Polynomial inequalities in quasidisks on weighted Bergman space
We continue studying on the Nikol’skii and Bernstein –Walsh type estimations for complex algebraic polynomials in the bounded and unbounded quasidisks on the weighted Bergman space.
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Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860507565742358528 |
|---|---|
| author | Abdullayev, G. A. Abdullayev, F. G. Tunç, E. Абдуллаєв, Г. А. Абдуллаєв, Ф. Г. Тунс, Е. |
| author_facet | Abdullayev, G. A. Abdullayev, F. G. Tunç, E. Абдуллаєв, Г. А. Абдуллаєв, Ф. Г. Тунс, Е. |
| author_sort | Abdullayev, G. A. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2019-12-05T09:24:56Z |
| description | We continue studying on the Nikol’skii and Bernstein –Walsh type estimations for complex algebraic polynomials in the bounded and unbounded quasidisks on the weighted Bergman space. |
| first_indexed | 2026-03-24T02:11:20Z |
| format | Article |
| fulltext |
UDC 517.5
F. G. Abdullayev (Mersin Univ., Turkey and Kyrgyz-Turkish Manas Univ., Bishkek, Kyrgystan),
T. Tunç (Mersin Univ., Turkey),
G. A. Abdullayev (Mersin Univ. Vocat. School Techn. Sci., Turkey)
POLYNOMIAL INEQUALITIES IN QUASIDISKS
ON WEIGHTED BERGMAN SPACE*
ПОЛIНОМIАЛЬНI НЕРIВНОСТI У КВАЗIДИСКАХ
НА ЗВАЖЕНИХ ПРОСТОРАХ БЕРГМАНА
We continue studying on the Nikol’skii and Bernstein – Walsh type estimations for complex algebraic polynomials in the
bounded and unbounded quasidisks on the weighted Bergman space.
Продовжено дослiдження оцiнок типу Нiкольського та Бернштейна – Уолша для комплексних алгебраїчних полiномiв
в обмежених та необмежених квазiдисках на зважених просторах Бергмана.
1. Introduction. Let \BbbC be a complex plane and \BbbC := \BbbC \cup \{ \infty \} ; G \subset \BbbC be a bounded Jordan region
with boundary L := \partial G such that 0 \in G, \Omega := \BbbC \setminus G = \mathrm{e}\mathrm{x}\mathrm{t}L, \Delta := \{ w : | w| > 1\} . Let w = \Phi (z)
be the univalent conformal mapping of \Omega onto \Delta such that \Phi (\infty ) = \infty and \mathrm{l}\mathrm{i}\mathrm{m}z\rightarrow \infty
\Phi (z)
z
> 0,
\Psi := \Phi - 1. For R > 1, we take LR := \{ z : | \Phi (z)| = R\} , GR := \mathrm{i}\mathrm{n}\mathrm{t}LR and \Omega R := \mathrm{e}\mathrm{x}\mathrm{t}LR . Let
\wp n denotes the class of all algebraic polynomials Pn(z) of degree at most n \in \BbbN := \{ 1, 2, . . .\} .
In this work we consider the following weight function h(z): Let \{ zj\} mj=1 be the fixed system of
distinct points on the curve L. For some fixed R0, 1 < R0 <\infty , consider generalized Jacobi weight
function h(z) which is 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 h0 is uniformly separated from zero on L, i.e., there
exists a constant c0 = c0(G) > 0 such that h0(z) \geq c0 > 0 for all z \in GR0 .
Let 0 < p \leq \infty . For the Jordan region G, we introduce
\| Pn\| p := \| Pn\| Ap(h,G) :=
\left( \int \int
G
h(z) | Pn(z)| p d\sigma z
\right) 1/p , 0 < p <\infty , (1.2)
\| Pn\| \infty := \| Pn\| A\infty (1,G) := \mathrm{m}\mathrm{a}\mathrm{x}
z\in G
| Pn(z)| , p = \infty . (1.3)
In this work, firstly we study the following Nikol’skii-type inequality:
\| Pn\| \infty \leq c1\lambda n(G, h, p) \| Pn\| p , (1.4)
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 the region G and the weight function h. The estimates
* This paper is supported by TUBITAK, project No. 115F652.
c\bigcirc F. G. ABDULLAYEV, T. TUNÇ, G. A. ABDULLAYEV, 2017
582 ISSN 1027-3190. Укр. мат. журн., 2017, т. 69, № 5
POLYNOMIAL INEQUALITIES IN QUASIDISKS ON WEIGHTED BERGMAN SPACE 583
of (1.4)-type for some 0 < p < \infty and h(z) were investigated in [21, p. 122 – 133], [14], [20]
(Sect. 5.3), [25], [2 – 8] (see also references therein) and others.
Secondly, we find pointwise estimations in unbounded region \Omega :
| Pn(z)| \leq c2
\eta n(G, h, p)
d(z, L)
\| Pn\| p | \Phi (z)|
n+1 , z \in \Omega , (1.5)
where c2 = c2(G, h, p) > 0 is a constant independent of n and Pn, and \eta n(G, h, p) \rightarrow \infty , n\rightarrow \infty ,
again depending on the region G and h. We note that, according to the maximum principle the
inequality (1.4) holds also for
\bigm| \bigm| \bigm| \bigm| Pn(z)
\Phi n+1(z)
\bigm| \bigm| \bigm| \bigm| for any points z \in \Omega . But we will try to have a more
accurate estimate for
\bigm| \bigm| \bigm| \bigm| Pn(z)
\Phi n+1(z)
\bigm| \bigm| \bigm| \bigm| of the (1.5)-type.
Analogous results of (1.5)-type for some norms and for different unbounded regions were ob-
tained by N. A. Lebedev, P. M. Tamrazov, V. K. Dzjadyk (see, for example, [17, p. 418 – 428]),
V. V. Andrievskii [13], N. Stylianopoulos [26], F. G. Abdullayev et al. [7, 8] and others.
Finally, combining obtained estimates for | Pn(z)| on bounded and unbounded regions, we get
the evaluation for | Pn(z)| in whole complex plane.
2. Definitions and main results. 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 depends on G in general and, on parameters inessential for the argument, otherwise, the
dependence will be explicitly stated. For any k \geq 0 and m > k, notation i = k,m means
i = k, k+1, . . . ,m. Let the function \varphi maps G conformally and univalently onto B := \{ w : | w| < 1\}
which is normalized by \varphi (0) = 0 and \varphi \prime (0) > 0; let \psi := \varphi - 1.
Definition 2.1 [22, p. 286 – 294]. A bounded Jordan region G is called a K-quasidisk, 0 \leq
\leq k < 1, if any conformal mapping \psi can be extended to a K-quasiconformal homeomorphism of
the plane \BbbC on the \BbbC , where K =
1 + k
1 - k
. In that case the curve L := \partial G is called a K-quasicircle.
Let G be a region bounded by two arcs of circle, symmetric with respect to the OX-axis and
OY -axis, each of the arcs crosses the OX-axis at \pm \varepsilon 0, where \varepsilon 0 > 0 and the angle between the arcs
is \pi (1 - k) where 0 \leq k < 1. This region is a K-quasidisk.
A region G (curve L) is called a quasidisk (quasicircle), if it is a K-quasidisk (K-quasicircle) for
some 0 \leq k < 1. A Jordan curve L is called a quasicircle or quasiconformal curve, if it is the image
of the unit circle under a quasiconformal mapping of \BbbC (see [18, p. 105; 22, p. 286]). On the other
hand, it is given some geometric criteria of quasiconformality of the curves (see also [10, p. 81; 23,
p. 107; 19, p. 341]). Now we give one of them.
