Interference of the weight and boundary contour for algebraic polynomials in the weighted Lebesgue spaces. I
We study the order of the height of the modulus of arbitrary algebraic polynomials with respect to the weighted Lebesgue space, where the contour and the weight functions have some singularities.
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| author | Abdullayev, F. G. Özkartepe, N. P. Абдуллаєв, Ф. Г. Озкартепе, Н. П. |
| author_facet | Abdullayev, F. G. Özkartepe, N. P. Абдуллаєв, Ф. Г. Озкартепе, Н. П. |
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| description | We study the order of the height of the modulus of arbitrary algebraic polynomials with respect to the weighted Lebesgue
space, where the contour and the weight functions have some singularities. |
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UDC 517.5
P. Özkartepe, F. G. Abdullayev (Mersin Univ., Turkey)
INTERFERENCE OF THE WEIGHT AND BOUNDARY CONTOUR
FOR ALGEBRAIC POLYNOMIALS IN THE WEIGHTED LEBESGUE SPACES. I
ПЕРЕШКОДИ, ПОВ’ЯЗАНI З ВАГОЮ ТА ГРАНИЧНИМ КОНТУРОМ,
ДЛЯ АЛГЕБРАЇЧНИХ ПОЛIНОМIВ У ЗВАЖЕНИХ ПРОСТОРАХ ЛЕБЕГА
We study the order of the height of the modulus of arbitrary algebraic polynomials with respect to the weighted Lebesgue
space, where the contour and the weight functions have some singularities.
Вивчається порядок висоти модуля довiльних алгебраїчних полiномiв вiдносно зважених просторiв Лебега, в яких
контур та ваговi функцiї мають деякi сингулярностi.
1. Introduction. Let \BbbC be a complex plane, \BbbC := \BbbC \cup \{ \infty \} ; G \subset \BbbC be a bounded Jordan region,
with 0 \in G and the boundary L := \partial G be a closed Jordan curve, \Omega := \BbbC \setminus G = \mathrm{e}\mathrm{x}\mathrm{t}L. Let \wp n
denotes the class of arbitrary algebraic polynomials Pn(z) of degree at most n \in \BbbN := \{ 1, 2, . . .\} .
Let 0 < p \leq \infty . For a rectifiable Jordan curve L, we denote
\| Pn\| \scrL p
:= \| Pn\| \scrL p(h,L)
:=
\left( \int
L
h(z) | Pn(z)| p | dz|
\right) 1/p
, 0 < p <\infty ,
\| Pn\| \scrL \infty
:= \| Pn\| \scrL \infty (1,L) := \mathrm{m}\mathrm{a}\mathrm{x}
z\in L
| Pn(z)| , p = \infty .
Clearly, \| \cdot \| \scrL p
is a quasinorm (i.e., a norm for 1 \leq p \leq \infty and a p-norm for 0 < p < 1).
Denoted by w = \Phi (z), the univalent conformal mapping of \Omega onto \Delta := \{ w : | w| > 1\} with
normalization \Phi (\infty ) = \infty , \mathrm{l}\mathrm{i}\mathrm{m}z\rightarrow \infty
\Phi (z)
z
> 0 and \Psi := \Phi - 1. For t \geq 1, we 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 the fixed system of distinct points on curve L which is located in the positive
direction. For some fixed R0, 1 < R0 < \infty , and z \in GR0 , consider generalized Jacobi weight
function h (z) which is defined as follows:
h(z) := h0(z)
m\prod
j=1
| z - zj | \gamma j , (1.1)
where \gamma j > - 1, for all j = 1, 2, . . . ,m, and 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.
In many problems of approximation theory, theory of polynomials and others, often need to study
the following inequality:
\| Pn\| \scrL \infty
\leq c\mu n(L, h, p) \| Pn\| \scrL p(h,L)
, (1.2)
c\bigcirc P. ÖZKARTEPE, F. G. ABDULLAYEV, 2016
ISSN 1027-3190. Укр. мат. журн., 2016, т. 68, № 10 1365
1366 P. ÖZKARTEPE, F. G. ABDULLAYEV
where c = c(G, p) > 0 is a constant which is independent of n and Pn, and \mu n(L, h, p) \rightarrow \infty , n\rightarrow
\rightarrow \infty , depending on the geometrical properties of curve L and weight function h in the neighborhood
of the points \{ zj\} mj=1 . In most cases, these problems can be divided in two parts. Firstly, the case
where the boundary curve and weight function do not have singularities and secondly, in case where
boundary curve or (and) weight function have an any singularities.
The first classical result of (1.2)-type, in case h(z) \equiv 1 and L = \{ z : | z| = 1\} for 0 < p < \infty
is found by Jackson in [13]. The other classical results are similar to (1.2) belongs to Szegö and
Zigmund, in [24]. The estimation of (1.2)-type for 0 < p <\infty and h(z) \equiv 1 where L is a rectifiable
Jordan curve is investigated by Suetin in [25], Mamedhanov in [16, 17], Nikol’skii in [19, p. 122 –
133], Pritsker in [22], Andrievskii in [11] (Theorem 6), Abdullayev et al. [2 – 7] and etc. There
are more references regarding the inequality of (1.2)-type, we can find in Milovanovic et al. [18]
(Sect. 5.3).
The question arises: how can “pay off” singularity curve and weight function, so that, the
estimation of (1.2) has coincided with the estimation of where the boundary curve and weight
functions are not any singularities.
Let a rectifiable Jordan curve be L, has a natural parametrization z = z(s), 0 \leq s \leq l := \mathrm{m}\mathrm{e}\mathrm{s}L.
It is said to be L \in C(1, \lambda ), 0 < \lambda < 1, if z(s) is continuously differentiable and z\prime (s) \in \mathrm{L}\mathrm{i}\mathrm{p}\lambda . Let
L belong to C(1, \lambda ) everywhere except for a single point z1 \in L, i.e., the derivative z\prime (s) satisfies
the Lipschitz condition on the [0, l] and z(0) = z(l) = z1, but z\prime (0) \not = z\prime (l). Assume that L has a
corner at z1 with exterior angle \nu \pi , 0 < \nu \leq 2, and denote the set of such curves by C(1, \lambda , \nu ).
Suetin, in [27], investigated this problem in case p = 2 for orthonormal on L polynomials Qn(z)
with the weight function h defined as in (1.1) and for the curve L \in C(1, \lambda , \nu ). He showed that
the condition of “pay off” singularity curve and weight function at the points z1 can be given as
following:
(1 + \gamma 1) \nu 1 = 1. (1.3)
Under this conditions, for Qn(z) in case L \in C(1, \lambda , \nu ) Suetin [27] provided the following
estimation:
| Qn(z)| \leq c(L)
\surd
n, z \in L, (1.4)
where c(L) > 0 is a constant independent on n.
