On the growth of derivatives of algebraic polynomials in a weighted Lebesgue space
UDC 517.5 We study growth rates of derivatives of an arbitrary algebraic polynomial in bounded and inbounded regions of the complex plane in weighted Lebesgue spaces.
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Institute of Mathematics, NAS of Ukraine
2022
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Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860512592485679104 |
|---|---|
| author | Аbdullayev, F. G. Imashkyzy, M. Аbdullayev, Fahreddin Аbdullayev, F. G. Imashkyzy, М. |
| author_facet | Аbdullayev, F. G. Imashkyzy, M. Аbdullayev, Fahreddin Аbdullayev, F. G. Imashkyzy, М. |
| author_sort | Аbdullayev, F. G. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2022-10-24T09:23:02Z |
| description | UDC 517.5
We study growth rates of derivatives of an arbitrary algebraic polynomial in bounded and inbounded regions of the complex plane in weighted Lebesgue spaces. |
| doi_str_mv | 10.37863/umzh.v74i5.7052 |
| first_indexed | 2026-03-24T03:31:14Z |
| format | Article |
| fulltext |
DOI: 10.37863/umzh.v74i5.7052
UDC 517.5
F. G. Abdullayev (Mersin Univ., Turkey and Kyrgyz-Turkish Manas Univ., Bishkek),
M. Imashkyzy (Kyrgyz-Turkish Manas Univ., Bishkek)
ON THE GROWTH OF DERIVATIVES OF ALGEBRAIC POLYNOMIALS
IN A WEIGHTED LEBESGUE SPACE
ПРО ЗРОСТАННЯ ПОХIДНИХ АЛГЕБРАЇЧНИХ ПОЛIНОМIВ
У ВАГОВОМУ ПРОСТОРI ЛЕБЕГА
We study growth rates of derivatives of an arbitrary algebraic polynomial in bounded and inbounded regions of the complex
plane in weighted Lebesgue spaces.
Вивчається зростання похiдних довiльних алгебраїчних полiномiв у обмежених i необмежених областях комплексної
площини у вагових просторах Лебега.
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 (without loss of generality, let 0 \in G); \Omega := \BbbC \setminus G = \mathrm{e}\mathrm{x}\mathrm{t}L. For t \in \BbbC
and \delta > 0, let \Delta (t, \delta ) := \{ w \in \BbbC : | w - t| > \delta \} , \Delta := \Delta (0, 1). Let \Phi : \Omega \rightarrow \Delta be the univalent
conformal mapping normalized by \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 .
Let \{ zj\} lj=1 \in L be the fixed system of distinct points. 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)
l\prod
j=1
| z - zj | \gamma j , (1)
where \gamma j > - 1 for all j = 1, 2, . . . , l, and h0 is uniformly separated from zero, i.e., there exists a
constant c0(L) > 0 such that, for all z \in GR0 , h0(z) \geq c0(L) > 0.
For each 0 < p \leq \infty and rectifiable Jordan curve L = \partial G, we introduce
\| Pn\| p := \| Pn\| \scrL p(h,L)
:=
\left( \int
L
h(z) | Pn(z)| p | dz|
\right) 1/p
<\infty , 0 < p <\infty , (2)
\| Pn\| \infty := \| Pn\| \scrL \infty (1,L) := \mathrm{m}\mathrm{a}\mathrm{x}
z\in L
| Pn(z)| , p = \infty ,
\scrL p(1, L) =: \scrL p(L).
As is known, in the theory of approximations on the complex plane, a special place is occupied
by the following well-known Bernstein – Walsh inequality [42]:
c\bigcirc F. G. ABDULLAYEV, M. IMASHKYZY, 2022
582 ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 5
ON THE GROWTH OF DERIVATIVES OF ALGEBRAIC POLYNOMIALS . . . 583
\| Pn\| C(GR) \leq | \Phi (z)| n \| Pn\| C(G) \forall Pn \in \wp n. (3)
So, for the points z \in G1+n - 1 , the \| Pn\| \infty have the same order of growth in GR and G with respect
to n. An analogue of this inequality in space \scrL p(L) is the following inequality [28]:
\| Pn\| \scrL p(LR) \leq | \Phi (z)| n+
1
p \| Pn\| \scrL p(L)
\forall Pn \in \wp n, p > 0. (4)
The estimate (4) has been generalized in [9] (Lemma 2.4) for weight function h(z) defined as in (1)
and was obtained
\| Pn\| \scrL p(h,LR) \leq R
n+ 1+\gamma \ast
p \| Pn\| \scrL p(h,L)
, \gamma \ast = \mathrm{m}\mathrm{a}\mathrm{x} \{ 0; \gamma j : 1 \leq j \leq l\} . (5)
If we replace the curve L with the region G and define two-dimensional analogs of the quantities
(2) (we denote them by \| Pn\| Ap(h,G) , \| Pn\| Ap(1,G) and Ap(G), respectively), then for them we
can also indicate the corresponding estimate of the type (5). For this, first of all, we will give the
following definition.
For any \delta > 0 and arbitrary t, w \in \BbbC , let B(t, \delta ) := \{ t : | t - w| < \delta \} and \varphi : G \rightarrow B :=
:= B(0, 1) := \{ w : | w| < 1\} be a conformal and univalent map which is normalized by \varphi (0) = 0
and \varphi \prime (0) > 0, \psi := \varphi - 1.
Following to [36, p. 286], a bounded Jordan region G is called a \kappa -quasidisk, 0 \leq \kappa < 1, if any
conformal mapping \psi can be extended to a K -quasiconformal, K =
1 + \kappa
1 - \kappa
, homeomorphism of the
plane \BbbC on the \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) with some 0 \leq \kappa < 1.
We denote this class as Q(\kappa ), 0 \leq \kappa < 1, and say that L = \partial G \in Q(\kappa ), if G \in Q(\kappa ), 0 \leq \kappa <
< 1. Further, we denote that G \in Q (L \in Q), if G \in Q(\kappa ) (L \in Q(\kappa )) for some 0 \leq \kappa < 1. It is
well-known that quasicircles can be non-rectifiable (see, for example, [26; 29, p. 104]). Additionally,
we say that L \in \widetilde Q(\kappa ), 0 \leq \kappa < 1, if L \in Q(\kappa ) and L is rectifiable.
In [2] given an analog of the estimates (3) and (5) for the quasidisks and h(z) defined as in (1)
for the \| Pn\|
Ap(h,G)
as follows:
\| Pn\|
Ap(h,GR)
\leq c1R
\ast n+1
p \| Pn\|
Ap(h,G)
, R > 1, p > 0,
where R\ast := 1 + c2(R - 1), c2 > 0 and c1 := c1(G, p, c2) > 0 are constants, independent from n
and R.
Further, for arbitrary Jordan region G and any Pn \in \wp n in [4] (Theorem 1.1) was proved that
\| Pn\|
Ap(GR)
\leq cR
n+2
p \| Pn\|
Ap(GR1
)
, p > 0,
is true for arbitrary R > R1 = 1 +
1
n
, where c =
\biggl(
2
ep - 1
\biggr) 1
p
\biggl[
1 +O
\biggl(
1
n
\biggr) \biggr]
, n \rightarrow \infty , is asymp-
totically sharp constant.