Let z1, z2 be an arbitrary points on L and L(z1, z2) denotes the subarc of L of shorter diameter
with endpoints z1 and z2. Lesley [19, p. 341] defined the curve L as “c-quasiconformal”, if for all
z1, z2 \in L and z \in L(z1, z2) there exists a constant c, independent from points z1, z2 and z, such
that
| z1 - z| + | z - z2|
| z1 - z2|
\leq c. (2.1)
A simple example of c-quasiconformal curves can be given a polygon whose smallest interior or
exterior opening angle 2 \mathrm{a}\mathrm{r}\mathrm{c}\mathrm{s}\mathrm{i}\mathrm{n}(1/c). It is well known that quasicircles can be nonrectifiable (see, for
example, [15; 18, p. 104]).
ISSN 1027-3190. Укр. мат. журн., 2017, т. 69, № 5
584 F. G. ABDULLAYEV, T. TUNÇ, G. A. ABDULLAYEV
In this work, we study similar problems to (1.4) and (1.5) for quasidisks. Now, we start to
formulate the new results.
Theorem 2.1. Let p > 0, G be a K-quasidisk for some 0 \leq k < 1 and h(z) be defined by (1.1).
Then, for any Pn \in \wp n, n \in \BbbN ,
| Pn(zj)| \leq c1n
(2+\gamma j)(1+k)/p \| Pn\| p , j = 1,m, (2.2)
and, consequently,
\| Pn\| \infty \leq c2n
(2+\gamma )(1+k)/p \| Pn\| p , (2.3)
where \gamma := \mathrm{m}\mathrm{a}\mathrm{x}
\bigl\{
0; \gamma j , j = 1,m
\bigr\}
.
For any \rho > 1 we divide \Omega \rho as \Omega \rho :=
m\bigcup
j=1
\Omega j
\rho , where \Omega j
\rho defined below in (4.6). For given
\vargamma > 0, we define
\nu n :=
\left\{
1, if \gamma j < \vargamma for all j = 1,m,
\mathrm{l}\mathrm{n}n, if exists j0, \gamma j0 = \vargamma and \gamma j < \vargamma for j \not = j0,
n\gamma j0/\vargamma - 1, if exists j0, \gamma j0 > \vargamma and \gamma j \leq \vargamma for j \not = j0.
(2.4)
Theorem 2.2. Let p > 0, L be a K-quasicircle for some 0 \leq k < 1 and h(z) be defined by
(1.1). Then there exists a constant c3 = c3(G, p, \gamma j) > 0 such that for any Pn \in \wp n, n \in \BbbN , and for
j = 1,m the inequality
| Pn(z)| \leq c3
\biggl( \surd
n\mu n
d(z, LR)
\biggr) 2/p
\| Pn\| p | \Phi (z)|
n+1 , z \in \Omega j
R, (2.5)
holds, where R = 1 +
\varepsilon 1
n
and \mu n = \nu n defined in (2.4) with \vargamma =
1
1 + k
.
According to the Bernstein lemma [30], the estimation (2.3) is again true for the z \in GR1 with a
different constant. Therefore, combining estimation (2.3) (for the z \in GR1) with (2.5), we obtain an
estimation for the growth of | Pn(z)| in the whole complex plane:
Corollary 2.1. Under the assumptions of Theorems 2.1 and 2.2, the following is true:
| Pn(z)| \leq c4 \| Pn\| p
\left\{
n(2+\gamma )(1+k)/p, z \in GR,\biggl( \surd
n\mu n
d(z, LR)
\biggr) 2/p
| \Phi (z)| n+1 , z \in \Omega j
R,
(2.6)
where c4 = c4(G, p, h) > 0.
Corollary 2.2. For any compact subset F \Subset \Omega and Pn \in \wp n, n \in \BbbN , we have
| Pn(z)| \leq c5
\biggl( \surd
n\mu n
d(z, L)
\biggr) 2/p
\| Pn\| p | \Phi (z)|
n+1 , z \in F, (2.7)
where c5 = c5(G,F ) > 0.
ISSN 1027-3190. Укр. мат. журн., 2017, т. 69, № 5
POLYNOMIAL INEQUALITIES IN QUASIDISKS ON WEIGHTED BERGMAN SPACE 585
From the conditions of the theorems, we see that they hold for K-quasidisks with 0 \leq k < 1.
But calculating the coefficient of quasiconformality for some curves is not an easy task. Therefore,
we define more general class of curves with another characteristics. One of them is the following:
Definition 2.2. We say that L = \partial G \in Q\alpha , 0 < \alpha \leq 1, if L is a quasicircle and \Phi \in \mathrm{L}\mathrm{i}\mathrm{p}\alpha
on \Omega .
We note that the classes Q\alpha are sufficiently large. A detailed information on it and the related
topics are contained in [?, ?, 19] (see also the references cited therein). We consider only some cases:
Remark 2.1. (a) If L is a Dini-smooth curve [?, p. 48], then L \in Q1.
(b) If L is a piecewise Dini-smooth curve and largest exterior angle on L has opening \alpha \pi ,
0 < \alpha \leq 1 [?, p. 52], then L \in Q\alpha .
(c) If L is a smooth curve having continuous tangent, then L \in Q\alpha for all 0 < \alpha < 1.
(d) If G is “L-shaped” region, then \Phi \in \mathrm{L}\mathrm{i}\mathrm{p}
2
3
, \Psi \in \mathrm{L}\mathrm{i}\mathrm{p}
1
2
.
(e) If L is quasismooth (in the sense of Lavrentiev), that is, for every pair z1, z2 \in L, there
exists a constant c > 1 such that s(z1, z2) \leq c| z1 - z2| , then \Phi \in \mathrm{L}\mathrm{i}\mathrm{p}\alpha for \alpha =
\pi
2(\pi - \mathrm{a}\mathrm{r}\mathrm{c}\mathrm{s}\mathrm{i}\mathrm{n}(1/c))
and \Psi \in \mathrm{L}\mathrm{i}\mathrm{p}\beta for \beta =
2
(1 + c)2
, where s(z1, z2) represents the smallest of the lengths of the arcs
joining z1 to z2 on L [?, 29].
(f) If L is a c-quasiconformal, then \Phi \in \mathrm{L}\mathrm{i}\mathrm{p}\alpha for \alpha =
\pi
2(\pi - \mathrm{a}\mathrm{r}\mathrm{c}\mathrm{s}\mathrm{i}\mathrm{n}(1/c))
and \Psi \in \mathrm{L}\mathrm{i}\mathrm{p}\beta for
\beta =
2(\mathrm{a}\mathrm{r}\mathrm{c}\mathrm{s}\mathrm{i}\mathrm{n}(1/c))2
\pi (\pi - \mathrm{a}\mathrm{r}\mathrm{c}\mathrm{s}\mathrm{i}\mathrm{n}(1/c))
.
For \partial G \in Q\alpha we have the following results:
Theorem 2.3. Let p > 0, L \in Q\alpha for some
1
2
\leq \alpha \leq 1 and h(z) be defined by (1.1). Then, for
any Pn \in \wp n, n \in \BbbN ,
| Pn(zj)| \leq c6n
(2+\gamma j)/p\alpha \| Pn\| p , j = 1,m, (2.8)
and, consequently,
\| Pn\| \infty \leq c7n
(2+\gamma )/p\alpha \| Pn\| p , (2.9)
where \gamma := \mathrm{m}\mathrm{a}\mathrm{x}
\bigl\{
0; \gamma j , j = 1,m
\bigr\}
.