In this work we study the estimations of the (1.2)-type for more general regions of the complex
plane and we obtain the analog of the equality (1.3) correnponding to the general case.
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, such dependence
will be explicitly stated.
For any k \geq 0 and m > k, notation i = k,m means i = k, k + 1, . . . ,m.
Before giving our new results, we need to give some definitions and the notations. Let z = \psi (w)
be the univalent conformal mapping of B := \{ w : | w| < 1\} onto the G normalized by \psi (0) = 0,
\psi \prime (0) > 0. By [20, p. 286 – 294], we say a bounded Jordan region G is called \kappa -quasidisk,
0 \leq \kappa < 1, if any conformal mapping \psi can be extended to a K -quasiconformal, K =
1 + \kappa
1 - \kappa
, the
ISSN 1027-3190. Укр. мат. журн., 2016, т. 68, № 10
INTERFERENCE OF THE WEIGHT AND BOUNDARY CONTOUR FOR ALGEBRAIC . . . 1367
homeomorphism of the plane \BbbC on plane \BbbC . In that case, the curve L := \partial G is called a \kappa -quasicircle.
The region G (curve L) is called a quasidisk (quasicircle), if it is \kappa -quasidisk (\kappa -quasicircle) for
some 0 \leq \kappa < 1.
We denoted the class of \kappa -quasicircle by Q(\kappa ), 0 \leq \kappa < 1, and L \in Q, if L \in Q(\kappa ), for some
0 \leq \kappa < 1. It is well-known that the quasicircle may not even be locally rectifiable in [14, p. 104].
Definition 2.1. It is said that L \in \widetilde Q(\kappa ), 0 \leq \kappa < 1, if L \in Q(\kappa ) and L is rectifiable.
Theorem 2.1. Let p > 0. Suppose that L \in \widetilde Q(\kappa ), for some 0 \leq \kappa < 1 and h(z) defined in
(1.1) for \gamma j = 0, for all j = 1,m. Then, for any Pn \in \wp n, n \in \BbbN , there exists c1 = c1(L, p) > 0
such that
\| Pn\| \scrL \infty
\leq c1n
1+\kappa
p \cdot \| Pn\| \scrL p(h0,L)
. (2.1)
Thus, Theorem 2.1 provides an opportunity to observe the growth of | Pn(z)| on the curve L.
Note that, Theorem 2.1 provided for L := \{ z : | z| = 1\} (i.e., \kappa = 0) in [13], for arbitrary rectufiable
curve L without weight function in [16], for polynomials in many variables in [22] (Theorem 1.1),
for the special curve in [4 – 7] and others.
From the conditions of the theorem, we see that, it holds for k-quasidisks with 0 \leq k < 1. But
calculating the coefficient of quasiconformality \kappa for some curves is not an easy task. Therefore, we
define a more general class of curves with another characteristic. One of them is the following:
Definition 2.2. We say that L \in Q\alpha , 0 < \alpha \leq 1, if L \in Q and \Phi \in \mathrm{L}\mathrm{i}\mathrm{p}\alpha , z \in \Omega .
We note that the class Q\alpha is sufficiently wide. A detailed account on it and the related topics are
contained in [15, 21, 28] (see also the references cited therein). We consider only some cases:
Remark 2.1. a) If L = \partial G is a Dini-smooth curve [21, p. 48], then L \in Q1.
b) If L = \partial G is a piecewise Dini-smooth curve and largest esterior angle at L has opening \alpha \pi ,
0 < \alpha \leq 1 [21, p. 52], then L \in Q\alpha .
c) If L = \partial G is a smooth curve having continuous tangent line, then L \in Q\alpha for all 0 < \alpha < 1.
d) If L is quasismooth (in the sense of Lavrentiev), that is, for every pair z1, z2 \in L, if s(z1, z2)
represents the smallest of the lengths of the arcs joining z1 to z2 on 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 =
1
2
\biggl(
1 - 1
\pi
\mathrm{a}\mathrm{r}\mathrm{c}\mathrm{s}\mathrm{i}\mathrm{n}
1
c
\biggr) - 1
[28].
e) If L is “c-quasiconformal” (see, for example, [15]), then \Phi \in \mathrm{L}\mathrm{i}\mathrm{p}\alpha for \alpha =
\pi
2
\biggl(
\pi - \mathrm{a}\mathrm{r}\mathrm{c}\mathrm{s}\mathrm{i}\mathrm{n}
1
c
\biggr) .
Also, if L is an asymptotic conformal curve, then \Phi \in \mathrm{L}\mathrm{i}\mathrm{p}\alpha for all 0 < \alpha < 1 [15].
Definition 2.3. It is said that L \in \widetilde Q\alpha , 0 < \alpha \leq 1, if L \in Q\alpha and L is rectifiable.
In this case we have the following theorem.
Theorem 2.2. Let p > 0. Suppose that L \in \widetilde Q\alpha , for some 0 < \alpha \leq 1 and h(z) defined as in
(1.1) with \gamma j = 0, for all j = 1,m. Then, for any Pn \in \wp n, n \in \BbbN , there exists c2 = c2(L, p) > 0
such that
\| Pn\| \scrL \infty
\leq c2 \| Pn\| \scrL p(h0,L)
\left\{
n
1
\alpha p ,
1
2
\leq \alpha \leq 1,
n
\delta
p , 0 < \alpha <
1
2
,
(2.2)
where \delta = \delta (L), \delta \in [1, 2] , is a certain number.
ISSN 1027-3190. Укр. мат. журн., 2016, т. 68, № 10
1368 P. ÖZKARTEPE, F. G. ABDULLAYEV
Therefore, according to (2.1), we can calculate \alpha in the right parts of estimation (2.2) for each
case, respectively.
Now, we assume that the the weight function h have “singularities” at the points \{ zi\} mi=1 , i.e.,
\gamma i \not = 0 for all i = 1,m. In this case, we have the following “local” (at the point zj \in L) estimations.