N. Stylianopoulos in [39] replaced the norm \| Pn\| C(G) with norm \| Pn\| A2(G) on the right-hand
side of (3) and found a new version of the Bernstein – Walsh lemma: Assume that L is quasiconformal
and rectifiable. Then there exists a constant c = c(L) > 0 depending only on L such that
ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 5
584 F. G. ABDULLAYEV, M. IMASHKYZY
| Pn(z)| \leq c
\surd
n
d(z, L)
\| Pn\| A2(G) | \Phi (z)|
n+1 , z \in \Omega ,
where d(z, L) := \mathrm{i}\mathrm{n}\mathrm{f} \{ | \zeta - z| : \zeta \in L\} , holds for every Pn \in \wp n.
In this paper, we continue the study of the problem on uniform and pointwise estimates of the
derivatives | P \prime
n(z)| in bounded (GR) and unbounded (\Omega R) regions of the complex plane for each
R \geq 1 and obtained estimates of the following type:\bigm| \bigm| P \prime
n(z)
\bigm| \bigm| \leq c4 \| Pn\| p
\Biggl\{
\lambda n(L, h, p), z \in GR,
\eta n(L, h, p, z), z \in \Omega R,
(6)
where \lambda n(.) and \eta n(.) \rightarrow \infty as n\rightarrow \infty , depending on the properties of the L, h.
Analogous results of (6)-type for | Pn(z)| , different weight function h, in unbounded region were
obtained in [5 – 17, 19, 20, 27, p. 418 – 428, 31, 35, 39].
Estimates of the (6)-type on points z \in G (also z \in GR), respect to norm \| Pn\| \scrL p(h,L)
or
\| Pn\| Ap(h,G) , p > 0, for some h(z) \equiv 1 or h(z) \not = 1 was studied since the beginning of the 20th
century in, for example, [25, 40], and has been studied by in [2, 3, 18, 22, 23, 24, 27, p. 418 – 428,
31], [32] (Sect. 5.3), [33, 34, p. 122 – 133, 38] (see also the references cited therein).
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.
In this work, we will try to get the result for more general curves, also including the above class
of curves. For this we need to give the following definitions of regions with some general functional
condition.
Definition. We say that L = \partial G = \partial \Omega \in Q\alpha , if L is a quasicircle and \Phi \in H\alpha (\Omega ) for some
0 < \alpha \leq 1, i.e., | \Phi (w) - \Phi (\tau )| \leq M | w - \tau | \alpha , 0 < \alpha \leq 1, for all | w| \geq 1, | \tau | \geq 1, and M > 0
constant independent of w and \tau .
Additionally, we say that L \in \widetilde Q\alpha , 0 < \alpha \leq 1, if L \in Q\alpha and L is rectifiable.
We note that the class Q\alpha is sufficiently large. A detailed account on it and the related topics are
contained in [30, 37, 41] (see also the references cited therein). We consider only some cases:
a) If L is a piecewise Dini-smooth curve and largest exterior angle on L has opening \alpha \pi ,
0 < \alpha \leq 1, [37, p. 52], then L \in \widetilde Q\alpha .
b) If L =: \partial G is a smooth curve having continuous tangent line, then L \in \widetilde Q\alpha for all 0 < \alpha < 1.
c) If G is “L-shaped” region, then L \in \widetilde Q 2
3
.
d) If L is quasismooth (in the sense of Lavrentiev), then L \in \widetilde Q\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
and c > 1 [41].
e) If L is ""c-quasiconformal”, then L \in Q\alpha for \alpha =
\pi
2
\biggl(
\pi - \mathrm{a}\mathrm{r}\mathrm{c}\mathrm{s}\mathrm{i}\mathrm{n}
1
c
\biggr) ; if L is an asymptotic
conformal curve, then L \in Q\alpha for all 0 < \alpha < 1 [30].
For 0 < \delta j < \delta 0 :=
1
4
\mathrm{m}\mathrm{i}\mathrm{n} \{ | zi - zj | : i, j = 1, 2, . . . , l, i \not = j\} , let \Omega (zj , \delta j) := \Omega \cap \{ z :
| z - zj | \leq \delta j\} , \delta := \mathrm{m}\mathrm{i}\mathrm{n}1\leq j\leq l\delta j . For L = \partial G, we set U\infty (L, \delta ) :=
\bigcup
\zeta \in L
U(\zeta , \delta ) is infinite open
cover of the curve L;
ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 5
ON THE GROWTH OF DERIVATIVES OF ALGEBRAIC POLYNOMIALS . . . 585
UN (L, \delta ) :=
\bigcup N
j=1
Uj(L, \delta ) \subset U\infty (L, \delta ) is finite open cover of the curve L; \Omega (\delta ) := \Omega (L, \delta ) :=
:= \Omega \cap UN (L, \delta ), \widehat \Omega := \Omega \setminus \Omega (\delta ); \Omega R(\delta ) := \Omega (LR, \delta ) := \Omega R \cap UN (LR, \delta ), \widehat \Omega R := \Omega R\setminus \Omega R(\delta ).
Now, we start to formulate the new results.
Theorem 1. Let p > 1, L \in \widetilde Q\alpha for some
1
2
\leq \alpha \leq 1, and h(z) be defined by (1). Then, for
any Pn \in \wp n, n \in \BbbN , we have
\bigm| \bigm| P \prime
n (z)
\bigm| \bigm| \leq c1
\bigm| \bigm| \Phi n+1(z)
\bigm| \bigm| \biggl\{ \| Pn\| p
d(z, L)
A1
n,p(z) +B1
n,1(z) | Pn (z)|
\biggr\}
, (7)
where c1 = c1(L, \gamma , p) > 0 is constant independent from n and z;
A1
n,p(z) :=
\left\{
n
\gamma \ast +1
\alpha p , 1 < p < 1 +
\gamma \ast + 1
\alpha
,
(n \mathrm{l}\mathrm{n}n)
1 - 1
p , p = 1 +
\gamma \ast + 1
\alpha
,
n
1 - 1
p , p > 1 +
\gamma \ast + 1
\alpha
,
B1
n,1(z) := n
1
\alpha , if z \in \Omega (\delta );
A1
n,p(z) :=
\left\{
n
\Bigl(
\gamma +1
p
- 1
\Bigr)
1
\alpha , 1 < p < 1 +
\gamma
1 + \alpha
,
(n \mathrm{l}\mathrm{n}n)
1 - 1
p , p = 1 +
\gamma
1 + \alpha
,
n
1 - 1
p , p > 1 +
\gamma \ast
1 + \alpha
,
B1
n,1(z) := n, if z \in \widehat \Omega (\delta ),
and \gamma \ast := \mathrm{m}\mathrm{a}\mathrm{x} \{ 0; \gamma \} .
Theorem 2. Let p > 1, L \in \widetilde Q\alpha for some
1
2
\leq \alpha \leq 1, and h(z) be defined by (1). Then, for
any Pn \in \wp n, n \in \BbbN , and z \in \Omega R, we have
| Pn (z)| \leq c2
\bigm| \bigm| \Phi n+1(z)
\bigm| \bigm|
d(z, L)
A2
n,p \| Pn\| p , (8)
where c2 = c2(L, \gamma , p) > 0 is constant independent from n and z;
A2
n,p :=
\left\{
n
\Bigl(
\gamma +1
p
- 1
\Bigr)
1
\alpha , 1 < p < 1 +
\gamma
1 + \alpha
, \gamma > 1 + \alpha ,
n
1 - 1
p (\mathrm{l}\mathrm{n}n)
1 - 1
p , p = 1 +
\gamma
1 + \alpha
, \gamma > 1 + \alpha ,
n
1 - 1
p , p > 1 +
\gamma
1 + \alpha
, \gamma > 1 + \alpha ,
n
1 - 1
p , p > 1, - 1 < \gamma \leq 1 + \alpha .