Theorem 2.4. Let p > 0, L \in Q\alpha for some
1
2
\leq \alpha \leq 1 and h(z) be defined by (1.1). Then,
there exists c8 = c8(G, p, \gamma j) > 0 such that for any Pn \in \wp n, n \in \BbbN ,
| Pn(z)| \leq c8
\biggl( \surd
n\delta n
d(z, LR)
\biggr) 2/p
\| Pn\| p | \Phi (z)|
n+1 , z \in \Omega j
R, j = 1,m, (2.10)
where R = 1 +
\varepsilon 1
n
and \delta n = \nu n defined in (2.4) with \vargamma = \alpha .
Corollary 2.3. Under the conditions Theorems 2.3 and 2.4, we have
| Pn(z)| \leq c9 \| Pn\| p
\left\{
n(2+\gamma )/\alpha p, z \in GR,\biggl( \surd
n\delta n
d(z, LR)
\biggr) 2/p
| \Phi (z)| n+1 , z \in \Omega j
R,
(2.11)
where c9 = c9(G, p, \gamma j) > 0.
ISSN 1027-3190. Укр. мат. журн., 2017, т. 69, № 5
586 F. G. ABDULLAYEV, T. TUNÇ, G. A. ABDULLAYEV
Corollary 2.4. Let p > 0, L be a c-quasiconformal for some c > 1 and h(z) be defined by
(1.1). Then, for any Pn \in \wp n, n \in \BbbN , the estimations (2.8), (2.9) and (2.10) hold with \alpha =
=
\pi
2(\pi - \mathrm{a}\mathrm{r}\mathrm{c}\mathrm{s}\mathrm{i}\mathrm{n}(1/c))
.
Note that, the estimations, analogous to Theorems 2.1 and 2.3 for p > 1 and to Theorems 2.2
and 2.4 for h(z) \equiv 1 were investigated in [6] ([Theorems 5.1 and 5.2) and [?], respectively.
Sharpness of the inequalities. The sharpness of the estimations (2.2) – (2.7) can be discussed by
comparing them with the following result.
Remark 2.2. (a) For any n \in \BbbN there exists a polynomial Q\ast
n \in \wp n such that the inequality
\| Q\ast
n\| \infty \geq c10n \| Q\ast
n\| A2(B) , (2.12)
is true for the unit disk B and the weight function h(z) \equiv 1.
(b) For any n \in \BbbN there exists a polynomial P \ast
n \in \wp n, a region G\ast
1 \subset \BbbC and a compact subset
F \ast \Subset \Omega \setminus G\ast
1 such that
| P \ast
n(z)| \geq c11
\surd
n
d(z, L)
\| P \ast
n\| A2(G\ast
1)
| \Phi (z)| n+1 for all z \in F. (2.13)
(c) For any n \in \BbbN there exists a polynomial T \ast
n \in \wp nsuch that the inequality
\| T \ast
n\| \infty \geq c12n
2 \| T \ast
n\| A2(h\ast ,B) (2.14)
is true for the unit disk B and the weight function h\ast (z) = | z - z1| 2.
3. Some auxiliary results. Throughout this paper, for a > 0 and b > 0, we use the expression
“a \preceq b” (order inequality), if a \leq cb. The expression “a \asymp b” means that “a \preceq b” and “b \preceq a”.
Let L be a K-quasiconformal curve, then there exists a quasiconformal reflection y(\cdot ) across L
such that y(G) = \Omega , y(\Omega ) = G and y(\cdot ) is fixed the points of L. There exists a quasiconformal
reflection y(\cdot ) satisfying the following conditions [10; 12, p. 26]:
| y(\zeta ) - z| \asymp | \zeta - z| for all z \in L and \varepsilon < | \zeta | < 1
\varepsilon
,
\bigm| \bigm| y \zeta
\bigm| \bigm| \asymp | y\zeta | \asymp 1, \varepsilon < | \zeta | < 1
\varepsilon
, (3.1)
\bigm| \bigm| y \zeta
\bigm| \bigm| \asymp | y(\zeta )| 2 , | \zeta | < \varepsilon ;
\bigm| \bigm| y \zeta
\bigm| \bigm| \asymp | \zeta | - 2 , | \zeta | > 1
\varepsilon
,
and for the Jacobian Jy = | yz| 2 - | y z| 2 of y(\cdot ), the relation
\bigm| \bigm| y \zeta
\bigm| \bigm| 2 \leq 1
1 - k2
Jy holds, where
k =
K2 - 1
K2 + 1
. Such quasiconformal reflection y(\cdot ) is called a regular quasiconformal reflection
across L.
Let L be a quasicircle and y(\cdot ) be a regular quasiconformal reflection across L. For any R > 1,
we put L\ast := y(LR), G
\ast := \mathrm{i}\mathrm{n}\mathrm{t}L\ast , \Omega \ast := \mathrm{e}\mathrm{x}\mathrm{t}L\ast , and denote by \Phi R the conformal mapping of \Omega \ast
onto \Delta with the normalization \Phi R(\infty ) = \infty , \mathrm{l}\mathrm{i}\mathrm{m}z\rightarrow \infty
\Phi R(z)
z
> 0, and let \Psi R := \Phi - 1
R . Moreover,
for any t > 1, we set L\ast
t := \{ z : | \Phi R(z)| = t\} , G\ast
t := \mathrm{i}\mathrm{n}\mathrm{t}L\ast
t , \Omega
\ast
t := \mathrm{e}\mathrm{x}\mathrm{t}L\ast
t . According to [11], for
all z \in L\ast and t \in L such that | z - t| = d(z, L), we have
ISSN 1027-3190. Укр. мат. журн., 2017, т. 69, № 5
POLYNOMIAL INEQUALITIES IN QUASIDISKS ON WEIGHTED BERGMAN SPACE 587
d(z, L) \asymp d(t, LR) \asymp d(z, L\ast
R), (3.2)
| \Phi R(z)| \leq | \Phi R(t)| \leq 1 + c(R - 1).
Lemma 3.1 [1]. Let L be a quasicircle, z1 \in L, z2, z3 \in \Omega \cap \{ z : | z - z1| \leq d(z1, Lr0)\} , where
Lr0 := \{ \zeta : | \varphi (\zeta )| = r0, 0 < r0 < 1\} and r0 := r0(G) is a constant, depending on G, 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 are
| z1 - z2| \asymp | z1 - z3| and | w1 - w2| \asymp | w1 - w3| .
(b) If | z1 - z2| \preceq | z1 - z3| , then\bigm| \bigm| \bigm| \bigm| w1 - w3
w1 - w2
\bigm| \bigm| \bigm| \bigm| \varepsilon \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| c ,
where \varepsilon < 1 and c > 1.
Lemma 3.2. Let G be a K-quasidisk for some 0 \leq k < 1. Then
| \Psi (w1) - \Psi (w2)| \succeq | w1 - w2| 1+k
for all w1, w2 \in \Delta .
This fact is derived to appropriate the results for the estimation | \Psi \prime (\tau )| (see [22, p. 287]
(Lemma 9.9) and [12] (Theorem 2.8)).
Lemma 3.3 ([?], Lemma 2.3). Let L be a quasicircle. For arbitrary R > 1, there exist numbers
\rho 1, \rho 2, \rho 3 and \rho 4 such that \rho 1 < \rho 2 and \rho 3 < \rho 4 and the following conditions are satisfied:
(1) G
\ast
\rho 1 \subseteq G \subseteq G
\ast
\rho 2 and G
\ast
\rho 3 \subseteq GR \subseteq G
\ast
\rho 4 ,
(2) \rho 1 - 1 \asymp \rho 2 - 1 \asymp \rho 3 - 1 \asymp \rho 4 - 1 \asymp R - 1.