Theorem 2.3. Let p > 0. Suppose that L \in \widetilde Q\alpha , for some
1
2
\leq \alpha \leq 1 and h(z) defined as in
(1.1). Then, for any \gamma i > - 1, i = 1,m, and Pn \in \wp n, n \in \BbbN , there exists c3 = c3(L, p, \gamma i, \alpha ) > 0
such that
| Pn(zi)| \leq c3n
\gamma i+1
\alpha p \| Pn\| \scrL p(h,L)
. (2.3)
Now, let’s introduce “special” singular points on the curve L. Let us give the following definition.
For \delta > 0 and z \in \BbbC let us set B(z, \delta ) := \{ \zeta : | \zeta - z| < \delta \} , \Omega (z, \delta ) := \Omega \cap B(z, \delta ).
Definition 2.4. We say that L \in Q\alpha ,\beta 1,...,\beta m , 0 < \beta i \leq \alpha \leq 1, i = 1,m, if
i) for every sequence noncrossing in pairs circles \{ D(\zeta i, \delta i)\} mi=1 restriction of the function \Phi on
\Omega (\zeta i, \delta i) belongs to \mathrm{L}\mathrm{i}\mathrm{p}\beta i (\Phi | \Omega (\zeta i, \delta i) \in \mathrm{L}\mathrm{i}\mathrm{p}\beta i), and restriction
\Phi
\bigm| \bigm| \bigm| \bigm| \bigm| \Omega \setminus
m\bigcup
i=1
\Omega (\zeta i, \delta i) \in \mathrm{L}\mathrm{i}\mathrm{p} \alpha ,
ii) there exists a sequence noncrossing in pairs circles \{ D(\zeta i, \delta
\ast
i )\}
m
i=1 , such that for all i = 1,m,
\delta \ast i > \delta i and \xi , z \in \Omega (\zeta i, \delta
\ast
i ), z \not = \zeta i \not = \xi , is fulfiled estimation
| \Phi (z) - \Phi (\xi )| \leq ki(z, \xi ) | z - \xi | \alpha , (2.4)
where
ki(z, \xi ) = ci\mathrm{m}\mathrm{a}\mathrm{x}
\Bigl(
| \xi - \zeta i| \beta i - \alpha ; | z - \zeta i| \beta i - \alpha
\Bigr)
,
and ci not depends on z and \xi .
Definition 2.5. We say that L \in \widetilde Q\alpha ,\beta 1,...,\beta m , 0 < \beta i \leq \alpha \leq 1, i = 1,m, if L \in Q\alpha ,\beta 1,...,\beta m ,
0 < \beta i \leq \alpha \leq 1, i = 1,m, and L = \partial G is rectifiable.
It is clear from Definition 2.4 (2.5), that each region L \in Q\alpha ,\beta 1,...,\beta m , 0 < \beta i \leq \alpha \leq 1, i = 1,m,
may have “singularities” at the points \{ zi\} mi=1 \in L. If a region L does not have such “singularities”,
i.e., if \beta i = \alpha for all i = 1,m, then it is written as G \in Q\alpha , 0 < \alpha \leq 1.
Throughout this work, we will assume that the points \{ zi\} mi=1 \in L are defined in (1.1) and
\{ \zeta i\} mi=1 \in L are defined in Definitions 2.4 and 2.5 coincides. Without the loss of generality, we also
will assume that the points \{ zi\} mi=1 are ordered in the positive direction on the curve L.
We state our new results. Our first results is related to the simple cases. Namely, let the curve L
and the weight function h has not the “singularities” at the points \{ zi\} mi=1 , i.e., \beta i = \alpha , and \gamma i = 0
for all i = 1,m. In this case, we have the following “local” (at the singuler points) and “global”
estimations.
Theorem 2.4. Let p > 0. Suppose that L \in \widetilde Q\alpha ,\beta 1,...,\beta m , for some
1
2
\leq \beta i \leq \alpha \leq 1, i = 1,m,
and h(z) defined as in (1.1). Then, for any \gamma i > - 1, i = 1,m, and Pn \in \wp n, n \in \BbbN , there exists
c4 = c4(L, p, \gamma i, \alpha , \beta i) > 0 such that
| Pn(zi)| \leq c4n
\gamma i+1
p\beta i \| Pn\| \scrL p(h,L)
(2.5)
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INTERFERENCE OF THE WEIGHT AND BOUNDARY CONTOUR FOR ALGEBRAIC . . . 1369
and
\| Pn\| \scrL \infty
\leq c4n
\widetilde \gamma +1
p\beta i \| Pn\| \scrL p(h,L)
, (2.6)
where \widetilde \gamma := \mathrm{m}\mathrm{a}\mathrm{x}
\bigl\{
0, \gamma i, i = 1,m
\bigr\}
.
Therefore, if contour L does not have any singular points, i.e., \beta i = \alpha , for i = 1,m, then we
have the Theorem 2.3.
Now, we conside the general case: assume that the curve L have “singularity” on the boundary
points \{ zi\} mi=1 , i.e., \beta i \not = \alpha , for all i = 1,m, and the weight function h have “singularity” at the
same points, i.e., \gamma i \not = 0 for some i = 1,m. For simplesity, let us suppose \gamma i \not = 0 for all i = 1,m.
In this case, we have the following “global” estimations.
Theorem 2.5. Let p > 0. Suppose that L \in \widetilde Q\alpha ,\beta 1,...,\beta m , for some
1
2
\leq \beta i \leq \alpha \leq 1, i = 1,m,
and h(z) defined as in (1.1) and
\gamma i + 1 =
\beta i
\alpha
, (2.7)
for each points \{ zi\} mi=1 . Then, for any Pn \in \wp n, n \in \BbbN , there exists c5 = c5(L, p, \gamma i, \alpha ) > 0 such
that
\| Pn\| \scrL \infty
\leq c5n
1
\alpha p \| Pn\| \scrL p(h,L)
. (2.8)
Comparing Theorem 2.5 with Theorem 2.2, it is seen that, if the equality (2.7) is satisfied, then
the growth rate of the polynomials Pn(z) on L does not depend on whether the weight function h(z)
and the boundary contour L have singularity or not. The condition (2.7) is called the condition of
“interference of singularities” of weight function h and contour L at the “singular” points \{ zi\} mi=1 .
Corollary 2.1. If L \in C(1, \lambda , \nu ), then L \in \widetilde Q\alpha ,\beta 1 for \alpha = 1 (2.1) and \beta 1 =
1
\nu
[15]. In this
case, for p = 2 from (2.7) and (2.8), we have
(\gamma 1 + 1) \nu 1 = 1,
\| Pn\| \scrL \infty
\leq c5
\surd
n \| Pn\| \scrL 2(h,L)
. (2.9)
For Pn \equiv Qn, the estimation (2.9) coincides from (1.4). Therefore, Theorem 2.5 is generalised the
result in [27] (Theorem 1).