Now, from Theorems 1 and 2, we can give an estimates for the | P \prime
n (z)| for z \in \Omega R.
Let p > 1, L \in \widetilde Q\alpha for some
1
2
\leq \alpha \leq 1, and h(z) be defined by (1). Then, for any
Pn \in \wp n, n \in \BbbN , we have
\bigm| \bigm| P \prime
n (z)
\bigm| \bigm| \leq c3
\bigm| \bigm| \Phi 2(n+1)(z)
\bigm| \bigm|
d(z, L)
\| Pn\| pA
3
n,p(z), (9)
ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 5
586 F. G. ABDULLAYEV, M. IMASHKYZY
where c3 = c3(L, \gamma , p) > 0 is constant independent from n and z;
A3
n,p(z) :=
\left\{
n
\gamma +1
p\alpha , 1 < p < 1 +
\gamma
1 + \alpha
, \gamma > 0,
n
1 - 1
p
+ 1
\alpha (\mathrm{l}\mathrm{n}n)
1 - 1
p , p = 1 +
\gamma
1 + \alpha
, \gamma > 0,
n
1 - 1
p
+ 1
\alpha , p > 1 +
\gamma
1 + \alpha
, \gamma > 0,
n
1 - 1
p
+ 1
\alpha , p > 1, - 1 < \gamma \leq 0,
if z \in \Omega (\delta );
A3
n,p(z) :=
\left\{
n
\Bigl(
\gamma +1
p
- 1
\Bigr)
1
\alpha
+1
, 1 < p < 1 +
\gamma
1 + \alpha
, \gamma > 1 + \alpha ,
n
2 - 1
p (\mathrm{l}\mathrm{n}n)
1 - 1
p , p = 1 +
\gamma
1 + \alpha
, \gamma > 1 + \alpha ,
n
2 - 1
p , p > 1 +
\gamma
1 + \alpha
, \gamma > 1 + \alpha ,
n
2 - 1
p , p > 1, - 1 < \gamma \leq 1 + \alpha ,
if z \in \widehat \Omega (\delta ).
Now, we will give estimate for | P \prime
n (z)| for bounded regions of the class \widetilde Q\alpha .
Theorem 3. Let p > 1, L \in \widetilde Q\alpha for some
1
2
\leq \alpha \leq 1, and h(z) be defined by (1). Then, for
any Pn \in \wp n, n \in \BbbN , we have
\bigm\| \bigm\| P \prime
n
\bigm\| \bigm\|
\infty \leq c4n
\Bigl(
\gamma \ast +1
p
+1
\Bigr)
1
\alpha \| Pn\| p , (10)
where c4 = c4(L, \gamma , p) > 0 is constant independent from n and z.
Remark 1. The inequalities (10) is sharp.
According to (3) (applied to the polynomial Qn - 1(z) := P \prime
n(z)), the estimation (10) is true also
for the points z \in G1+\varepsilon 0n - 1 with a different constant. Therefore, combining estimations (9) and (10)
(for the z \in GR,) we will obtain estimation on the growth of | P \prime
n(z)| in the whole complex plane.
Theorem 4. Let p > 1, L \in \widetilde Q\alpha for some
1
2
\leq \alpha \leq 1, and h(z) be defined by (1). Then, for
any Pn \in \wp n, n \in \BbbN , we have
\bigm| \bigm| P \prime
n (z)
\bigm| \bigm| \leq c5 \| Pn\| p
\left\{
n
\Bigl(
\gamma \ast +1
p
+1
\Bigr)
1
\alpha , z \in GR,\bigm| \bigm| \Phi 2(n+1)(z)
\bigm| \bigm|
d(z, L)
A3
n,p(z), z \in \Omega R,
where c5 = c5(L, \gamma , p) > 0 is constant independent from n and z, A3
n,p(z) defined as in Theorem 2
for each z \in \Omega R.
3. Some auxiliary results. Throughout this paper we denote ``a \preceq b"" and ``a \asymp b"" are
equivalent to a \leq cb and c1a \leq b \leq c2a for some constants c, c1, c2, respectively.
Lemma 1 [1]. Let G be a quasidisk, z1 \in L, z2, z3 \in \Omega \cap \{ z : | z - z1| \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, are
| z1 - z2| \asymp | z1 - z3| and | w1 - w2| \asymp | w1 - w3| .
b) If | z1 - z2| \preceq | z1 - z3| , then
ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 5
ON THE GROWTH OF DERIVATIVES OF ALGEBRAIC POLYNOMIALS . . . 587\bigm| \bigm| \bigm| \bigm| w1 - w3
w1 - w2
\bigm| \bigm| \bigm| \bigm| c1 \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| c2 ,
where 0 < r0 < 1 a constant, depending on G.
Corollary 1. Under the conditions of Lemma 1, we have
| w1 - w2| c1 \preceq | z1 - z2| \preceq | w1 - w2| \varepsilon ,
where \varepsilon = \varepsilon (G) < 1.
Lemma 2. Let L \in Q\alpha for some
1
2
\leq \alpha \leq 1. Then, for all w1, w2 \in \Omega \prime , we have
| \Psi (w1) - \Psi (w2)| \succeq | w1 - w2|
1
\alpha .
This fact it follows from of an appropriate result for the mapping f \in
\sum
(\kappa ) [36, p. 287] and
estimation for the \Psi \prime [21] (Theorem 2.8):
d(\Psi (\tau ) , L) \asymp
\bigm| \bigm| \Psi \prime (\tau )
\bigm| \bigm| (| \tau | - 1). (11)
4. Proof of theorems. Proof of Theorem 1. Let L \in \widetilde Q\alpha ,
1
2
\leq \alpha \leq 1, 0 < \beta \leq 1 and
R = 1 +
1
n
, R1 := 1 +
R - 1
2
. For z \in \Omega , let us set
Hn (z) :=
Pn (z)
\Phi n+1(z)
.
Let us represent the derivative of Hn (z) as follows:
H \prime
n (z) =
P \prime
n (z)
\Phi n+1(z)
+ Pn (z)
\Bigl(
\Phi - (n+1)(z)
\Bigr) \prime
, z \in \Omega .
Then \bigm| \bigm| P \prime
n (z)
\bigm| \bigm| \leq \bigm| \bigm| \Phi n+1(z)
\bigm| \bigm| \biggl\{ \bigm| \bigm| \bigm| \bigm| \biggl( Pn (z)
\Phi n+1(z)
\biggr) \prime \bigm| \bigm| \bigm| \bigm| + | Pn (z)|
\bigm| \bigm| \bigm| \bigm| \Bigl( \Phi - (n+1)(z)
\Bigr) \prime
\bigm| \bigm| \bigm| \bigm| \biggr\} . (12)
Therefore, to estimate | P \prime
n (z)| we must evaluate
A)
\bigm| \bigm| \bigm| \bigm| \biggl( Pn (z)
\Phi n+1(z)
\biggr) \prime \bigm| \bigm| \bigm| \bigm| and B)
\bigm| \bigm| \bigm| \bigm| \Bigl( \Phi - (n+1)(z)
\Bigr) \prime
\bigm| \bigm| \bigm| \bigm| , z \in \Omega .