Lemma 3.4 ([4], Lemma 3.3). Let L be a quasicircle; h(z) is defined as in (1.1). Then, for
arbitrary Pn(z) \in \wp n, any R > 1 and n \in \BbbN , we have
\| Pn\| Ap(h,GR) \preceq [1 + c(R - 1)]n+1/p \| Pn\| p , p > 0.
4. Proofs of the main results. 4.1. Proofs of Theorems 2.1 and 2.3. We begin with the proof
of the Theorem 2.3. Let L \in Q\alpha for some
1
2
\leq \alpha \leq 1, R = 1 +
1
n
and R1 := 1 + \varepsilon 1(R - 1).
Denote by \{ \zeta j\} , 1 \leq j \leq m \leq n, the zeros of Pn(z) lying on \Omega (if such zeros exist) and let
Bm(z) :=
m\prod
j=1
\~Bj(z) =
m\prod
j=1
\Phi (z) - \Phi (\zeta j)
1 - \Phi (\zeta j)\Phi (z)
(4.1)
denote a Blaschke function with respect of the zeros of Pn(z). For any p > 0 and z \in \Omega , let us set
Hn,p(z) :=
\biggl(
Pn(z)
Bm(z)\Phi n+1(z)
\biggr) p/2
. (4.2)
The function Hn,p(z), Hn,p(\infty ) = 0, is analytic in \Omega , continuous on \Omega and does not have zeros in
\Omega . We take an arbitrary continuous branch of the Hn,p(z) and for this branch we maintain the same
designation. Cauchy integral representation for the region \Omega is given as:
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588 F. G. ABDULLAYEV, T. TUNÇ, G. A. ABDULLAYEV
Hn,p(z) = - 1
2\pi i
\int
LR1
Hn,p(\zeta )
d\zeta
\zeta - z
, z \in \Omega R1 .
Since | Bm(\zeta )| = 1 for \zeta \in L, then for arbitrary \varepsilon , 0 < \varepsilon < \varepsilon 1, there exists a circle | w| = 1 +
\varepsilon
n
,
such that for any j = 1,m the following is satisfied:\bigm| \bigm| \~Bj(\zeta )
\bigm| \bigm| > 1 - \varepsilon , \zeta \in L1+\varepsilon /n.
Then | Bm(\zeta )| > (1 - \varepsilon )m \succeq 1 for \zeta \in LR1 and | Bm(z)| \leq 1 for z \in \Omega R1 . On the other hand,
| \Phi (\zeta )| = R1 > 1 for \zeta \in LR1 . Therefore for any z \in \Omega R1 from the inequality
| Hn,p(z)| \leq
1
2\pi
\int
LR1
\bigm| \bigm| Hn,p(\zeta )
\bigm| \bigm| | d\zeta |
| \zeta - z|
we have
| Pn(z)| p/2 \leq
\bigm| \bigm| Bm(z) \Phi n+1(z)
\bigm| \bigm| p/2
2\pi
\int
LR1
\bigm| \bigm| \bigm| \bigm| Pn(\zeta )
Bm(\zeta ) \Phi n+1(\zeta )
\bigm| \bigm| \bigm| \bigm| p/2 | d\zeta |
| \zeta - z|
\preceq
\preceq
\bigm| \bigm| \Phi n+1(z)
\bigm| \bigm| p/2 \int
LR1
\bigm| \bigm| Pn(\zeta )
\bigm| \bigm| p/2 | d\zeta |
| \zeta - z|
. (4.3)
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 |
| \zeta - z|
\right)
2
=
\left( \int
LR1
h1/2(\zeta ) | Pn(\zeta )| p/2
| d\zeta |
h1/2(\zeta )| \zeta - z|
\right)
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
=: \scrA n\scrB n(w), (4.4)
where fn,p(t) := h1/p(\Psi (t))Pn(\Psi (t))(\Psi \prime (t))2/p, | t| = R1.
For the estimate integral \scrA n, 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
\scrA n =
\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.
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POLYNOMIAL INEQUALITIES IN QUASIDISKS ON WEIGHTED BERGMAN SPACE 589
On the other hand, by applying mean value estimation, we have\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 ,
and therefore
\scrA n \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.
By taking into account the fact that at most two of the discs with center t\prime k are intersecting, we obtain
\scrA n \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 3.4, we get
\scrA n \preceq n
\int \int
GR\setminus G
h(\zeta ) | Pn(\zeta )| p d\sigma \zeta \preceq n \| Pn\| pp . (4.5)
To estimate the integral \scrB n(w), we introduce for wj := \Phi (zj), \varphi j := \mathrm{a}\mathrm{r}\mathrm{g}wj , j = 1,m, and for any
\rho > 1
\Delta 1(\rho ) :=
\biggl\{
t = rei\theta : r > \rho ,
\varphi m + \varphi 1
2
\leq \theta <
\varphi 1 + \varphi 2
2
\biggr\}
,
\Delta j(\rho ) :=
\biggl\{
t = rei\theta : r > \rho ,
\varphi j - 1 + \varphi j
2
\leq \theta <
\varphi j + \varphi j+1
2
\biggr\}
, j = 2,m - 1,
\Delta m(\rho ) :=
\biggl\{
t = rei\theta : r > \rho ,
\varphi m - 1 + \varphi m
2
\leq \theta <
\varphi m + \varphi 1
2
\biggr\}
.
Let \Omega j := \Psi (\Delta j), \Omega
j
\rho := \Psi (\Delta j(\rho )), \Delta j := \Delta j(1) and
Lj := L \cap \Omega
j
, Lj
\rho := L\rho \cap \Omega
j
\rho . (4.6)
Then we get
\scrB n(w) =
\int
| t| =R1
| dt|
h(\Psi (t)) | \Psi (t) - \Psi (w)| 2
\preceq
\preceq
m\sum
j=1
\int
\Phi (Lj
R1
)
| dt| \prod m
j=1
| \Psi (t) - \Psi (wj)| \gamma j | \Psi (t) - \Psi (w)| 2
\asymp
\asymp
m\sum
j=1
\int
\Phi (Lj
R1
)
| dt|
| \Psi (t) - \Psi (wj)| \gamma j | \Psi (t) - \Psi (w)| 2
:=
m\sum
j=1
Bn,j(w), (4.7)
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590 F. G. ABDULLAYEV, T. TUNÇ, G. A. ABDULLAYEV
since the points \{ zj\} mj=1 \in L are distinct. It remains to estimate the integrals Bn,j(w) for each
j = 1,m.
Firstly, we assume that z \in LR. For simplicity of our next calculations, we assume that m = 1.
We put \Phi (L1
R1
) =
3\bigcup
i=1
Ki(R1), where
K1(R1) :=
\Bigl\{
t \in \Phi (L1
R1
) : | t - w1| <
c1
n
\Bigr\}
,
K2(R1) :=
\Bigl\{
t \in \Phi (L1
R1
) :
c1
n
\leq | t - w1| < c2
\Bigr\}
,
K3(R1) :=
\bigl\{
t \in \Phi (L1
R1
) : c2 \leq | t - w1| < c3 < \mathrm{d}\mathrm{i}\mathrm{a}\mathrm{m}G
\bigr\}
and \Phi (L1
R) =
3\bigcup
i=1
Ki(R), where
K1(R) :=
\biggl\{
\tau \in \Phi (L1
R) : | \tau - w1| <
2c1
n
\biggr\}
,
K2(R) :=
\biggl\{
\tau \in \Phi (L1
R) :
2c1
n
\leq | \tau - w1| < c2
\biggr\}
,
K3(R) :=
\bigl\{
\tau \in \Phi (L1
R) : c2 \leq | \tau - w1| < c3 < \mathrm{d}\mathrm{i}\mathrm{a}\mathrm{m}G
\bigr\}
.