2.1. Sharpness of estimates. The sharpness of the estimations (2.1), (2.2) and for some special
cases can be discussed by comparing them with the following results:
Remark 2.2. For any n \in \BbbN , there exists polynomials P \ast
n \in \wp n, weight functions h\ast and the
constants c6 = c6(L) > 0 such that, for L := \{ z : | z| = 1\} we have
\| P \ast
n\| C(L) \geq c6n
1
p \| P \ast
n\| \scrL p(h\ast , L) .
3. Some auxiliary results. For a > 0 and b > 0, we shall 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.
The following definitions of the K -quasiconformal curves are well-known (see, for example, [9;
14, p. 97; 23]):
ISSN 1027-3190. Укр. мат. журн., 2016, т. 68, № 10
1370 P. ÖZKARTEPE, F. G. ABDULLAYEV
Definition 3.1. The Jordan arc (or curve) 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 line segment (or circle).
Let F (L) denotes the set of all sense preserving plane homeomorphisms f of the region D \supset L
such that f(L) is a line segment (or circle) and lets define
KL := \mathrm{i}\mathrm{n}\mathrm{f} \{ K(f) : f \in F (L)\} ,
where K(f) is the maximal dilatation of a such mapping f. L is a quasiconformal curve, if KL <\infty ,
and L is a K -quasiconformal curve, if KL \leq K.
Remark 3.1. It is well-known that, if we are not interested with the coefficients of quasiconfor-
mality of the curve, then the definitions of “quasicircle” and “quasiconformal curve” are identical.
However, if we are also interested with the coefficients of quasiconformality of the given curve, then
we will consider that if the curve L is K -quasiconformal, then it is \kappa -quasicircle with \kappa =
K2 - 1
K2 + 1
.
By Remark 3.1, for simplicity, we will use both terms, depending on the situation.
For z \in \BbbC and M \subset \BbbC , we set
d(z,M) = \mathrm{d}\mathrm{i}\mathrm{s}\mathrm{t} (z,M) := \mathrm{i}\mathrm{n}\mathrm{f} \{ | z - \zeta | : \zeta \in M\} .
Lemma 3.1 [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 and similarly
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, c > 1, 0 < r0 < 1 are constants, depending on G and Lr0 := \{ z = \psi (w) : | w| = r0\} .
Lemma 3.2. Let G \in Q(\kappa ) for some 0 \leq \kappa < 1. Then
| \Psi (w1) - \Psi (w2)| \succeq | w1 - w2| 1+\kappa
for all w1, w2 \in \Delta .
This fact follows from [20, p. 287] (Lemma 9.9) and the estimation for the \Psi
\prime
(see, for example,
[10], Theorem 2.8)
\bigm| \bigm| \Psi \prime (\tau )
\bigm| \bigm| \asymp d(\Psi (\tau ) , L)
| \tau | - 1
. (3.1)
Let \{ zj\} mj=1 be a fixed the system of the points on L and the weight function h (z) defined as
(1.1).
Lemma 3.3 [6]. Let L be a rectifiable Jordan curve, h(z) defined as in (1.1). Then, for arbitrary
Pn(z) \in \wp n, any R > 1 and n \in \BbbN , we have
\| Pn\| \scrL p(h,LR) \leq R
n+ 1+\gamma \ast
p \| Pn\| \scrL p(h,L)
, p > 0. (3.2)
Remark 3.2. In case of h(z) \equiv 1, the estimation (3.2) has been proved in [12].
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INTERFERENCE OF THE WEIGHT AND BOUNDARY CONTOUR FOR ALGEBRAIC . . . 1371
4. Proof of theorems. 4.1. Proof of Theorem 2.4. Suppose that L \in \widetilde Q\alpha ,\beta 1,...,\beta m , for some
1
2
\leq \beta i \leq \alpha \leq 1, i = 1,m, be given and h(z) defined as in (1.1). Let w = \varphi R(z) be the univalent
conformal mapping of GR, R > 1, 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) lying on GR. Let
Bm,R(z) :=
m\prod
j=1
Bj,R(z) =
m\prod
j=1
\varphi R(z) - \varphi R(\zeta j)
1 - \varphi R(\zeta j)\varphi R(z)
, (4.1)
denotes a Blashke function with respect to zeros \{ \zeta j\} , 1 \leq j \leq m \leq n, of Pn(z) [29]. Clearly,
| Bm,R(z)| \equiv 1, z \in LR, (4.2)
and
| Bm,R(z)| < 1, z \in GR. (4.3)
For any p > 0 and z \in GR let us set
Tn (z) :=
\biggl[
Pn (z)
Bm,R(z)
\biggr] p/2
. (4.4)
The function Tn (z) is analytic in GR, continuous on GR and does not have zeros in GR. We take
an arbitrary continuous branch of the Tn (z) and for this branch we maintain the same designation.
Then, the Cauchy integral representation for the Tn (z) in GR gives
Tn (z) =
1
2\pi i
\int
LR
Tn (\zeta )
d\zeta
\zeta - z
, z \in GR, (4.5)
or
Tn (zj) =
1
2\pi i
\int
LR
Tn (\zeta )
d\zeta
\zeta - zj
.
Now, let z \in L. Multiplying the numerator and the determinator of the integrand by h1/2(\zeta ),
according to the Hölder inequality, from (4.2) and (4.3), we obtain
| Pn (zj)| \leq
\biggl(
1
2\pi
\biggr) 2/p
\left( \int
LR
h(\zeta ) | Pn(\zeta )| p | d\zeta |
\right)
1/p
\times
\times
\left( \int
LR
| d\zeta | \prod m
j=1
| \zeta - zj | 2+\gamma j
\right)
1/p
=:
\biggl(
1
2\pi
\biggr) 2/p
In,1 \times In,2, (4.6)
where
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1372 P. ÖZKARTEPE, F. G. ABDULLAYEV
In,1 := \| Pn\| \scrL p(h,LR) , In,2 :=
\left( \int
LR
| d\zeta | \prod m
j=1
| \zeta - zj | 2+\gamma j
\right)
1/p
.
