![MathJax Logo](/templates/jsp/_style2/_tandf/pb2/images/math-jax.gif)
ABSTRACT
In this paper, we study the growth of the mth derivative of an arbitrary algebraic polynomial in bounded and unbounded general domains of the complex plane in weighted Lebesgue spaces. Further, we obtain estimates for the derivatives at the closure of this regions. As a result, estimates for derivatives on the entire complex plane were found.
1. Introduction
Let be a complex plane and
be a bounded Jordan region with boundary
(without loss of generality, let
;
. For
and
, let
. Let
be the univalent conformal mapping normalized by
and
;
. For
, we take
,
, and
.
Let denotes the class of all algebraic polynomials
of degree at most
.
Let be the fixed system of distinct points. For some fixed
,
, and
, consider generalized Jacobi weight function
:
(1)
(1) where
, for all
and some constant
,
.
For each and rectifiable Jordan curve
, we introduce:
(2)
(2) In the theory of approximations of a function of a complex variable, the following Bernstein–Walsh inequality is often used [Citation1]:
(3)
(3) It follows that the quantity
has the same order of growth in
and
with respect to n for all constants
.
An analogue of this inequality in space is the following inequality [Citation2]:
This estimate has been generalized in [Citation3, Lemma 2.4] for weight function
, defined as in (Equation1
(1)
(1) ), as follows:
(4)
(4) If we replace the curve L to the region G and define two-dimensional analogues of the quantities (Equation2
(2)
(2) ) (we denotes them by
,
and
respectively), then for them we can also indicate the corresponding estimate of the type (Equation4
(4)
(4) ). For this, first we will give the following definition.
For any and arbitrary
let
and
be a conformal and univalent map which is normalized by
and
;
.
Definition 1.1
[Citation4, p.286]
A bounded Jordan region G is called a κ-quasidisk, , if any conformal mapping ψ can be extended to a K-quasiconformal,
, homeomorphism of the plane
on
. In that case the curve
is called a κ-quasicircle. The region G (curve L) is called a quasidisk (quasicircle), if it is κ-quasidisk (κ-quasicircle) with some
.
Denote by , class of κ-quasidisks and say that
, if
. Further, we denote that
, if
for some
. Quasicircles can be non-rectifiable (see, for example, [[Citation5],[Citation6, p.104]]). Since the object of study will be integrals along a curve, then we will say that
, if
and
is rectifiable. Correspondingly,
, if
for some
.
The analogue of the estimates (Equation3(3)
(3) ) and (Equation4
(4)
(4) ) for arbitrary quasidisks and
defined as in (Equation1
(1)
(1) ) for the
can be given as follows [Citation7]:
(5)
(5) where
and
constants, independent of n and R. This estimate was generalized for arbitrary Jordan region G and
in [Citation8, Theorem 1.1] as follows:
where
, is asymptotically sharp constant.
N. Stylianopoulos [Citation9] replaced the norm with norm
on the right-hand side of (Equation3
(3)
(3) ) and found a new version of the Bernstein–Walsh Lemma: Let
. Then there exists a constant
depending only on L such that
holds for every
, where
.
In this paper, we continue the study of the problem on uniform and pointwise estimates of the derivatives ,
, in
and
,
, and the following types of estimates will be found:
(6)
(6) where
and
, as
, depending on the properties of the L, h.
Analogous results of (Equation6(6)
(6) )-type for m = 0, different weight function h, in unbounded region were obtained in [Citation3,Citation9–22,Citation23, p.418–428,Citation24] and others.
Estimates of the (Equation6(6)
(6) )-type for
, for the norms
or
, p>0, for some h (
or
) was studied since the beginning of the twentieth century [Citation25–27] and has been studied in [Citation7,Citation23, p.418–428,Citation24,Citation28–36,Citation37, Sect. 5.3,Citation38,Citation39, p.122–133,Citation40] (see also the references cited therein) and others.
2. Definitions and main results
Throughout this paper, are positive and
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
and m>k, notation
means
.
Let S be rectifiable Jordan curve or arc and let , be the natural parametrization of S. A Jordan curve or arc is called smooth, if S has a continuous tangent
at every point
. The class of such curves or arcs is denoted by
.
Let ,
be arbitrary points on L and
denotes the subarc of L of shorter diameter with endpoints
and
. The curve L is a quasicircle if and only if (three-point property).
Lesley [Citation41, p.341] said that the curve L is ‘c-quasiconformal’, if there exists the constant c>0, independent from points
,
and z such that
. The Jordan curve L is called asymptotically conformal [Citation42, Citation43], if
, as
. According to the geometric criteria of quasiconformality of the curves [ [Citation43, p.107],[Citation44, p.81]], every asymptotically conformal curve is a quasicircle. Every smooth curve is asymptotically conformal but corners are not allowed. The asymptotically conformal curves can be non-rectifiable.