Let w \in \Phi (L1
R) an arbitrary fixed point. We will estimate the following integral for each cases
with w \in Ki(R) and t \in Ki(R1), i = 1, 2, 3:
Bn,1(w) :=
\int
\Phi (L1
R1
)
| dt|
| \Psi (t) - \Psi (w1)| \gamma 1 | \Psi (t) - \Psi (w)| 2
=
=
3\sum
i=1
\int
Ki(R1)
| dt|
| \Psi (t) - \Psi (w1)| \gamma 1 | \Psi (t) - \Psi (w)| 2
=:
=:
3\sum
i=1
Bi
n,1(w). (4.8)
Case 1. Let w \in K1(R). We put K1
j (R1) :=
\bigl\{
t \in \Phi (L1
R1
) : | t - w1| < | t - w|
\bigr\}
, K2
j (R1) :=
:= Kj(R)\setminus K1
j (R1), j = 1, 2. Since \Phi \in \mathrm{L}\mathrm{i}\mathrm{p}\alpha on \Omega , according to Lemma 3.1 we have
B1
n,1(w) :=
\int
K1
1 (R1)
| dt|
| \Psi (t) - \Psi (w1)| 2+\gamma 1
+
\int
K2
1 (R1)
| dt|
| \Psi (t) - \Psi (w)| 2+\gamma 1
\preceq
\preceq
\int
K1
1 (R1)
| dt|
| t - w1| (2+\gamma 1)/\alpha
+
\int
K2
1 (R1)
| dt|
| t - w| (2+\gamma 1)/\alpha
\preceq n(2+\gamma 1)/\alpha - 1, \gamma 1 > 0,
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POLYNOMIAL INEQUALITIES IN QUASIDISKS ON WEIGHTED BERGMAN SPACE 591
B1
n,1(w) =
\int
K1(R1)
| \Psi (t) - \Psi (w1)| - \gamma 1 | dt|
| \Psi (t) - \Psi (w)| 2
\preceq
\int
K1(R1)
| dt|
| t - w| 2/\alpha
\preceq n2/\alpha - 1, \gamma 1 \leq 0,
B2
n,1(w) :=
2\sum
i=1
\int
Ki
2(R1)
| dt|
| \Psi (t) - \Psi (w1)| \gamma 1 | \Psi (t) - \Psi (w)| 2
\preceq
\preceq
\int
K1
2 (R1)
| dt|
| t - w1| (2+\gamma 1)/\alpha
+
\int
K2
2 (R1)
| dt|
| t - w| (2+\gamma 1)/\alpha
\preceq n(2+\gamma 1)/\alpha - 1, \gamma 1 > 0,
B2
n,1(w) =
\int
K2(R1)
| \Psi (t) - \Psi (w1)| - \gamma 1 | dt|
| \Psi (t) - \Psi (w)| 2
\preceq
\int
K2(R1)
| dt|
| t - w| 2/\alpha
\preceq n2/\alpha - 1, \gamma 1 \leq 0.
Since | t - w1| \geq c2 and | t - w| \geq | | t - w1| - | w - w1| | \geq c2 -
2c1
n
\succeq 1, for t \in K3(R1) and
w \in K1(R), then we obtain
B3
n,1(w) \preceq
\int
K3(R1)
| dt| \preceq | K3(R1)| \preceq 1, \gamma 1 > 0,
B3
n,1(w) \preceq
\int
K3(R1)
| dt|
| \Psi (t) - \Psi (w)| 2
\preceq
\int
K3(R1)
| dt| \preceq 1, \gamma 1 \leq 0.
Case 2. Let w \in K2(R). Then
B1
n,1(w) \preceq
\int
K1(R1)
| dt|
| t - w1| \gamma 1/\alpha | t - w| 2/\alpha
\preceq
\preceq 1
(R1 - 1)\gamma 1/\alpha
\Bigl( c2
n
\Bigr) 2/\alpha \int
K1(R1)
| dt| \preceq n(2+\gamma 1)/\alpha - 1, \gamma 1 > 0,
B1
n,1(w) \preceq
\int
K1(R1)
| dt|
| t - w| 2/\alpha
\preceq n2/\alpha - 1, \gamma 1 \leq 0,
B2
n,1(w) :=
2\sum
i=1
\int
Ki
2(R1)
| dt|
| \Psi (t) - \Psi (w1)| \gamma 1 | \Psi (t) - \Psi (w)| 2
\preceq
\preceq
\int
K1
2 (R1)
| dt|
| t - w1| (2+\gamma 1)/\alpha
+
\int
K2
2 (R1)
| dt|
| t - w| (2+\gamma 1)/\alpha
\preceq n(2+\gamma 1)/\alpha - 1, \gamma 1 > 0,
B2
n,1(w) =
\int
K2(R1)
| \Psi (t) - \Psi (w1)| - \gamma 1 | dt|
| \Psi (t) - \Psi (w)| 2
\preceq
\int
K2(R1)
| dt|
| t - w| 2/\alpha
\preceq n2/\alpha - 1, \gamma 1 \leq 0.
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592 F. G. ABDULLAYEV, T. TUNÇ, G. A. ABDULLAYEV
Since | t - w1| \geq c2 and | t - w| \geq 1 for t \in K3(R1) and w \in K2(R), then
B3
n,1(w) :=
\int
K3(R1)
| dt|
| \Psi (t) - \Psi (w1)| \gamma 1 | \Psi (t) - \Psi (w)| 2
\preceq
\preceq
\biggl(
1
c2
\biggr) \gamma 1/\alpha \int
K3(R1)
| dt|
| t - w| 2/\alpha
\preceq n2/\alpha - 1, \gamma 1 > 0,
B3
n,1(w) =
\int
K3(R1)
| \Psi (t) - \Psi (w1)| - \gamma 1 | dt|
| \Psi (t) - \Psi (w)| 2
\preceq
\preceq
\int
K3(R1)
| dt|
| \Psi (t) - \Psi (w)| 2
\preceq
\int
K3(R1)
| dt|
| t - w| 2/\alpha
\preceq n2/\alpha - 1, \gamma 1 \leq 0.
Case 3. Let w \in K3(R). Then
B1
n,1(w) \preceq
\int
K1(R1)
| dt|
| t - w1| \gamma 1/\alpha
\preceq n\gamma 1/\alpha - 1, \gamma 1 > 0,
B1
n,1(w) \preceq
\int
K1(R1)
| dt|
| t - w| 2/\alpha
\preceq
\biggl(
c2 -
2c1
n
\biggr) - 2/\alpha \int
K1(R1)
| dt| \preceq 1, \gamma 1 \leq 0,
B2
n,1(w) \preceq
\int
K2(R1)
| dt|
| \Psi (t) - \Psi (w1)| 2+\gamma 1
\preceq
\preceq
\int
K2(R1)
| dt|
| t - w1| (2+\gamma 1)/\alpha
\preceq n(2+\gamma 1)/\alpha - 1, \gamma 1 > 0,
B2
n,1(w) \preceq
\int
K2(R1)
| dt| \preceq 1, \gamma 1 \leq 0,
B3
n,1(w) \preceq
\int
K3(R1)
| dt|
| t - w| 2/\alpha
\preceq n2/\alpha - 1, \gamma 1 > 0,
B3
n,1(w) \preceq
\int
K3(R1)
| dt|
| t - w| 2/\alpha
\preceq n2/\alpha - 1, \gamma 1 \leq 0.