Then, by Lemma 3.3, for any points \{ zj\} mj=1 \in L, we have
| Pn (zj)| \preceq \| Pn\| \scrL p
(In,2)
1/p . (4.7)
To estimate the integral In,2, we introduce
wj := \Phi (zj), \varphi j := \mathrm{a}\mathrm{r}\mathrm{g}wj , L
j := L \cap \Omega
j
, Lj
R := LR \cap \Omega
j
, j = 1,m, (4.8)
where \Omega j := \Psi (\Delta
\prime
j),
\Delta
\prime
1 :=
\biggl\{
t = \mathrm{R}\mathrm{e}i\theta : R > 1,
\varphi m + \varphi 1
2
\leq \theta <
\varphi 1 + \varphi 2
2
\biggr\}
,
\Delta
\prime
m :=
\biggl\{
t = \mathrm{R}\mathrm{e}i\theta : R > 1,
\varphi m - 1 + \varphi m
2
\leq \theta <
\varphi m + \varphi 1
2
\biggr\}
,
(4.9)
and, for j = 2,m - 1,
\Delta
\prime
j :=
\biggl\{
t = \mathrm{R}\mathrm{e}i\theta : R > 1,
\varphi j - 1 + \varphi j
2
\leq \theta <
\varphi j + \varphi j+1
2
\biggr\}
. (4.10)
Then, we get
(In,2)
p =
m\sum
i=1
\int
Li
R
| d\zeta | \prod m
j=1
| \zeta - zj | 2+\gamma j
\asymp
m\sum
i=1
\int
Li
R
| d\zeta |
| \zeta - zi| 2+\gamma i
=:
m\sum
i=1
Iin,2, (4.11)
where
Iin,2 :=
\int
Li
R
| d\zeta |
| \zeta - zi| 2+\gamma i
, i = 1,m, (4.12)
since the points \{ zj\} mj=1 \in L are distinct. It remains to estimate the integrals Iin,2 for each i = 1,m.
For simplicity of our next calculations, we assume that
m = 1, R = 1 +
\varepsilon 0
n
. (4.13)
Let the numbers \delta 1, \delta \ast 1 , 0 < \delta 1 < \delta \ast 1 < \delta 0 < \mathrm{d}\mathrm{i}\mathrm{a}\mathrm{m}G, are choosen from Definition 2.4. By denoted
l1R,1 := L1
R \cap \Omega (z1, \delta 1), l1R,2 := L1
R\setminus l1R,1, F 1
R,i := \Phi (l1R,i), i = 1, 2,
we get
I1n,2 :=
\int
L1
R
| d\zeta |
| \zeta - z1| 2+\gamma 1
=
\int
l1R,1
| d\zeta |
| \zeta - z1| 2+\gamma 1
+
\int
l1R,2
| d\zeta |
| \zeta - z1| 2+\gamma 1
. (4.14)
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INTERFERENCE OF THE WEIGHT AND BOUNDARY CONTOUR FOR ALGEBRAIC . . . 1373
By applying the Lemma 3.1, we have\int
l1R,1
| d\zeta |
| \zeta - z1| 2+\gamma 1
=
\int
\Phi (l1R,1)
d(\Psi (\tau ), L) | d\tau |
| \Psi (\tau ) - \Psi (w1)| 2+\gamma 1 (| \tau | - 1)
\preceq
\preceq
\int
\Phi (l1R,1)
| d\tau |
| \Psi (\tau ) - \Psi (w1)| 1+\gamma 1 (| \tau | - 1)
\preceq
\preceq n
\int
\Phi (l1R,1)
| d\tau |
| \tau - w1|
\gamma 1+1
\beta 1
\preceq n
\gamma 1+1
\beta 1 , (4.15)
\int
l1R,2
| d\zeta |
| \zeta - z1| 2+\gamma 1
\preceq (\delta 1)
2+\gamma 1\mathrm{m}\mathrm{e}\mathrm{s} l1R,1 \preceq 1. (4.16)
Then, from (4.14), we get
I1n,2 \preceq n
\gamma 1+1
\beta 1 . (4.17)
By combining the relations (4.7) – (4.17), we obtain
| Pn (z1)| \preceq n
\gamma 1+1
p\beta 1 \| Pn\| \scrL p
,
and, according to (4.13), we completed the proof.
4.2. Proof of Theorem 2.5. Suppose that L \in \widetilde Q\alpha ,\beta 1,...,\beta m , for some
1
2
\leq \beta i \leq \alpha \leq 1, i = 1,m,
be given and h(z) defined as in (1.1). Let w = \varphi R(z), Bm,R(z) and Tn (z) be defined as in begining
to proof of the Theorem 2.4 by (4.1) and (4.4). Then Cauchy integral representation for the Tn (z)
in GR gives
Tn (z) =
1
2\pi i
\int
LR
Tn (\zeta )
d\zeta
\zeta - z
, z \in GR, (4.18)
or \bigm| \bigm| \bigm| \bigm| \bigm|
\biggl[
Pn (z)
Bm,R(z)
\biggr] p/2 \bigm| \bigm| \bigm| \bigm| \bigm| \leq 1
2\pi
\int
LR
\bigm| \bigm| \bigm| \bigm| Pn (\zeta )
Bm,R(\zeta )
\bigm| \bigm| \bigm| \bigm| p/2 | d\zeta |
| \zeta - z|
\leq
\int
LR
| Pn (\zeta )|
p/2 | d\zeta |
| \zeta - z|
,
since | Bm,R(\zeta )| = 1, for \zeta \in LR. Lets now z \in L. Multiplying the numerator and determinator of
the integrand by h1/2(\zeta ), by the Hölder inequality, we obtain
\bigm| \bigm| \bigm| \bigm| Pn (z)
Bm,R(z)
\bigm| \bigm| \bigm| \bigm| p/2 \leq 1
2\pi
\left( \int
LR
h(\zeta ) | Pn(\zeta )| p | d\zeta |
\right)
1/2
\times
\times
\left( \int
LR
| d\zeta | \prod m
j=1
| \zeta - zj | \gamma j | \zeta - z| 2
\right)
1/2
=:
1
2\pi
Jn,1 \times Jn,2, (4.19)
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1374 P. ÖZKARTEPE, F. G. ABDULLAYEV
where
Jn,1 :=
\left( \int
LR
h(\zeta ) | Pn(\zeta )| p | d\zeta |
\right)
1/2
, Jn,2 :=
\left( \int
LR
| d\zeta | \prod m
j=1
| \zeta - zj | \gamma j | \zeta - z| 2
\right)
1/2
.