A. Since the function Hn (z) :=
Pn (z)
\Phi n+1(z)
, Hn (\infty ) = 0, is analytic in \Omega , continuous on \Omega , then
Cauchy integral representation for the derivatives gives
H \prime
n (z) = - 1
2\pi i
\int
LR1
Hn (\zeta )
d\zeta
(\zeta - z)2
, z \in \Omega R.
Then
ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 5
588 F. G. ABDULLAYEV, M. IMASHKYZY\bigm| \bigm| \bigm| \bigm| \biggl( Pn (z)
\Phi n+1(z)
\biggr) \prime \bigm| \bigm| \bigm| \bigm| \leq 1
2\pi
\int
LR1
\bigm| \bigm| \bigm| \bigm| Pn (\zeta )
\Phi n+1(\zeta )
\bigm| \bigm| \bigm| \bigm| | d\zeta |
| \zeta - z| 2
\leq 1
2\pi d(z, LR1)
\int
LR1
| Pn (\zeta )|
| d\zeta |
| \zeta - z|
. (13)
Denote by
An(z) :=
\int
LR1
| Pn (\zeta )|
| d\zeta |
| \zeta - z|
, (14)
and estimate this integral. For this we give some notations.
Let wj := \Phi (zj), \varphi j := \mathrm{a}\mathrm{r}\mathrm{g}wj . Without loss of generality, we will assume that \varphi l < 2\pi . For
\eta := \mathrm{m}\mathrm{i}\mathrm{n}
\bigl\{
\eta j , j = 1, l
\bigr\}
, where \eta j = \mathrm{m}\mathrm{i}\mathrm{n}t\in \partial \Phi (\Omega (zj ,\delta j)) | t - wj | > 0, let us set
\Delta (\eta j) := \{ t : | t - wj | \leq \eta j\} \subset \Phi (\Omega (zj , \delta j)),
\Delta (\eta ) :=
l\bigcup
j=1
\Delta j(\eta ), \widehat \Delta j = \Delta \setminus \Delta (\eta j), \widehat \Delta (\eta ) := \Delta \setminus \Delta (\eta ), \Delta \prime
1 := \Delta \prime
1(1),
\Delta \prime
1(\rho ) :=
\biggl\{
t = Rei\theta : R \geq \rho > 1,
\varphi 0 + \varphi 1
2
\leq \theta <
\varphi 1 + \varphi 2
2
\biggr\}
,
\Delta \prime
j := \Delta \prime
j(1), \Delta \prime
j(\rho ) :=
\biggl\{
t = Rei\theta : R \geq \rho > 1,
\varphi j - 1 + \varphi j
2
\leq \theta <
\varphi j + \varphi 0
2
\biggr\}
, j = 2, 3, . . . , l,
where \varphi 0 = 2\pi - \varphi l, \Omega j := \Psi (\Delta \prime
j), L
j
R1
:= LR1 \cap \Omega j , \Omega =
\bigcup l
j=1
\Omega j .
For simplicity of calculations, we can limit ourselves to only one point on the boundary, which
the weight function has singularity, i.e., let h(z) be defined as in (1) for l = 1 and we put \gamma 1 =: \gamma . To
estimate An(z), multiplying the numerator and denominator of the integrand by h
1
p (\zeta ) and applying
the Hölder inequality and (5), we obtain
An(z) =
\int
LR1
| Pn (\zeta )|
| d\zeta |
| \zeta - z|
=
\int
LR1
h
1
p (\zeta ) | Pn (\zeta )|
| d\zeta |
h
1
p (\zeta ) | \zeta - z|
\leq
\leq \| Pn\| p
\left( \int
LR1
| d\zeta |
h
q
p (\zeta ) | \zeta - z| q
\right)
1
q
,
1
p
+
1
q
= 1.
Replacing the variable \tau = \Phi (\zeta ), we get
An(z) \preceq \| Pn\| p
\left( \int
| \tau | =R1
| \Psi \prime (\tau )| | d\tau |
| \Psi (\tau ) - \Psi (w1)| \gamma (q - 1) | \Psi (\tau ) - \Psi (w)| q
\right)
1
q
, z = \Psi (w). (15)
To estimate the integral on the right-hand side, we put
F 1
R1
:= \Phi (LR1) := \Delta \prime
1 \cap \{ \tau : | \tau | = R1\} ,
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ON THE GROWTH OF DERIVATIVES OF ALGEBRAIC POLYNOMIALS . . . 589
E11
R1
:=
\bigl\{
\tau : \tau \in F 1
R1
, | \tau - w1| < c1(R1 - 1)
\bigr\}
,
E12
R1
:=
\bigl\{
\tau : \tau \in F 1
R1
, c1(R1 - 1) \leq | \tau - w1| < \eta
\bigr\}
,
E13
R1
:= \{ \tau : \tau \in \Phi (LR1), \eta \leq | \tau - w1| < \eta \ast \} ,
where 0 < c1 < \eta is chosen so that \{ \tau : | \tau - w1| < c1(R1 - 1)\} \cap \Delta \not = \varnothing and \Phi (LR1) =
=
\bigcup 3
k=1
E1k
R1
. Taking into consideration these notations, from (15) we have
An(z) \preceq \| Pn\| p
3\sum
k=1
Jk
n(z), (16)
where \Bigl(
Jk
n(z)
\Bigr) q
:=
\int
E1k
R1
| \Psi \prime (\tau )| | d\tau |
| \Psi (\tau ) - \Psi (w1)| \gamma (q - 1) | \Psi (\tau ) - \Psi (w)| q
, k = 1, 2, 3.
For any k = 1, 2, denote by
E1k
R1,1 :=
\Bigl\{
\tau \in E1k
R1
: | \Psi (\tau ) - \Psi (w1)| \geq | \Psi (\tau ) - \Psi (w)|
\Bigr\}
, E1k
R1,2 := E1k
R1
\setminus E1k
R1,1,
\Bigl(
I(E1k
R1,1)
\Bigr) q
:=
\left\{
\int
E1k
R1,1
| \Psi \prime (\tau )| | d\tau |
| \Psi (\tau ) - \Psi (w)| \gamma (q - 1)+q
, if \gamma \geq 0,
\int
E1k
R1,1
| \Psi (\tau ) - \Psi (w1)| ( - \gamma )(q - 1) | \Psi \prime (\tau )| | d\tau |
| \Psi (\tau ) - \Psi (w)| q
, if \gamma < 0,
(17)
\Bigl(
I(E1k
R1,2)
\Bigr) q
:=
\int
E1k
R1,2
| \Psi \prime (\tau )| | d\tau |
| \Psi (\tau ) - \Psi (w1)| \gamma (q - 1)+q
, k = 1, 2,
and estimate the last integrals. Given the possible values \gamma ( - 1 < \gamma < 0 and \gamma \geq 0), we will
consider the cases separately.