Following [Citation4, p.163], we say that a bounded Jordan curve L is λ-quasismooth (in the sense of Lavrentiev) curve, if for every pair , there exists a constant
such that
where
is the linear measure (length) of
. The region G is called a λ-quasismooth region, if
is a λ-quasismooth curve.
According to the ‘three-point’ criterion [Citation6, p.100], every piecewise -curve (without cusps) and quasismoth curve are quasiconformal.
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 quasidisks with some general functional conditions.
Definition 2.1
We say that , if
is a quasicircle and
for some
.
Additionally, we say that if
and L is rectifiable.
We note, that the class is sufficiently large. A detailed account of it and the related topics are contained in [?, Citation41,Citation45] (see also the references cited therein).
Definition 2.2
We say that , if
,
and
for some
and
.
We note, that the class and
(also
) is sufficiently large. This can be seen from the following:
If L is a piecewise Dini-smooth curve and the largest exterior (interior) angle on L has opening
(
),
, then
[[Citation41],[Citation43, p.52]]. If L is a smooth curve having continuous tangent line, then
for all
.
If G is ‘L-shaped’ region, then
for
and
.
If L is quasismooth, then
for
and
[Citation45,Citation46].
If L is ‘c-quasiconformal’, then
for
and
[Citation41].
If L is an asymptotically conformal curve, then
for all
[Citation41].
Clearly, that for any
and
. The class
is defined as a subclass of the class
with the additional functional condition. It turns out that more accurate estimates can be obtained for this class of curves.
For , let
; For
we set:
-infinite open cover of the curve
-finite open cover of the curve L;
;
,
.
Now, we start to formulate the new results. Firstly we give a recurrent estimate for ,
.
Theorem 2.1
Let p>1; for some
and
be defined by (Equation1
(1)
(1) ). Then, for any
, and every
, we have:
(7)
(7) where
is a constant independent of n and z;
and
(8)
(8)
Theorem 2.2
Let p>1; for some
and
be defined by (Equation1
(1)
(1) ). Then, for any
, and every
, we have:
(9)
(9) where
is a constant independent of n and z;
and
is defined as in (Equation8
(8)
(8) ).
The estimates (Equation7(7)
(7) ) and (Equation9
(9)
(9) ) allow us to obtain sequentially estimates for
,
. First, we will take m = 2 and j = 1, 2. Then, in these cases we need estimates for
and
. Estimates for
are carried out sequentially by applying the estimates (Equation7
(7)
(7) ) and (Equation9
(9)
(9) ).
2.1. Estimate for ![](//:0)
![](//:0)
Theorem 2.3
Let p>1; for some
and
be defined by (Equation1
(1)
(1) ). Then, for any
, and
, we have:
(10)
(10) where
is a constant independent of n and z;
Theorem 2.4
[Citation11, Th. 1.6 (Cor. 1.7)]
Let p>1; for some
and
be defined by (Equation1
(1)
(1) ). Then, for any
, and
, we have:
(11)
(11) where
is a constant independent of n and z;
2.2. Estimate for ![](//:0)
![](//:0)
According to Theorems 2.1, 2.2 and 2.3, we can give an estimate for at each point
.
Theorem 2.5
Let p>1; for some
and
be defined by (Equation1
(1)
(1) ). Then, for any
, we have:
(12)
(12) where
is a constant independent of n and z;
Theorem 2.6
Let p>1; for some
and
be defined by (Equation1
(1)
(1) ). Then, for any
, we have:
(13)
(13) where
is a constant independent of n and z;
2.3. Estimate for ![](//:0)
![](//:0)
Considering the estimates obtained for (Theorems 2.5, 2.6) and for
(Theorem 2.3) in Theorems 2.1, 2.2 respectively, we have the following two theorems:
Theorem 2.7
Let p>1; for some
and
be defined by (Equation1
(1)
(1) ). Then, for any
, we have:
(14)
(14) where
is a constant independent of n and z;
Theorem 2.8
Let p>1; for some
and
be defined by (Equation1
(1)
(1) ). Then, for any
, and
, we have:
(15)
(15) where
is a constant independent of n and z;
3. Estimate ![](//:0)
, for bounded regions
Now, we can state estimates for ,
, in the bounded regions of the classes
and
.
Theorem 3.1
Let p>1; for some
and
be defined by (Equation1
(1)
(1) ). Then, for any
, and every
, we have:
(16)
(16) where
is a constant independent of n and z, and
is defined as in (Equation8
(8)
(8) ).
Theorem 3.2
Let p>1; for some
and
be defined by (Equation1
(1)
(1) ). Then, for any
, and every
, we have:
(17)
(17) where
is a constant independent of n and z, and
is defined as in (Equation8
(8)
(8) ).
Remark 3.1
The inequalities (Equation16(16)
(16) ) and (Equation17
(17)
(17) ) is sharp.