Then, since | Bm,R(z)| < 1, for z \in L, from Lemma 3.3, we have
| Pn (z)| \preceq (Jn,1Jn,2)
2/p \preceq \| Pn\| p (Jn,2)
2/p , z \in L. (4.20)
The integral Jn,2 we estimate analogous to the integral In,2. By using designations (4.8) – (4.10), we
obtain
(Jn,2)
2 =
m\sum
i=1
\int
Li
R
| d\zeta | \prod m
j=1
| \zeta - zj | \gamma j | \zeta - z| 2
\asymp
m\sum
i=1
\int
Li
R
| d\zeta |
| \zeta - zi| \gamma i | \zeta - z| 2
=:
m\sum
i=1
J i
n,2, (4.21)
where
J i
n,2 :=
\int
Li
R
| d\zeta |
| \zeta - zi| \gamma i | \zeta - z| 2
, i = 1,m, (4.22)
since the points \{ zj\} mj=1 \in L are distinct. It remains to estimate the integrals J i
n,2 for each i = 1,m.
As we have assumed in (4.13) for simplicity of calculations, here, we also assume that
m = 1, R = 1 +
\varepsilon 0
n
. (4.23)
Let the numbers \delta 1, \delta \ast 1 , 0 < \delta 1 < \delta \ast 1 < \delta 0 < \mathrm{d}\mathrm{i}\mathrm{a}\mathrm{m}G, are choosen from Definition 2.4. We denote
L1
R,1 := L1
R \cap \Omega (z1, \delta 1),
L1
R,2 := L1
R \cap (\Omega (z1, \delta
\ast
1)\setminus \Omega (z1, \delta 1)),
L1
R,3 := LR\setminus (L1
R,1 \cup L1
R,2), F 1
R,i := \Phi (L1
R,i),
L1
1 := L1 \cap D(z1, \delta 1),
L1
2 := L1 \cap (D(z1, \delta
\ast
1)\setminus D(z1, \delta 1)),
L1
3 := L\setminus (L1
1 \cup L1
2), F 1
i := \Phi (L1
i ), i = 1, 2, 3.
By taking into consideration these designations and by replacing the variable \tau = \Phi (\zeta ), from (3.1)
and (4.12), we have
J1
n,2 \asymp
3\sum
i=1
\int
F 1
R,i
| \Psi \prime (\tau )| | d\tau |
| \Psi (\tau ) - \Psi (w1)| \gamma 1 | \Psi (\tau ) - \Psi (w\prime )| 2
\asymp
\asymp
3\sum
i=1
\int
F 1
R,i
d(\Psi (\tau ), L) | d\tau |
| \Psi (\tau ) - \Psi (wi)| \gamma 1 | \Psi (\tau ) - \Psi (w\prime )| 2 (| \tau | - 1)
=:
3\sum
i=1
J(F 1
R,i). (4.24)
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INTERFERENCE OF THE WEIGHT AND BOUNDARY CONTOUR FOR ALGEBRAIC . . . 1375
So, we need to evaluate the integrals J(F 1
R,i) for each i = 1, 2, 3. Therefore, we will continue in the
following manner. Let
\| Pn\| \infty =:
\bigm| \bigm| Pn
\bigl(
z\prime
\bigr) \bigm| \bigm| , z\prime \in L, (4.25)
and w\prime = \Phi (z\prime ). There are two possible cases: the point z\prime may lie on L1 or L2.
1) Suppose first that z\prime \in L1. If z\prime \in L1
i , then w\prime \in F 1
i for i = 1, 2, 3. Let’s F 1,1
R,j :=
:=
\Bigl\{
\tau \in F 1
R,j : | \tau - w1| \geq | \tau - w\prime |
\Bigr\}
, F 1,2
R,j := F 1
R,j\setminus F
1,1
R,j , j = 1, 2. Consider the individual cases.
1.1) Let z\prime \in L1
1. Applying Lemma 3.1, we have
J(F 1
R,1) =
\int
F 1
R,1
d(\Psi (\tau ), L) | d\tau |
| \Psi (\tau ) - \Psi (w1)| \gamma 1 | \Psi (\tau ) - \Psi (w\prime )| 2 (| \tau | - 1)
\preceq
\preceq n
\int
F 1,1
R,1
| d\tau |
| \Psi (\tau ) - \Psi (w\prime )| \gamma 1+1 + n
\int
F 1,2
R,1
| d\tau |
| \Psi (\tau ) - \Psi (w1)| \gamma 1+1 \preceq
\preceq n
\int
F 1,1
R,1
| d\tau |
| \tau - w\prime |
\gamma 1+1
\beta
- 1
\alpha | \tau - w\prime |
1
\alpha
+ n
\int
F 1,2
R,1
| d\tau |
| \tau - w1|
\gamma 1+1
\beta
- 1
\alpha
| \tau - w1|
1
\alpha
\preceq
\preceq n
\int
F 1,1
R,1
| d\tau |
| \tau - w\prime |
1
\alpha
+ n
\int
F 1,2
R,1
| d\tau |
| \tau - w1|
1
\alpha
\preceq n
1
\alpha , (4.26)
J(F 1
R,2) =
\int
F 1
R,2
d(\Psi (\tau ), L) | d\tau |
| \Psi (\tau ) - \Psi (w1)| \gamma 1 | \Psi (\tau ) - \Psi (w\prime )| 2 (| \tau | - 1)
\preceq
\preceq
\int
F 1
R,2
d(\Psi (\tau ), L) | d\tau |
| \Psi (\tau ) - \Psi (w\prime )| 2 (| \tau | - 1)
\left\{ (\delta 1)
- \gamma 1 , \gamma 1 \geq 0,\bigl(
2\mathrm{d}\mathrm{i}\mathrm{a}\mathrm{m}G
\bigr) - \gamma 1 , - 1 < \gamma 1 < 0,
\preceq
\preceq
\int
F 1
R,2
| d\tau |
| \Psi (\tau ) - \Psi (w\prime )| (| \tau | - 1)
. (4.27)
Setting in (2.4) z := z\prime , \xi : \zeta = \Psi (\tau ) and according to | \zeta - z1| > | z\prime - z1| , we obtain\bigm| \bigm| \zeta - z\prime
\bigm| \bigm| \alpha \succeq \mathrm{m}\mathrm{a}\mathrm{x}
\Bigl\{
| \zeta - z1| \alpha - \beta 1 ;
\bigm| \bigm| z\prime - z1
\bigm| \bigm| \alpha - \beta 1
\Bigr\} \bigm| \bigm| w\prime - \tau
\bigm| \bigm| =
= | \zeta - z1| \alpha - \beta 1
\bigm| \bigm| w\prime - \tau
\bigm| \bigm| \geq \delta \alpha - \beta 1
1
\bigm| \bigm| w\prime - \tau
\bigm| \bigm| \succeq \bigm| \bigm| w\prime - \tau
\bigm| \bigm| . (4.28)
Then, from (4.27), we get
J(F 1
R,2) \preceq n
\int
F 1
R,2
| d\tau |
| \Psi (\tau ) - \Psi (w\prime )|
\preceq n
\int
F 1
R,2
| d\tau |
| \tau - w\prime |
1
\alpha
\preceq n
1
\alpha ,
J(F 1
R,3) =
\int
F 1
R,3
d(\Psi (\tau ), L) | d\tau |
| \Psi (\tau ) - \Psi (w1)| \gamma 1 | \Psi (\tau ) - \Psi (w\prime )| 2 (| \tau | - 1)
\preceq
(4.29)
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1376 P. ÖZKARTEPE, F. G. ABDULLAYEV
\preceq n
\delta \ast 1 - \delta 1
\int
F 1
R,3
| d\tau |
\Biggl\{
(\delta 1)
- \gamma 1 , \gamma 1 \geq 0,\bigl(
2\mathrm{d}\mathrm{i}\mathrm{a}\mathrm{m}G
\bigr) - \gamma 1 , - 1 < \gamma 1 < 0,
\preceq n.