1. Let \gamma \geq 0. If z \in \Omega (\delta ), applying Lemma 2 and (11), we get
\bigl(
I(E11
R1,1)
\bigr) q \preceq n
\int
E11
R1,1
| d\tau |
| \tau - w|
[\gamma (q - 1)+q - 1]
\alpha
\preceq n1+
\gamma (q - 1)+q - 1
\alpha \mathrm{m}\mathrm{e}\mathrm{s}E11
R1,1 \preceq n
\gamma (q - 1)+q - 1
\alpha ,
I(E11
R1,1) \preceq n
\gamma +1
p\alpha ,
\bigl(
I(E11
R1,2)
\bigr) q \preceq n
\int
E11
R1,2
| d\tau |
| \tau - w1|
\gamma (q - 1)+q - 1
\alpha
\preceq n1+
\gamma (q - 1)+q - 1
\alpha \mathrm{m}\mathrm{e}\mathrm{s}E11
R1,2 \preceq n
\gamma (q - 1)+q - 1
\alpha ,
I(E11
R1,2) \preceq n
\gamma +1
p\alpha ,
(18)
ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 5
590 F. G. ABDULLAYEV, M. IMASHKYZY
\bigl(
I(E12
R1,1)
\bigr) q \preceq \int
E12
R1,1
d(\Psi (\tau ) , L) | d\tau |
(| \tau | - 1) | \Psi (\tau ) - \Psi (w)| \gamma (q - 1)+q
\preceq
\preceq n
\left\{
n
\gamma (q - 1)+q - 1
\alpha
- 1, [\gamma (q - 1) + q - 1] > \alpha ,
\mathrm{l}\mathrm{n}n, [\gamma (q - 1) + q - 1] = \alpha ,
1, [\gamma (q - 1) + q - 1] < \alpha ,
(19)
I(E12
R1,1) \preceq
\left\{
n
\gamma +1
\alpha p , p < 1 +
\gamma + 1
\alpha
,
(n \mathrm{l}\mathrm{n}n)
1 - 1
p , p = 1 +
\gamma + 1
\alpha
,
n
1 - 1
p , p > 1 +
\gamma + 1
\alpha
,
\bigl(
I(E12
R1,2)
\bigr) q \preceq n
\left\{
n
\gamma (q - 1)+q - 1
\alpha
- 1, [\gamma (q - 1) + q - 1] > \alpha ,
\mathrm{l}\mathrm{n}n, [\gamma (q - 1) + q - 1] = \alpha ,
1, [\gamma (q - 1) + q - 1] < \alpha ,
(20)
I(E12
R1,2) \preceq
\left\{
n
\gamma +1
\alpha p , p < 1 +
\gamma + 1
\alpha
,
(n \mathrm{l}\mathrm{n}n)
1 - 1
p , p = 1 +
\gamma + 1
\alpha
,
n
1 - 1
p , p > 1 +
\gamma + 1
\alpha
.
For \tau \in E13
R1
we see that \eta < | \tau - w1| < 2\pi \.R1, | \tau - w| \geq \eta - c1. Therefore, | \Psi (\tau ) - \Psi (w1)| \succeq 1
from Lemma 1 and, for | \tau - w1| \geq \eta , | \Psi (\tau ) - \Psi (w)| \succeq | \tau - w|
1
\alpha from Lemma 2. Then, for
w \in \Delta (w1, \eta ), applying (11), we have
\bigl(
J3
2 (z)
\bigr) q \preceq \int
E13
R1
d(\Psi (\tau ) , L) | d\tau |
(| \tau | - 1) | \Psi (\tau ) - \Psi (w)| q
\preceq
\left\{
n
q - 1
\alpha , q - 1 > \alpha ,
n \mathrm{l}\mathrm{n}n, q - 1 = \alpha ,
n, q - 1 < \alpha ,
\preceq n
\int
E11
R1,1
| d\tau |
| \tau - w|
q - 1
\alpha
,
J3
2 (z) \preceq
\left\{
n
1
p\alpha , p < 1 +
1
\alpha
,
(n \mathrm{l}\mathrm{n}n)
1 - 1
p , p = 1 +
1
\alpha
,
n
1 - 1
p , p > 1 +
1
\alpha
,
z \in \Omega (\delta ), (21)
J3
2 (z) \preceq n(
1
\alpha
- 1)(1 - 1
p
)
, z \in \widehat \Omega (\delta ).
Combining (18) – (21), for p > 1, \gamma \geq 0 and z \in \Omega (\delta ), we get
ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 5
ON THE GROWTH OF DERIVATIVES OF ALGEBRAIC POLYNOMIALS . . . 591
3\sum
k=1
Jk
n(z) \preceq
\left\{
n
\gamma +1
\alpha p , 1 < p < 1 +
\gamma + 1
\alpha
,
n
1 - 1
p (\mathrm{l}\mathrm{n}n)
1 - 1
p , p = 1 +
\gamma + 1
\alpha
,
n
1 - 1
p , p > 1 +
\gamma + 1
\alpha
.
(22)
If z \in \widehat \Omega (\delta ), then\bigl(
J1
2 (z)
\bigr) q \preceq n
\int
E11
R1
| d\tau |
| \tau - w1|
\gamma (q - 1) - 1
\alpha
\preceq n1+
\gamma (q - 1) - 1
\alpha \mathrm{m}\mathrm{e}\mathrm{s}E11
R1
\preceq n
\gamma (q - 1) - 1
\alpha , J1
2 (z) \preceq n
\Bigl(
\gamma +1
p
- 1
\Bigr)
1
\alpha ,
(23)
\bigl(
J2
2 (z)
\bigr) q \preceq n
\int
E11
R1
| d\tau |
| \tau - w1|
\gamma (q - 1) - 1
\alpha
\preceq
\left\{
n
\gamma (q - 1) - 1
\alpha , \gamma (q - 1) - 1 > \alpha ,
n \mathrm{l}\mathrm{n}n, \gamma (q - 1) - 1 = \alpha ,
n, \gamma (q - 1) - 1 < \alpha ,
J2
2 (z) \preceq
\left\{
n
\Bigl(
\gamma +1
p
- 1
\Bigr)
1
\alpha , 1 < p < 1 +
\gamma
1 + \alpha
,
(n \mathrm{l}\mathrm{n}n)
1 - 1
p , p = 1 +
\gamma
1 + \alpha
,
n
1 - 1
p , p > 1 +
\gamma
1 + \alpha
,
\bigl(
J3
2 (z)
\bigr) q \preceq \int
E13
R1
d(\Psi (\tau ) , L)
| \tau | - 1
| d\tau | \preceq n(
1
\alpha
- 1), J3
2 (z) \preceq n
( 1
\alpha
- 1)
\Bigl(
1 - 1
p
\Bigr)
,
(24)
and, from (16) – (24), we obtain
An(z) \preceq \| Pn\| p
\left\{
n
\gamma +1
\alpha p , 1 < p < 1 +
\gamma + 1
\alpha
,
n
1 - 1
p (\mathrm{l}\mathrm{n}n)
1 - 1
p , p = 1 +
\gamma + 1
\alpha
,
n
1 - 1
p , p > 1 +
\gamma + 1
\alpha
,
if z \in \Omega (\delta ), (25)
An(z) \preceq \| Pn\| p
\left\{
n
\Bigl(
\gamma +1
p
- 1
\Bigr)
1
\alpha , 1 < p < 1 +
\gamma
1 + \alpha
,
(n \mathrm{l}\mathrm{n}n)
1 - 1
p , p = 1 +
\gamma
1 + \alpha
,
n
1 - 1
p , p > 1 +
\gamma
1 + \alpha
,
if z \in \widehat \Omega (\delta ). (26)
2. If \gamma < 0, for w \in \Delta (w1, \eta ) \cap \Omega R(\delta ) such that | \Psi (\tau ) - \Psi (w1)| \preceq | \Psi (\tau ) - \Psi (w)| , according
to Lemma 1, analogously we have
ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 5
592 F. G. ABDULLAYEV, M. IMASHKYZY
\bigl(
I(E11
R1,1)
\bigr) q \preceq n
\int
E11
R1,1
| d\tau |
| \tau - w|
q - 1
\alpha
\preceq n1+
q - 1
\alpha \mathrm{m}\mathrm{e}\mathrm{s}E11
R1
\preceq n
q - 1
\alpha , I(E11
R1,1) \preceq n
1
p\alpha , (27)
\bigl(
I(E11
R1,2)
\bigr) q \preceq n
\int
E11
R1,2
| d\tau |
| \tau - w1|
\gamma (q - 1)+q - 1
\alpha
\preceq n1+
\gamma (q - 1)+q - 1
\alpha \mathrm{m}\mathrm{e}\mathrm{s}E11
R1,2 \preceq n
\gamma (q - 1)+q - 1
\alpha ,
I(E11
R1,2) \preceq n
\gamma +1
p\alpha .