By combining the estimates obtained in the Cases 1 – 3 with (4.3) – (4.5), (4.7) and (4.8), for any
p > 0 and for all z \in LR, we obtain
| Pn(z)| \preceq \Gamma n \| Pn\| p , (4.9)
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POLYNOMIAL INEQUALITIES IN QUASIDISKS ON WEIGHTED BERGMAN SPACE 593
where
\Gamma n :=
\left\{
m\sum
j=1
n
\gamma j+2
p\alpha , if at least for one j, \gamma j > 0, j = 1,m,
n2/p\alpha , if \gamma j \leq 0, for all j = 1,m.
The estimation (4.9) satisfied for all z \in LR. We show that it occurs on L. For R > 1, let w = \varphi R(z)
denotes the univalent conformal mapping of GR onto B normalized by \varphi R(0) = 0, \varphi \prime
R(0) > 0, and
let \{ \xi j\} , 1 \leq j \leq m \leq n, zeros of Pn(z), lying on GR (if such zeros exist). Let
Bm,R(z) :=
m\prod
j=1
\~Bj,R(z) =
m\prod
j=1
\varphi R(z) - \varphi R(\xi j)
1 - \varphi R(\xi j)\varphi R(z)
(4.10)
denotes a Blaschke function with respect to zeros \{ \xi j\} , 1 \leq j \leq m \leq n, of Pn(z). Clearly,
| Bm,R(z)| \equiv 1, z \in LR; | Bm,R(z)| < 1, z \in GR.
For any z \in GR, let us set
Hn(z) :=
Pn(z)
Bm,R(z)
.
The function Hn(z) is analytic in GR, continuous on GR and does not have zeros in GR. Then,
applying maximal modulus principle to Hn(z), we have\bigm| \bigm| \bigm| \bigm| Pn(z)
Bm,R(z)
\bigm| \bigm| \bigm| \bigm| \leq \mathrm{m}\mathrm{a}\mathrm{x}
\zeta \in GR
\bigm| \bigm| \bigm| \bigm| Pn(\zeta )
Bm,R(\zeta )
\bigm| \bigm| \bigm| \bigm| \leq \mathrm{m}\mathrm{a}\mathrm{x}
\zeta \in LR
| Pn(\zeta )| \preceq \Gamma n \| Pn\| p \forall z \in L,
and, therefore, we find
\mathrm{m}\mathrm{a}\mathrm{x}
z\in L
| Pn(z)| \preceq n(\gamma +2)/\alpha p \| Pn\| p \forall p > 0,
where \gamma := \mathrm{m}\mathrm{a}\mathrm{x}
\bigl\{
0; \gamma j , j = 1,m
\bigr\}
, and the proof (2.9) is completed.
Now, we will begin to proof (2.8). For the arbitrary fixed R = 1 +
1
n
, let us set L\ast := y(LR).
According to Lemma 3.3, the number \rho 1 := 1 + c1(R - 1) can be chosen as G
\ast
\rho 1 \subseteq G. Let
\rho := 1 +
\rho 1 - 1
2
. Let \{ \zeta j\} , 1 \leq j \leq m \leq n, zeros of Pn(z) lying in \Omega \ast and let
B\ast
m(z) :=
m\prod
j=1
B\ast
j,R(z) =
m\prod
j=1
\Phi R(z) - \Phi R(\zeta j)
1 - \Phi R(\zeta j)\Phi R(z)
, z \in \Omega \ast ,
denotes a Blaschke function with respect of the zeros \{ \zeta j\} of Pn(z) in \Omega \ast . For any p > 0 and
z \in \Omega \ast , we define
S\ast
n,p(z) :=
\Biggl(
Pn(z)
B\ast
m(z)\Phi n+1
R (z)
\Biggr) p/2
.
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594 F. G. ABDULLAYEV, T. TUNÇ, G. A. ABDULLAYEV
The function S\ast
n,p(z) is analytic in \Omega \ast , continuous on \Omega \ast , S\ast
n,p(\infty ) = 0 and does not have zeros in
\Omega \ast . We take an arbitrary continuous branch of the S\ast
n,p(z) and for this branch we maintain the same
designation. Then, by the Cauchy integral formula for the region \Omega \ast
R1
, we have
S\ast
n,p(z) = - 1
2\pi i
\int
L\ast
\rho
S\ast
n,p(\zeta )
d\zeta
\zeta - z
, z \in \Omega \ast
R1
. (4.11)
Since | B\ast
m(\zeta )| = 1, for \zeta \in L, then, for arbitrary \varepsilon , 0 < \varepsilon < \varepsilon 1, there exists a circle | w| = 1+
\varepsilon
n
,
such that for any j = 1,m the following is satisfied:\bigm| \bigm| B\ast
j,R(\zeta )
\bigm| \bigm| > 1 - \varepsilon , \zeta \in L1+\varepsilon /n.
Then | B\ast
m(\zeta )| > (1 - \varepsilon )m \succeq 1 for \zeta \in L\ast
\rho and | B\ast
m(z)| \leq 1 for z \in \Omega \ast
\rho . On the other hand,
| \Phi R(\zeta )| = R > 1 for \zeta \in L\ast
\rho . Therefore, for any z = zj , j = 1,m, from (4.11) we get
| Pn (zj)| p/2 \leq
\bigm| \bigm| B\ast
m(z)\Phi n+1
R (zj)
\bigm| \bigm| p/2
2\pi
\int
L\ast
\rho
\bigm| \bigm| \bigm| \bigm| \bigm| Pn(\zeta )
B\ast
m(\zeta ) \Phi n+1
R (\zeta )
\bigm| \bigm| \bigm| \bigm| \bigm|
p/2
| d\zeta |
| \zeta - zj |
\preceq
\preceq
\bigm| \bigm| \Phi n+1
R (zj)
\bigm| \bigm| p/2 \int
L\ast
\rho
| Pn(\zeta )| p/2
| d\zeta |
| \zeta - zj |
. (4.12)
According to Lemma 3.3, there exists a number \rho 2 > \rho 1 such that G \subseteq G
\ast
\rho 2 and \rho 2 - 1 \asymp R - 1.
Using (3.2) we have | \Phi R(zj)| \leq \rho 2 \leq 1+ c(R - 1), and so,
\bigm| \bigm| \Phi n+1
R (zj)
\bigm| \bigm| \preceq 1. Therefore, from (4.12)
we have
| Pn (zj)| p/2 \preceq
\int
L\ast
\rho
| Pn(\zeta )| p/2
| d\zeta |
| \zeta - zj |
. (4.13)
Let
bm(z) :=
m\prod
j=1
b\ast j (z) =
m\prod
j=1
\Phi R(z) - \Phi R(zj)
1 - \Phi R(zj)\Phi R(z)
denote a Blaschke function for the weight function h(z) with respect to they singular points \{ zj\} \in L,
j = 1,m. Multiplying the numerator and the denominator of the last integrand by
\prod m
j=1
\bigm| \bigm| \bigm| \bigm| \bigm| \zeta - zj
b\ast j (\zeta )
\bigm| \bigm| \bigm| \bigm| \bigm|
\gamma j/2
,
replacing the variable w = \Phi R(z) and applying the Hölder inequality, from (4.13) we obtain
| Pn (zj)| p/2 \preceq
\left( \int
| t| =\rho
m\prod
j=1
\bigm| \bigm| \bigm| \bigm| \bigm| \Psi R(t) - \Psi R(wj)
b\ast j (\Psi R(t))
\bigm| \bigm| \bigm| \bigm| \bigm|
\gamma j
| Pn (\Psi R(t))| p
\bigm| \bigm| \Psi \prime
R(t)
\bigm| \bigm| 2 | dt|
\right)
1/2
\times
\times
\left( \int
| t| =\rho
m\prod
j=1
\bigm| \bigm| \bigm| \bigm| \bigm| \Psi R(t) - \Psi R(wj)
b\ast j (\Psi R(t))
\bigm| \bigm| \bigm| \bigm| \bigm|
- \gamma j | dt|
| \Psi R(t) - \Psi R(wj)| 2
\right)
1/2
. (4.14)
ISSN 1027-3190. Укр. мат. журн., 2017, т. 69, № 5
POLYNOMIAL INEQUALITIES IN QUASIDISKS ON WEIGHTED BERGMAN SPACE 595
Since
\bigm| \bigm| b\ast j (\zeta )\bigm| \bigm| = 1, j = 1,m, for \zeta \in L\ast , then, for arbitrary \varepsilon > 0 there exists a circle
| w| = 1 +
\varepsilon
n
, such that for any j = 1, 2, . . . ,m the following is satisfied:
\bigm| \bigm| b\ast j (\zeta )\bigm| \bigm| > 1 - \varepsilon . Then\bigm| \bigm| b\ast j (\zeta )\bigm| \bigm| > (1 - \varepsilon ) \succeq 1 for \zeta \in L\ast
\rho and
\bigm| \bigm| b\ast j (z)\bigm| \bigm| \leq 1 for z \in \Omega \ast
\rho . According this reason, from (4.14)
we get
| Pn (zj)| p/2 \preceq
\left( \int
L\ast
\rho
| gn,p(t)| p | dt|
\right)
1/2\left( \int
L\ast
\rho
m\prod
j=1
| dt|
| \Psi R(t) - \Psi R(wj)| 2+\gamma j
\right)
1/2
:= Jn,1Jn,2, (4.15)
where
gn,p(t) :=
m\prod
j=1
\bigl[
\Psi R(t) - \Psi R(wj)
\bigr] \gamma j/pPn (\Psi R(t))
\bigl[
\Psi \prime
R(t)
\bigr] 2/p
, | t| = \rho .