1.2) Let z\prime \in L1
2. Analogously to case 1.1, we have
J(F 1
R,1) =
\int
F 1
R,1
d(\Psi (\tau ), L) | d\tau |
| \Psi (\tau ) - \Psi (w1)| \gamma 1 | \Psi (\tau ) - \Psi (w\prime )| 2 (| \tau | - 1)
\preceq
\preceq n
\int
F 1,1
R,2
| d\tau |
| \Psi (\tau ) - \Psi (w\prime )| \gamma 1+1 + n
\int
F 1,2
R,2
| d\tau |
| \Psi (\tau ) - \Psi (w1)| \gamma 1+1 \preceq
\preceq n
\int
F 1,1
R,2
| d\tau |
| \tau - w\prime |
\gamma 1+1
\beta
- 1
\alpha | \tau - w\prime |
1
\alpha
+ n
\int
F 1,2
R,2
| d\tau |
| \tau - w1|
\gamma 1+1
\beta
- 1
\alpha
| \tau - w1|
1
\alpha
\preceq
\preceq n
\int
F 1,1
R,2
| d\tau |
| \tau - w\prime |
1
\alpha
+ n
\int
F 1,2
R,2
| d\tau |
| \tau - w1|
1
\alpha
\preceq n
1
\alpha . (4.30)
For z\prime \in L1
2, applying (2.4), we see that\bigm| \bigm| \zeta - z\prime
\bigm| \bigm| \alpha \succeq \mathrm{m}\mathrm{a}\mathrm{x}
\Bigl\{
| \zeta - z1| \alpha - \beta 1 ;
\bigm| \bigm| z\prime - z1
\bigm| \bigm| \alpha - \beta 1
\Bigr\} \bigm| \bigm| w\prime - \tau
\bigm| \bigm| \geq
\geq \delta \alpha - \beta 1
1
\bigm| \bigm| w\prime - \tau
\bigm| \bigm| \succeq \bigm| \bigm| w\prime - \tau
\bigm| \bigm| ,
and, consequently, we obtain
J(F 1
R,2) =
\int
F 1
R,2
d(\Psi (\tau ), L) | d\tau |
| \Psi (\tau ) - \Psi (w1)| \gamma 1 | \Psi (\tau ) - \Psi (w\prime )| 2 (| \tau | - 1)
\preceq
\preceq
\int
F 1
R,2
d(\Psi (\tau ), L) | d\tau |
| \Psi (\tau ) - \Psi (w\prime )| 2 (| \tau | - 1)
\left\{ (\delta 1)
- \gamma 1 , \gamma 1 \geq 0,\bigl(
2\mathrm{d}\mathrm{i}\mathrm{a}\mathrm{m}G
\bigr) - \gamma 1 , - 1 < \gamma 1 < 0,
\preceq
\preceq n
\int
F 1
R,2
| d\tau |
| \Psi (\tau ) - \Psi (w\prime )|
\preceq n
\int
F 1
R,2
| d\tau |
| \tau - w\prime |
1
\alpha
\preceq n
1
\alpha , (4.31)
J(F 1
R,3) =
\int
F 1
R,3
d(\Psi (\tau ), L) | d\tau |
| \Psi (\tau ) - \Psi (w1)| \gamma 1 | \Psi (\tau ) - \Psi (w\prime )| 2 (| \tau | - 1)
\preceq
\preceq n
(\delta \ast 1)
\alpha - \beta 1
\int
F 1
R,3
| d\tau |
| \tau - w\prime |
1
\alpha
\left\{ (\delta 1)
- \gamma 1 , \gamma 1 \geq 0,\bigl(
2\mathrm{d}\mathrm{i}\mathrm{a}\mathrm{m}G
\bigr) - \gamma 1 , - 1 < \gamma 1 < 0,
\preceq
\preceq n
\int
F 1
R,3
| d\tau |
| \tau - w\prime |
1
\alpha
\preceq n
1
\alpha . (4.32)
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INTERFERENCE OF THE WEIGHT AND BOUNDARY CONTOUR FOR ALGEBRAIC . . . 1377
1.3) Let z\prime \in L1
3. In this situation for the integral J(F 1
R,1), we get
J(F 1
R,1) =
\int
F 1
R,1
d(\Psi (\tau ), L) | d\tau |
| \Psi (\tau ) - \Psi (w1)| \gamma 1 | \Psi (\tau ) - \Psi (w\prime )| 2 (| \tau | - 1)
\preceq
\preceq n
\int
F 1
R,1
| d\tau |
| \Psi (\tau ) - \Psi (w1)| \gamma 1+1 \preceq n
\int
F 1
R,1
| d\tau |
| \tau - w1|
\gamma 1+1
\beta 1
\preceq
\preceq n.n
\gamma 1+1
\beta 1
- 1
= n
1
\alpha . (4.33)
Applying (2.4), we see that\bigm| \bigm| \zeta - z\prime
\bigm| \bigm| \alpha \succeq \mathrm{m}\mathrm{a}\mathrm{x}
\Bigl\{
| \zeta - z1| \alpha - \beta 1 ;
\bigm| \bigm| z\prime - z1
\bigm| \bigm| \alpha - \beta 1
\Bigr\} \bigm| \bigm| w\prime - \tau
\bigm| \bigm| \geq
\geq \delta \alpha - \beta 1
1
\bigm| \bigm| w\prime - \tau
\bigm| \bigm| \succeq \bigm| \bigm| w\prime - \tau
\bigm| \bigm| ,
so, we get
J(F 1
R,2) =
\int
F 1
R,2
d(\Psi (\tau ), L) | d\tau |
| \Psi (\tau ) - \Psi (w1)| \gamma 1 | \Psi (\tau ) - \Psi (w\prime )| 2 (| \tau | - 1)
\preceq
\preceq n
\int
F 1
R,2
| d\tau |
| \Psi (\tau ) - \Psi (w\prime )|
\left\{ (\delta 1)
- \gamma 1 , \gamma 1 \geq 0,\bigl(
2 \mathrm{d}\mathrm{i}\mathrm{a}\mathrm{m}G
\bigr) - \gamma 1 , - 1 < \gamma 1 < 0,
\preceq
\preceq n
\int
F 1
R,2
| d\tau |
| \tau - w\prime |
1
\alpha
\preceq n
1
\alpha (4.34)
and
J(F 1
R,3) =
\int
F 1
R,3
d(\Psi (\tau ), L) | d\tau |
| \Psi (\tau ) - \Psi (w1)| \gamma 1 | \Psi (\tau ) - \Psi (w\prime )| 2 (| \tau | - 1)
\preceq
\preceq n
(\delta \ast 1)
\alpha - \beta 1
\int
F 1
R,3
| d\tau |
| \tau - w\prime |
1
\alpha
\left\{ (\delta \ast 1)
- \gamma 1 , \gamma 1 \geq 0,\bigl(
2 \mathrm{d}\mathrm{i}\mathrm{a}\mathrm{m}G
\bigr) - \gamma 1 , - 1 < \gamma 1 < 0,
\preceq
\preceq n
\int
F 1
R,3
| d\tau |
| \tau - w\prime |
1
\alpha
\preceq n
1
\alpha . (4.35)