(28)
For \tau \in E12
R1
we see that | \tau - w1| < \eta and, from Lemma 1, | \Psi (\tau ) - \Psi (w1)| \preceq 1. Then, for
w \in \Delta (w1, \eta ) \cap \Omega R(\delta ) such that | \Psi (\tau ) - \Psi (w1)| \preceq | \Psi (\tau ) - \Psi (w)| , applying Lemma 2, we get
\bigl(
I(E12
R1,1)
\bigr) q \preceq n
\int
E12
R1,1
| d\tau |
| \tau - w|
q - 1
\alpha
\preceq
\left\{
n
q - 1
\alpha , q - 1 > \alpha ,
(n \mathrm{l}\mathrm{n}n)
1 - 1
p , q - 1 = \alpha ,
n
1 - 1
p , q - 1 < \alpha ,
I(E12
R1,1) \preceq
\left\{
n
1
\alpha p , 1 < p < 1 +
1
\alpha
,
(n \mathrm{l}\mathrm{n}n)
1 - 1
p , p = 1 +
1
\alpha
,
n
1 - 1
p , p > 1 +
1
\alpha
,
(29)
\bigl(
I(E12
R1,2)
\bigr) q \preceq n
\int
E12
R1,2
| d\tau |
| \tau - w|
q+\gamma (q - 1) - 1
\alpha
\preceq
\left\{
n
q+\gamma (q - 1) - 1
\alpha , q + \gamma (q - 1) - 1 > \alpha ,
n \mathrm{l}\mathrm{n}n, q + \gamma (q - 1) - 1 = \alpha ,
n, q + \gamma (q - 1) - 1 < \alpha ,
I(E12
R1,2) \preceq
\left\{
n
\gamma +1
p\alpha , 1 < p < 1 +
\gamma + 1
\alpha
,
(n \mathrm{l}\mathrm{n}n)
1 - 1
p , p = 1 +
\gamma + 1
\alpha
,
n
1 - 1
p , p > 1 +
\gamma + 1
\alpha
.
(30)
For \tau \in E13
R1
and each w \in \Delta (w1, \eta ) \cap \Omega R(\delta ) we see that \eta < | \tau - w1| < 2\pi R1. Therefore, from
Lemma 1 and applying (11), we obtain
\bigl(
I(E13
R1
)
\bigr) q \preceq n
\int
E13
R1
| d\tau |
| \tau - w|
q - 1
\alpha
\preceq
\left\{
n
q - 1
\alpha , q - 1 > \alpha ,
(n \mathrm{l}\mathrm{n}n) , q - 1 = \alpha ,
n, q - 1 < \alpha ,
I(E13
R1
) \preceq
\left\{
n
1
\alpha p , 1 < p < 1 +
1
\alpha
,
(n \mathrm{l}\mathrm{n}n)
1 - 1
p , p = 1 +
1
\alpha
,
n
1 - 1
p , p > 1 +
1
\alpha
.
(31)
ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 5
ON THE GROWTH OF DERIVATIVES OF ALGEBRAIC POLYNOMIALS . . . 593
Therefore, combining (27) – (31), in case of \gamma < 0 for z \in \Omega (\delta ), we have
3\sum
k=1
Jk
n(z) \preceq
\left\{
n
1
\alpha p , 1 < p < 1 +
1
\alpha
,
(n \mathrm{l}\mathrm{n}n)
1 - 1
p , p = 1 +
1
\alpha
,
n
1 - 1
p , p > 1 +
1
\alpha
.
(32)
If z \in \widehat \Omega (\delta ), then | w - w1| \geq \eta . From Lemma 2 and from (11), we get\bigl(
J1
2 (z)
\bigr) q \preceq n
\int
E11
R1
| \Psi (\tau ) - \Psi (w1)| ( - \gamma )(q - 1)+1 | d\tau | \preceq n\mathrm{m}\mathrm{e}\mathrm{s}E11
R1
\preceq 1, J1
2 (z) \preceq 1,
\bigl(
J2
2 (z)
\bigr) q \preceq \int
E12
R1
d(\Psi (\tau ) , L)
| \tau | - 1
| d\tau | \preceq n, J2
2 (z) \preceq n
1 - 1
p , (33)
\bigl(
J3
2 (z)
\bigr) q \preceq \int
E13
R1
d(\Psi (\tau ) , L) | d\tau |
| \tau | - 1
\preceq n, J3
2 (z) \preceq n
1 - 1
p .
Therefore, combining the last tree estimates, in case of \gamma < 0, for z \in \widehat \Omega (\delta ), we obtain
3\sum
k=1
Jk
n(z) \preceq n
1 - 1
p . (34)
Then, for - 1 < \gamma < 0, from (16), (32) and (34), we have
An(z) \preceq \| Pn\| p
\left\{
n
1
\alpha p , 1 < p < 1 +
1
\alpha
, z \in \Omega (\delta ),
(n \mathrm{l}\mathrm{n}n)
1 - 1
p , p = 1 +
1
\alpha
, z \in \Omega (\delta ),
n
1 - 1
p , p > 1 +
1
\alpha
, z \in \Omega (\delta ),
n
1 - 1
p , p > 1, z \in \widehat \Omega (\delta ).
(35)
Therefore, combining (22) and (35), for any \gamma > - 1, p > 1, we obtain
An(z) \preceq \| Pn\| p
\left\{
n
\gamma \ast +1
\alpha p , 1 < p < 1 +
\gamma \ast + 1
\alpha
,
(n \mathrm{l}\mathrm{n}n)
1 - 1
p , p = 1 +
\gamma \ast + 1
\alpha
,
n
1 - 1
p , p > 1 +
\gamma \ast + 1
\alpha
,
if z \in \Omega (\delta ), (36)
An(z) \preceq \| Pn\| p
\left\{
n
\Bigl(
\gamma +1
p
- 1
\Bigr)
1
\alpha , 1 < p < 1 +
\gamma
1 + \alpha
,
(n \mathrm{l}\mathrm{n}n)
1 - 1
p , p = 1 +
\gamma
1 + \alpha
,
n
1 - 1
p , p > 1 +
\gamma \ast
1 + \alpha
,
if z \in \widehat \Omega (\delta ), (37)
\gamma \ast := \mathrm{m}\mathrm{a}\mathrm{x} \{ 0; \gamma \} .
ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 5
594 F. G. ABDULLAYEV, M. IMASHKYZY
B. Now, we begin to estimate the
\bigm| \bigm| \bigm| \bigl( \Phi - (n+1)(z)
\bigr) \prime \bigm| \bigm| \bigm| .
Since \Phi (\infty ) = \infty , then Cauchy integral representation for the region \Omega R gives\Bigl(
\Phi - (n+1)(z)
\Bigr) \prime
= - 1
2\pi i
\int
LR1
\Phi - (n+1)(\zeta )d\zeta
(\zeta - z)2
, z \in \Omega R.
Replacing the variable \tau = \Phi (\zeta ) and according to (11), we have
\bigm| \bigm| \bigm| \bigm| \Bigl( \Phi - (n+1)(z)
\Bigr) \prime
\bigm| \bigm| \bigm| \bigm| \preceq n
\int
| \tau | =R1
| d\tau |
| \tau - w|
1
\alpha
\preceq
\left\{ n
1
\alpha , if z \in \Omega (\delta ),
n, if z \in \widehat \Omega (\delta ). (38)
Combining estimates (12) – (16), (36), (37) and (38), we get
\bigm| \bigm| P \prime
n (z)
\bigm| \bigm| \leq \bigm| \bigm| \Phi n+1(z)
\bigm| \bigm| \left[ An(z)
d(z, L)
+ | Pn (z)|
\left\{ n
1
\alpha , if z \in \Omega (\delta ),
n, if z \in \widehat \Omega (\delta )
\right] , (39)
where for any \gamma > - 1, p > 1, z \in \Omega (\delta ) and z \in \widehat \Omega (\delta ) An(z) defined as in (36), (37), respectively.
Theorem 1 is proved.
Proof of Theorem 2. Now let us start the evaluations of | Pn (z)| . Let L \in \widetilde Q\alpha for some
1
2
\leq \alpha \leq 1.
Since the function Hn (z) :=
Pn (z)
\Phi n+1(z)
, Hn (\infty ) = 0, is analytic in \Omega , continuous on \Omega , then
Cauchy integral representation for the region \Omega R1 gives
Hn (z) = - 1
2\pi i
\int
LR1
Hn (\zeta )
d\zeta
\zeta - z
, z \in \Omega R.
Then \bigm| \bigm| \bigm| \bigm| Pn (z)
\Phi n+1(z)
\bigm| \bigm| \bigm| \bigm| \leq 1
2\pi
\int
LR1
\bigm| \bigm| \bigm| \bigm| Pn (\zeta )
\Phi n+1(\zeta )
\bigm| \bigm| \bigm| \bigm| | d\zeta |
| \zeta - z|
\leq 1
2\pi d(z, LR1)
\int
LR1
| Pn (\zeta )| | d\zeta | ,
and, so,
| Pn (z)| \preceq
\bigm| \bigm| \Phi n+1(z)
\bigm| \bigm|
d(z, LR1)
\int
LR1
| Pn (\zeta )| | d\zeta | . (40)
Denote by
An :=
\int
LR1
| Pn (\zeta )| | d\zeta | , (41)
and repeating estimate (14) for An(z), for any \gamma > - 1, p > 1, z \in \Omega R, for An, we obtain
ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 5
ON THE GROWTH OF DERIVATIVES OF ALGEBRAIC POLYNOMIALS . . . 595
An \preceq \| Pn\| p
\left\{
n
\Bigl(
\gamma +1
p
- 1
\Bigr)
1
\alpha , 1 < p < 1 +
\gamma
1 + \alpha
, \gamma > 1 + \alpha ,
n
1 - 1
p (\mathrm{l}\mathrm{n}n)
1 - 1
p , p = 1 +
\gamma
1 + \alpha
, \gamma > 1 + \alpha ,
n
1 - 1
p , p > 1 +
\gamma
1 + \alpha
, \gamma > 1 + \alpha ,
n
1 - 1
p , p > 1, - 1 < \gamma \leq 1 + \alpha .
(42)
Combining estimates (40) – (42), we have
| Pn (z)| \preceq
\bigm| \bigm| \Phi n+1(z)
\bigm| \bigm|
d(z, LR1)
An,
where An \preceq \| Pn\| p
\left\{
n
\Bigl(
\gamma +1
p
- 1
\Bigr)
1
\alpha , 1 < p < 1 +
\gamma
1 + \alpha
, \gamma > 1 + \alpha ,
n
1 - 1
p (\mathrm{l}\mathrm{n}n)
1 - 1
p , p = 1 +
\gamma
1 + \alpha
, \gamma > 1 + \alpha ,
n
1 - 1
p , p > 1 +
\gamma
1 + \alpha
, \gamma > 1 + \alpha ,
n
1 - 1
p , p > 1, - 1 < \gamma \leq 1 + \alpha .
Theorem 2 is proved.
Proof of Theorem 3. From (36), (37) and (39), we get
\bigm| \bigm| P \prime
n (z)
\bigm| \bigm| \preceq \bigm| \bigm| \Phi n+1(z)
\bigm| \bigm|
d(z, L)
\left[ An(z) + | Pn (z)|
\left\{ n
1
\alpha , if z \in \Omega (\delta ),
n, if z \in \widehat \Omega (\delta )
\right] ,
where, for any \gamma > - 1, p > 1,
An(z) \preceq \| Pn\| p
\left\{
n
\gamma \ast +1
\alpha p , 1 < p < 1 +
\gamma \ast + 1
\alpha
,
(n \mathrm{l}\mathrm{n}n)
1 - 1
p , p = 1 +
\gamma \ast + 1
\alpha
,
n
1 - 1
p , p > 1 +
\gamma \ast + 1
\alpha
,
if z \in \Omega (\delta ),
An(z) \preceq \| Pn\| p
\left\{
n
\Bigl(
\gamma +1
p
- 1
\Bigr)
1
\alpha , 1 < p < 1 +
\gamma
1 + \alpha
,
(n \mathrm{l}\mathrm{n}n)
1 - 1
p , p = 1 +
\gamma
1 + \alpha
,
n
1 - 1
p , p > 1 +
\gamma \ast
1 + \alpha
,
if z \in \widehat \Omega (\delta ),
and \gamma \ast := \mathrm{m}\mathrm{a}\mathrm{x} \{ 0; \gamma \} .
Taking into account estimates for | Pn (z)| from (8) and combining with (38) gives a proofs of
need estimates
ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 5
596 F. G. ABDULLAYEV, M. IMASHKYZY
\bigm| \bigm| P \prime
n (z)
\bigm| \bigm| \preceq \bigm| \bigm| \Phi 2(n+1)(z)
\bigm| \bigm|
d(z, L)
\left\{
n
\gamma +1
p\alpha , 1 < p < 1 +
\gamma
1 + \alpha
, \gamma > 0,
n
1 - 1
p
+ 1
\alpha (\mathrm{l}\mathrm{n}n)
1 - 1
p , p = 1 +
\gamma
1 + \alpha
, \gamma > 0,
n
1 - 1
p
+ 1
\alpha , p > 1 +
\gamma
1 + \alpha
, \gamma > 0,
n
1 - 1
p
+ 1
\alpha , p > 1, - 1 < \gamma \leq 0,
if z \in \Omega (\delta ),
\bigm| \bigm| P \prime
n (z)
\bigm| \bigm| \preceq \bigm| \bigm| \Phi 2(n+1)(z)
\bigm| \bigm|
d(z, L)
\left\{
n
\Bigl(
\gamma +1
p
- 1
\Bigr)
1
\alpha
+1
, 1 < p < 1 +
\gamma
1 + \alpha
, \gamma > 1 + \alpha ,
n
2 - 1
p (\mathrm{l}\mathrm{n}n)
1 - 1
p , p = 1 +
\gamma
1 + \alpha
, \gamma > 1 + \alpha ,
n
2 - 1
p , p > 1 +
\gamma
1 + \alpha
, \gamma > 1 + \alpha ,
n
2 - 1
p , p > 1, - 1 < \gamma \leq 1 + \alpha ,
if z \in \widehat \Omega (\delta ).