The integral Jn,1 we will estimate analogously to estimation of the integral \scrA n from (4.4). For this,
we separate the circle | t| = \rho to n equal parts \eta n with \mathrm{m}\mathrm{e}\mathrm{s} \eta n =
2\pi \rho
n
and by applying the mean
value theorem to the integral Jn,1, we have
(Jn,1)
2 =
\int
| t| =\rho
| gn,p(t)| p | dt| =
n\sum
k=1
\int
\eta k
| gn,p(t)| p | dt| =
n\sum
k=1
\bigm| \bigm| gn,p \bigl( t\prime k\bigr) \bigm| \bigm| p\mathrm{m}\mathrm{e}\mathrm{s} \eta k, t\prime k \in \eta k.
On the other hand, by applying mean value estimation\bigm| \bigm| gn,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
| gn,p(\xi )| p d\sigma \xi ,
we obtain
(Jn,1)
2 \preceq
n\sum
k=1
\mathrm{m}\mathrm{e}\mathrm{s} \eta k
\pi
\bigl( \bigm| \bigm| t\prime k\bigm| \bigm| - 1
\bigr) 2 \int \int
| \xi - t\prime k| <| t\prime k| - 1
| gn,p(\xi )| p d\sigma \xi , t\prime k \in \eta k.
By taking into account, at most two of the discs with center t\prime k are intersecting, we have
(Jn,1)
2 \preceq \mathrm{m}\mathrm{e}\mathrm{s} \eta 1
(| t\prime 1| - 1)2
\int \int
1<| \xi | <\rho 1
| gn,p(\xi )| p d\sigma \xi \preceq n
\int \int
1<| \xi | <\rho 1
| gn,p(\xi )| p d\sigma \xi .
By replacing the variable w = \Phi R(z) and according to inclusion G
\ast
\rho 1 \subseteq G, for Jn,1 we get
(Jn,1)
2 \preceq n
\int \int
G\ast
\rho 1
\setminus G\ast
h(\zeta ) | Pn(\zeta )| p d\sigma \zeta \preceq n \| Pn\| pp . (4.16)
Let’s estimate
(Jn,2)
2 =
\int
| t| =\rho
m\prod
j=1
| dt|
| \Psi R(t) - \Psi R(wj)| 2+\gamma j
.
ISSN 1027-3190. Укр. мат. журн., 2017, т. 69, № 5
596 F. G. ABDULLAYEV, T. TUNÇ, G. A. ABDULLAYEV
Since the points \{ zj\} mj=1 \in L are distinct, we obtain
(Jn,2)
2 \asymp
\int
| t| =\rho
| dt|
| \Psi R(t) - \Psi R(wj)| 2+\gamma j
.
Now, since \Phi \in \mathrm{L}\mathrm{i}\mathrm{p}\alpha , we have
(Jn,2)
2 \preceq
\int
| t| =\rho
| dt|
| t - wj | (2+\gamma j)/\alpha
\preceq n(2+\gamma j)/\alpha - 1. (4.17)
Combining estimations (4.13) – (4.17), we get
| Pn (zj)| \preceq n(2+\gamma j)/\alpha p \| Pn\| p ,
and the proof (2.8) is completed.
The proof of the Theorem 2.1 (estimations (2.2) and (2.3)) obtained analogously to the proof of
Theorem 2.3. In this case, we change estimation under the condition \Phi \in \mathrm{L}\mathrm{i}\mathrm{p}\alpha with estimation from
the Lemma 3.2.
4.2. Proofs of Theorems 2.2 and 2.4. Let us begin to proof of Theorem 2.4. Let z \in \Omega R be an
arbitrary fixed point. Then z \in \Omega j
R for some j = 1,m. From (4.3) we have
| Pn(z)| p/2 \preceq
\bigm| \bigm| \Phi n+1(z)
\bigm| \bigm| p/2
d(z, LR)
\int
LR
\bigm| \bigm| Pn(\zeta )
\bigm| \bigm| p/2| d\zeta | . (4.18)
Analogously to the estimations (4.4) – (4.9), for each j = 1,m we obtain\left( \int
LR
| Pn(\zeta )| p/2 | d\zeta |
\right)
2
\preceq n \| Pn\| pp
\int
| t| =R
| dt|
| \Psi (t) - \Psi (wj)| \gamma j
.
Further \int
| t| =R
| dt|
| \Psi (t) - \Psi (wj)| \gamma j
\preceq
\int
| t| =R
| dt|
| t - wj | \gamma j/\alpha
\preceq \delta n. (4.19)
Therefore, from (4.18) we get
| Pn(z)| \preceq
\biggl( \surd
n\delta n
d(z, LR)
\biggr) 2/p
| \Phi (z)| n+1 \| Pn\| p , z \in \Omega j
R,
and we obtain the proof of (2.5).
The proof of the Theorem 2.2 will be obtained from (4.19), according to Lemma 3.2\int
| t| =R
| dt|
| \Psi (t) - \Psi (wj)| \gamma j
\preceq
\int
| t| =R
| dt|
| t - wj | \gamma j(1+\kappa )
\preceq \mu n.
ISSN 1027-3190. Укр. мат. журн., 2017, т. 69, № 5
POLYNOMIAL INEQUALITIES IN QUASIDISKS ON WEIGHTED BERGMAN SPACE 597
4.3. Proof of Remark 2.2. The estimations (2.12) and (2.13) are shown in [?] (Theorem 17).
Let us prove (2.14). Denote by \{ Kn(z)\} , \mathrm{d}\mathrm{e}\mathrm{g}Kn = n, n = 0, 1, 2, . . . , the system of orthonormal
polynomials with the weight function h(z) for the region G, i.e., Kn(z) := \alpha nz
n +\alpha n - 1z
n - 1 + . . .
. . .+ \alpha 0, \alpha n > 0 and \int \int
G
h(z)Kn(z)Km(z) d\sigma z = \delta n,m,
where \delta n,m is the Kronecker’s symbol.