By the relations (4.24) – (4.35), we obtain
J1
n,2 \preceq n
1
\alpha . (4.36)
Therefore, in case of z\prime \in L1 for each \gamma 1 > - 1 and for all z \in L, from (4.7), (4.11) and (4.36)
we get
ISSN 1027-3190. Укр. мат. журн., 2016, т. 68, № 10
1378 P. ÖZKARTEPE, F. G. ABDULLAYEV
| Pn (z)| \preceq n
1
p\alpha \| Pn\| p . (4.37)
Theorem 2.5 is proved.
4.3. Proof of Remark 2.2. a) Let L := \{ z : | z| = 1\} , h\ast (z) \equiv 1 and P \ast
n(z) =
\sum n
j=1
zj .
Then L \in \widetilde Q1, | P \ast
n(z)| \leq
\sum n
j=1
| z| j = n, | z| = 1. On the other hand, | P \ast
n(1)| = n. Therefore,
\| P \ast
n\| \scrL \infty
= n. \| P \ast
n\| \scrL 2(1,L)
=
\surd
2\pi n. Then
\| P \ast
n\| \scrL \infty
= n =
\surd
n\surd
2\pi
\| P \ast
n\| \scrL 2(1,L)
\geq 1\surd
2\pi
\surd
n \| P \ast
n\| \scrL 2(1,L)
.
Theorem 2.1 follows from [8] (Theorem 2.2). Theorem 2.3 is obtained from Theorem 2.4 for the
case \beta i = \alpha , i = 1,m. For
1
2
< \alpha \leq 1, the Theorem 2.2 we get from the Theorem 2.3 in the case
\gamma i = 0, i = 1,m. Case of \alpha =
1
2
follows by using the estimation d(z, LR) \succeq n - 2, where is true for
arbitrary continuum with connected complement (see, for example, [10], Corollary 2.7).
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INTERFERENCE OF THE WEIGHT AND BOUNDARY CONTOUR FOR ALGEBRAIC . . . 1379
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Received 23.02.16
ISSN 1027-3190. Укр. мат. журн., 2016, т. 68, № 10
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| id | umjimathkievua-article-1926 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T02:15:23Z |
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| publisher | Institute of Mathematics, NAS of Ukraine |
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| resource_txt_mv | umjimathkievua/fe/ce82c654dbc8e87dcdf8b4423f86fffe.pdf |
| spelling | umjimathkievua-article-19262019-12-05T09:31:57Z Interference of the weight and boundary contour for algebraic polynomials in the weighted Lebesgue spaces. I Перешкоди, пов’язанi з вагою та граничним контуром, для алгебраїчних полiномiв у зважених просторах Лебега Abdullayev, F. G. Özkartepe, N. P. Абдуллаєв, Ф. Г. Озкартепе, Н. П. We study the order of the height of the modulus of arbitrary algebraic polynomials with respect to the weighted Lebesgue space, where the contour and the weight functions have some singularities. Вивчається порядок висоти модуля довiльних алгебраїчних полiномiв вiдносно зважених просторiв Лебега, в яких контур та ваговi функцiї мають деякi сингулярностi. Institute of Mathematics, NAS of Ukraine 2016-10-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/1926 Ukrains’kyi Matematychnyi Zhurnal; Vol. 68 No. 10 (2016); 1365-1379 Український математичний журнал; Том 68 № 10 (2016); 1365-1379 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/1926/908 Copyright (c) 2016 Abdullayev F. G.; Özkartepe N. P. |
| spellingShingle | Abdullayev, F. G. Özkartepe, N. P. Абдуллаєв, Ф. Г. Озкартепе, Н. П. Interference of the weight and boundary contour for algebraic polynomials in the weighted Lebesgue spaces. I |
| title | Interference of the weight and boundary contour for algebraic polynomials in the weighted Lebesgue spaces. I |
| title_alt | Перешкоди, пов’язанi з вагою та граничним контуром, для алгебраїчних полiномiв у зважених просторах Лебега |
| title_full | Interference of the weight and boundary contour for algebraic polynomials in the weighted Lebesgue spaces. I |
| title_fullStr | Interference of the weight and boundary contour for algebraic polynomials in the weighted Lebesgue spaces. I |
| title_full_unstemmed | Interference of the weight and boundary contour for algebraic polynomials in the weighted Lebesgue spaces. I |
| title_short | Interference of the weight and boundary contour for algebraic polynomials in the weighted Lebesgue spaces. I |
| title_sort | interference of the weight and boundary contour for algebraic polynomials in the weighted lebesgue spaces. i |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/1926 |
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