In conclusion, note that in proofs everywhere there is a quantity d(z, LR1). Let us show that
d(z, LR1) \succeq d(z, L) holds for all z \in \Omega R. For the points z /\in \Omega (LR1 , d(LR1 , LR)), we have
d(z, LR1) \succeq \delta \succeq d(z, L). Now, let z \in \Omega (LR1 , d(LR1 , LR)). Denote by \xi 1 \in LR1 the point such
that d(z, LR1) = | z - \xi 1| and point \xi 2 \in L such that d(z, L) = | z - \xi 2| , and for w = \Phi (z),
t1 = \Phi (\xi 1), t2 = \Phi (\xi 2), we get | w - w1| \geq | | w - w2| - | w2 - w1| | \geq
\bigm| \bigm| \bigm| \bigm| | w - w2| -
1
2
| w - w2|
\bigm| \bigm| \bigm| \bigm| \geq
\geq 1
2
| w - w2| . Then, according to Lemma 1, we obtain d(z, LR1) \succeq d(z, L).
Theorem 3 is proved.
Proof of Theorem 4. Let p > 1, U := B(z, d(z, LR1)) and z \in L is an arbitrary fixed point. By
Cauchy integral formulas for derivatives, we have
P \prime
n(z) =
1
2\pi i
\int
\partial U
Pn(t)
(t - z)2
dt.
Then, and applying (3), we obtain\bigm| \bigm| P \prime
n(z)
\bigm| \bigm| \leq 1
2\pi
\mathrm{m}\mathrm{a}\mathrm{x}
z\in \partial U
| Pn(t)|
\int
\partial U
| dt|
| t - z| 2
\preceq \mathrm{m}\mathrm{a}\mathrm{x}
t\in G
| Pn(t)|
1
d(z, LR1)
.
Now, applying [11] (Theorem 2.4) and using Lemma 2, we get\bigm| \bigm| P \prime
n(z)
\bigm| \bigm| \preceq n
\gamma \ast +1
p\alpha \| Pn\| p n
1
\alpha \preceq n
\Bigl(
\gamma \ast +1
p
+1
\Bigr)
1
\alpha \| Pn\| p.
Sharpness of the inequality (10) can be argued as follows.
These inequalities can be interpreted as a combination of the well known sharp Markov inequali-
ties \| P \prime
n\| \infty \preceq n \| Pn\| \infty whit inequality for \| Pn\| \infty in terms of the norm \| Pn\| p . The sharpness of the
last inequality can be verified in the following examples: for the polynomial Tn(z) = 1+z+. . .+zn,
h\ast (z) = h0(z), h
\ast \ast (z) = | z - 1| \gamma , \gamma > 0, L := \{ z : | z| = 1\} and any n \in \BbbN there exist
c6 = c6(h
\ast , p) > 0, c7 = c7(h
\ast \ast , p) > 0 such that
a) \| T\| \infty \geq c6n
1
p \| T\| \scrL p(h\ast , L) , p > 1,
b) \| T\| \infty \geq c7n
\gamma +1
p \| T\| \scrL p(h\ast \ast , L) , p > \gamma + 1.
ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 5
ON THE GROWTH OF DERIVATIVES OF ALGEBRAIC POLYNOMIALS . . . 597
Really, if L := \{ z : | z| = 1\} , then L \in \widetilde Q1. Let a) h\ast (z) \equiv 1; b) h\ast \ast (z) = | z - 1| \gamma , \gamma > 0.
Obviously,
| T (z)| \leq
n - 1\sum
j=0
\bigm| \bigm| zj\bigm| \bigm| = n, | z| = 1, | T (1)| = n.
Then \| T\| \infty = n.
On the other hand, according to [40, p. 236], we have
\| T\| \scrL p(h\ast ,L) \asymp n
1 - 1
p , p > 1, and \| T\| \scrL p(h\ast \ast ,L) \asymp n
1 - \gamma +1
p , p > \gamma + 1.
Therefore,
a) \| T\| \infty = n \asymp n
1
p \| T\| \scrL p(h\ast ,L) , p > 1;
b) \| T\| \infty = n = n \cdot n1 -
\gamma +1
p \cdot n
\gamma +1
p
- 1 \asymp n
\gamma +1
p \| T\| \scrL p(h\ast \ast ,L) , p > \gamma + 1.
Theorem 4 is proved.
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|
| id | umjimathkievua-article-7052 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T03:31:14Z |
| publishDate | 2022 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/9e/60a6f0bb79c881a4333742b197dad29e.pdf |
| spelling | umjimathkievua-article-70522022-10-24T09:23:02Z On the growth of derivatives of algebraic polynomials in a weighted Lebesgue space On the growth of derivatives of algebraic polynomials in a weighted Lebesgue space Аbdullayev, F. G. Imashkyzy, M. Аbdullayev, Fahreddin Аbdullayev, F. G. Imashkyzy, М. Algebraic polynomial, Quasiconformal mapping, Quasicircle. UDC 517.5 We study growth rates of derivatives of an arbitrary algebraic polynomial in bounded and inbounded regions of the complex plane in weighted Lebesgue spaces. УДК 517.5 Про зростання похiдних алгебраїчних полiномiву ваговому просторi ЛебегаВивчається зростання похiдних довiльних алгебраїчних полiномiв у обмежених i необмежених областях комплексної площини у вагових просторах Лебега. Institute of Mathematics, NAS of Ukraine 2022-06-17 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/7052 10.37863/umzh.v74i5.7052 Ukrains’kyi Matematychnyi Zhurnal; Vol. 74 No. 5 (2022); 582 - 598 Український математичний журнал; Том 74 № 5 (2022); 582 - 598 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/7052/9233 Copyright (c) 2022 Fahreddin Аbdullayev, Meerim Imashkyzy |
| spellingShingle | Аbdullayev, F. G. Imashkyzy, M. Аbdullayev, Fahreddin Аbdullayev, F. G. Imashkyzy, М. On the growth of derivatives of algebraic polynomials in a weighted Lebesgue space |
| title | On the growth of derivatives of algebraic polynomials in a weighted Lebesgue space |
| title_alt | On the growth of derivatives of algebraic polynomials in a weighted Lebesgue space |
| title_full | On the growth of derivatives of algebraic polynomials in a weighted Lebesgue space |
| title_fullStr | On the growth of derivatives of algebraic polynomials in a weighted Lebesgue space |
| title_full_unstemmed | On the growth of derivatives of algebraic polynomials in a weighted Lebesgue space |
| title_short | On the growth of derivatives of algebraic polynomials in a weighted Lebesgue space |
| title_sort | on the growth of derivatives of algebraic polynomials in a weighted lebesgue space |
| topic_facet | Algebraic polynomial Quasiconformal mapping Quasicircle. |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/7052 |
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