand, 1966.
13. 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.
14. Andrievskii V. V. Weighted polynomial inequalities in the complex plan // J. Approxim. Theory. – 2012. – 164, № 9. –
P. 1165 – 1183.
15. Batchayev I. M. Integral representations in domains with quasiconformal boundary and some of their applications:
Avtoref. dis. cand. fiz.-mat. nauk. – Baku, 1981 (in Russian).
16. Dzyadyk V. K., Shevchuk I. A. Theory of uniform approximation of functions by polynomials. – Berlin; New York:
Walter de Gruyter, 2008.
17. Gaier D. On the convergence of the Bieberbach polynomials in regions with corners // Constr. Approxim. – 1988. –
4. – P. 289 – 305.
18. Lehto O., Virtanen K. I. Quasiconformal mapping in the plane. – Berlin: Springer Verlag, 1973.
19. Mergelyan S. N. Some questions of constructive functions theory // Proc. Steklov Inst. Math. – 1951. – 37. – P. 1 – 92
(in Russian).
20. Milovanovic G. V., Mitrinovic D. S., Rassias Th. M. Topics in polynomials: extremal problems, inequalities, zeros. –
Singapore: World Sci., 1994.
21. Nikol’skii S. M. Approximation of function of several variable and imbeding theorems. – New York: Springer-Verlag,
1975.
22. Rickman S. Characterization of quasiconformal arcs // Ann. Acad. Sci. Fenn. Math. – 1966. – 395.
23. Pritsker I. Comparing norms of polynomials in one and several variables // J. Math. Anal. and Appl. – 1997. – 216. –
P. 685 – 695.
24. Tamrazov P. M. Smoothness and polynomial approximations. – Kiev: Naukova Dumka, 1975 (in Russian).
25. Tunç T., Şimşek D., Oruç E. Pointwise Bernstein – Walsh-type inequalities in regions with interior zero angles in the
Bergman space // Trans. NAS of Azerb. Ser. Phys.-Tech. and Math. Sci. – 2017. – 37, № 1. – P. 1 – 12.
26. Walsh J. L. Interpolation and approximation by rational functions in the complex domain. – Amer. Math. Soc., 1960.
27. Warschawski S. E. Über das randverhalten der ableitung der ableitung der abbildungsfunktion bei konformer
abbildung // Math. Z. – 1932. – 35. – S. 321 – 456
Received 08.06.17
ISSN 1027-3190. Укр. мат. журн., 2018, т. 70, № 3
|
| id | umjimathkievua-article-1559 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T02:08:05Z |
| publishDate | 2018 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/41/cf0e712ed605a3a230ee413a5ab47441.pdf |
| spelling | umjimathkievua-article-15592019-12-05T09:18:29Z Polynomial inequalities in regions with interior zero angles in the Bergman space Полiномiальнi нерiвностi в областях iз внутрiшнiми нульовими кутами у просторi Бергмана Abdullayev, F. G. Balci, S. Imash, kyzy M. Абдуллаєв, Ф. Г. Балчи, С. Імаш, кизи М. We investigate the order of growth of the moduli of arbitrary algebraic polynomials in the weighted Bergman space $A_p(G, h),\; p > 0$, in regions with interior zero angles at finitely many boundary points. We obtain estimations for algebraic polynomials in bounded regions with piecewise smooth boundary. Вивчається порядок зростання модулiв довiльних алгебраїчних полiномiв у ваговому просторi Бергмана $A_p(G, h)$, $\;p > 0$, в областях iз внутрiшнiми нульовими кутами у скiнченнiй кiлькостi точок. Отримано оцiнки для алгебраїчних полiномiв в обмежених областях з кусково-гладкою межею. Institute of Mathematics, NAS of Ukraine 2018-03-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/1559 Ukrains’kyi Matematychnyi Zhurnal; Vol. 70 No. 3 (2018); 318-336 Український математичний журнал; Том 70 № 3 (2018); 318-336 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/1559/541 Copyright (c) 2018 Abdullayev F. G.; Balci S.; Imash kyzy M. |
| spellingShingle | Abdullayev, F. G. Balci, S. Imash, kyzy M. Абдуллаєв, Ф. Г. Балчи, С. Імаш, кизи М. Polynomial inequalities in regions with interior zero angles in the Bergman space |
| title | Polynomial inequalities in regions with interior zero
angles in the Bergman space |
| title_alt | Полiномiальнi нерiвностi в областях
iз внутрiшнiми нульовими кутами у просторi Бергмана |
| title_full | Polynomial inequalities in regions with interior zero
angles in the Bergman space |
| title_fullStr | Polynomial inequalities in regions with interior zero
angles in the Bergman space |
| title_full_unstemmed | Polynomial inequalities in regions with interior zero
angles in the Bergman space |
| title_short | Polynomial inequalities in regions with interior zero
angles in the Bergman space |
| title_sort | polynomial inequalities in regions with interior zero
angles in the bergman space |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/1559 |
| work_keys_str_mv | AT abdullayevfg polynomialinequalitiesinregionswithinteriorzeroanglesinthebergmanspace AT balcis polynomialinequalitiesinregionswithinteriorzeroanglesinthebergmanspace AT imashkyzym polynomialinequalitiesinregionswithinteriorzeroanglesinthebergmanspace AT abdullaêvfg polynomialinequalitiesinregionswithinteriorzeroanglesinthebergmanspace AT balčis polynomialinequalitiesinregionswithinteriorzeroanglesinthebergmanspace AT ímaškizim polynomialinequalitiesinregionswithinteriorzeroanglesinthebergmanspace AT abdullayevfg polinomialʹninerivnostivoblastâhizvnutrišniminulʹovimikutamiuprostoribergmana AT balcis polinomialʹninerivnostivoblastâhizvnutrišniminulʹovimikutamiuprostoribergmana AT imashkyzym polinomialʹninerivnostivoblastâhizvnutrišniminulʹovimikutamiuprostoribergmana AT abdullaêvfg polinomialʹninerivnostivoblastâhizvnutrišniminulʹovimikutamiuprostoribergmana AT balčis polinomialʹninerivnostivoblastâhizvnutrišniminulʹovimikutamiuprostoribergmana AT ímaškizim polinomialʹninerivnostivoblastâhizvnutrišniminulʹovimikutamiuprostoribergmana |