Let \wp p
n := \wp p
n,h = \wp p
n,h,G be the space \wp n with the norm (1.2) for p \geq 1. Similarly, \wp \infty
n := \wp \infty
n,G
be the space \wp n with the norm (1.3).
Consider a sequence of lineer operators In,h : \wp 2
n,h \rightarrow \wp \infty
n,G
, In,h(Pn) = Pn, with the norms
\| In,h\| := \mathrm{s}\mathrm{u}\mathrm{p}
\Bigl\{
\| Pn\| \infty : Pn \in \wp n, \| Pn\| A2(h,G) \leq 1
\Bigr\}
.
In [3] (Theorem 1), it was proved the following theorem.
Theorem A. Suppose that there exists \xi \in L such that \| Kn\| \infty \asymp | Kn(\xi )| and, for a certain
number \beta \geq 0, \| Kn\| \infty \asymp n\beta . Then \| In,h\| \asymp n\beta +1/2.
Let G = B and h(z) = | z - 1| 2 . Then [27, p. 76]
Kn(z) =
2\sqrt{}
\pi (n+ 1)(n+ 2)(n+ 3)
n\sum
j=0
(j + 1)zj(1 + z + . . .+ zn - j).
Therefore,
\| Kn\| A\infty (1,B) = Kn(1) =
2\sqrt{}
\pi (n+ 1)(n+ 2)(n+ 3)
n\sum
j=0
(j + 1)(n - j + 1) =
=
1
3
\surd
\pi
\sqrt{}
(n+ 1)(n+ 2)(n+ 3).
On the other hand, according to [3] (Lemma 1), we obtain
\| In,h\| =
\left( n\sum
j=0
| Kj(1)| 2
\right) 1/2 = 1
3
\surd
\pi
\left( n\sum
j=0
(j + 1)(j + 2)(j + 3)
\right) 1/2 =
=
1
6
\surd
\pi
\sqrt{}
(n+ 1)(n+ 2)(n+ 3)(n+ 4).
Therefore, we can choose T \ast
n \in \wp n such that \| In,h\| = \| T \ast
n\| A\infty (1,B) .
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.
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.
ISSN 1027-3190. Укр. мат. журн., 2017, т. 69, № 5
598 F. G. ABDULLAYEV, T. TUNÇ, G. A. ABDULLAYEV
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 singularity 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. Pol. Math. – 2014. – 111. – P. 39 – 58.
8. Abdullayev F. G., Özkartepe N. P. On the behavior of the algebraic polynomial in unbounded regions with piecewise
Dini-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. Ahlfors L. Lectures on quasiconformal mappings. – Princeton, NJ: Van Nostrand, 1966.
11. Andrievskii V. V. Constructive characterization of the harmonic functions in domains with quasiconformal bound-
ary // Quasiconformal Continuation and Approximation by Function in the Set of the Complex Plane. – Kiev, 1985
(in Russian).
12. 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.
13. Andrievskii V. V. Weighted polynomial inequalities in the complex plane // J. Approxim. Theory. – 2012. – 164,
№ 9. – P. 1165 – 1183.
14. Batchayev I. M. Integral representations in domains with quasiconformal boundary and some of their applications:
Avtoref. dis. . . . cand. fiz.-mat. nauk. – Baku, 1981. – 16 p. (in Russian).
15. Belinskii P. P. General properties of quasiconformal mappings. – Novosibirsk: Nauka, 1974 (in Russian).
16. Becker J., Pommerenke C. Uber die quasikonforme Fortsetzung schlichten Funktionen // Math. Z. – 1978. – 161. –
S. 69 – 80.
17. Dzjadyk V. K. Introduction to the theory of uniform approximation of function by polynomials. – Moskow: Nauka,
1977.
18. Lehto O., Virtanen K. I. Quasiconformal mapping in the plane. – Berlin: Springer-Verlag, 1973.
19. Lesley F. D. Hölder continuity of conformal mappings at the boundary via the strip method // Indiana Univ. Math. J. –
1982. – 31. – P. 341 – 354.
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. Pommerenke Ch. Univalent functions. – Göttingen: Vandenhoeck and Ruprecht, 1975.
23. Pommerenke Ch. Boundary behaviour of conformal maps. – Berlin: Springer-Verlag, 1992.
24. Pommerenke Ch., Warschawski S. E. On the quantitative boundary behavior of conformal maps // Comment. Math.
Helv. – 1982. – 57. – P. 107 – 129.
25. Pritsker I. Comparing norms of polynomials in one and several variables // J. Math. Anal. and Appl. – 1997. – 216. –
P. 685 – 695.
26. Stylianopoulos N. Strong asymptotics for Bergman polynomials over domains with corners and applications // Const.
Approxim. – 2013. – 33. – P. 59 – 100.
27. Suetin P. K. Orthogonal polynomials over an area and Bieberbach polynomials // Tr. Mat. Inst. Akad. Nauk SSSR. –
1971. – 100. – P. 1 – 92 (in Russian).
28. Warschawski S. E. On differentiability at the boundary in conformal mapping // Proc. Amer. Math. Soc. – 1961. –
12. – P. 614 – 620.
29. Warschawski S. E. On Hölder continuity at the boundary in conformal maps // J. Math. and Mech. – 1968. – 18. –
P. 423 – 427.
30. Walsh J. L. Interpolation and approximation by rational functions in the complex domain // Amer. Math. Soc., 1960.
Received 11.09.16
ISSN 1027-3190. Укр. мат. журн., 2017, т. 69, № 5
|
| id | umjimathkievua-article-1719 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T02:11:20Z |
| publishDate | 2017 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/e7/e96684cd3f913deb071ec418a3d7d4e7.pdf |
| spelling | umjimathkievua-article-17192019-12-05T09:24:56Z Polynomial inequalities in quasidisks on weighted Bergman space Полiномiальнi нерiвностi у квазiдисках на зважених просторах Бергмана Abdullayev, G. A. Abdullayev, F. G. Tunç, E. Абдуллаєв, Г. А. Абдуллаєв, Ф. Г. Тунс, Е. We continue studying on the Nikol’skii and Bernstein –Walsh type estimations for complex algebraic polynomials in the bounded and unbounded quasidisks on the weighted Bergman space. Продовжено дослiдження оцiнок типу Нiкольського та Бернштейна – Уолша для комплексних алгебраїчних полiномiв в обмежених та необмежених квазiдисках на зважених просторах Бергмана. Institute of Mathematics, NAS of Ukraine 2017-05-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/1719 Ukrains’kyi Matematychnyi Zhurnal; Vol. 69 No. 5 (2017); 582-598 Український математичний журнал; Том 69 № 5 (2017); 582-598 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/1719/701 Copyright (c) 2017 Abdullayev G. A.; Abdullayev F. G.; Tunç E. |
| spellingShingle | Abdullayev, G. A. Abdullayev, F. G. Tunç, E. Абдуллаєв, Г. А. Абдуллаєв, Ф. Г. Тунс, Е. Polynomial inequalities in quasidisks on weighted Bergman space |
| title | Polynomial inequalities in quasidisks on weighted
Bergman space |
| title_alt | Полiномiальнi нерiвностi у квазiдисках на зважених просторах Бергмана |
| title_full | Polynomial inequalities in quasidisks on weighted
Bergman space |
| title_fullStr | Polynomial inequalities in quasidisks on weighted
Bergman space |
| title_full_unstemmed | Polynomial inequalities in quasidisks on weighted
Bergman space |
| title_short | Polynomial inequalities in quasidisks on weighted
Bergman space |
| title_sort | polynomial inequalities in quasidisks on weighted
bergman space |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/1719 |
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