4. Estimates for ![](//:0)
and ![](//:0)
in the whole complex plane
According to (Equation3(3)
(3) ) (applied to the polynomial
), the estimations (Equation16
(16)
(16) ) and (Equation17
(17)
(17) ) are true also for the points
,
, with a different constant. Therefore, combining these estimations with (Equation12
(12)
(12) ), (Equation13
(13)
(13) ), (Equation14
(14)
(14) ), (Equation15
(15)
(15) ), we will obtain an estimation on the growth for
and
, respectively, in the whole complex plane:
Theorem 4.1
Let p>1; for some
and
be defined by (Equation1
(1)
(1) ). Then, for any
, we have:
where
is a constant independent of n and z;
defined as in Theorem 2.5 for all
.
Theorem 4.2
Let p>1; for some
and
be defined by (Equation1
(1)
(1) ). Then, for any
, we have:
where
is a constant independent of n and z;
defined as in Theorem 2.6 for all
.
Theorem 4.3
Let p>1; for some
and
be defined by (Equation1
(1)
(1) ). Then, for any
, we have:
where
is a constant independent of n and z;
is defined as in Theorem 2.7 for all
.
Theorem 4.4
Let p>1; for some
and
be defined by (Equation1
(1)
(1) ). Then, for any
, we have:
where
is a constant independent of n and z;
defined as in Theorem 2.8 for all
.
Thus, using Theorems 2.1, 2.2 and estimating sequentially for each
, and combining the obtained estimates with Theorems 3.1 and 3.2, we can obtain estimates for the
, for each points
.
5. Some auxiliary results
Throughout this paper we denote ‘’ and ‘
’ are equivalent to
and
for some constants
, respectively.
Lemma 5.1
[Citation47]
Let G be a quasidisk, ,
;
, j = 1, 2, 3. Then
The statements
and
are equivalent. Therefore,
and
also are equivalent.
If
, then
where
a constant, depending on G.
Corollary 5.1
Under the conditions of Lemma 5.1, we have:
where
.
Lemma 5.2
Let for some
. Then, for all
, we have:
This fact follows from an appropriate result for the mapping [Citation4, p.287] and estimation for
[Citation48, Th.2.8]:
(18)
(18)
Lemma 5.3
[Citation28]
Let L = G be a rectifiable Jordan curve and ,
, be arbitrary polynomial and weight function
satisfy the condition (Equation1
(1)
(1) ). Then for any R>1, p>0 and
6. Proofs of theorems
Proofs
Proofs of Theorems 2.1 and 2.2
The proofs of Theorems 2.1 and 2.2 will be simultaneously.
Let ,
, and let
,
. For
, let us define
. Then, the mth derivative of
has a representation:
where
. Passing to the modules in both parts, we get:
(19)
(19) Therefore, to estimate
for
, it suffices to estimate the statements: (A)
,
; (B)
.
Now let us start with evaluations (A) and (B) for the cases and
, separately.
(A) Since the function is analytic in Ω, continuous on
and
, then Cauchy integral representation for the mth derivatives gives:
Then,
(20)
(20) Denote by
(21)
(21) and estimate this integral. For this, we give some notations.
Let . Without loss of generality, we will assume that
. For
, where
, let us set:
For simplicity of calculations, we can restrict ourselves by considering one point on the boundary. So, let the weight function
be defined as in (Equation1
(1)
(1) ) for l = 1 and we put:
. To estimate
, multiplying the numerator and denominator of the integrand by
and applying the Hölder inequality, we obtain:
Using Lemma 5.3 and after replacing the variable
, we get:
To estimate the last integral, we put:
(22)
(22) where
chosen so that
and
. Then, from (Equation22
(22)
(22) ) we have:
(23)
(23) For any
denote by
(24)
(24) and estimate these integrals separately.
1. Let for some
.
1.1. Let If
, applying Lemma 5.2 to (Equation24
(24)
(24) ), we get:
(25)
(25)
(26)
(26)
(27)
(27) For
, we have
,
. Then,
, by Lemma 5.1 and for
,
, by Lemma 5.2. Applying (Equation18
(18)
(18) ), for
, we get:
(28)
(28) Combining (Equation25
(25)
(25) –Equation28
(28)
(28) ), for
and
, we get:
(29)
(29) If
, then
(30)
(30) Combining (Equation23
(23)
(23) )–(Equation30
(30)
(30) ), we obtain:
(31)
(31)
(32)
(32) 1.2. If
, for
, such that
, according Lemma 5.1, we have:
(33)
(33)
(34)
(34) For
we have
and, so,
, from Lemma 5.1. Then, for
, such that
, applying Lemma 5.2, we get:
(35)
(35)
(36)
(36) For
and each
we have
. Therefore, from Lemma 5.1 and applying (Equation18
(18)
(18) ), we get:
(37)
(37) Further, combining (Equation33
(33)
(33) )–(Equation37
(37)
(37) ) in case of
, for
, we have:
(38)
(38) If
, then
and from Lemma 5.2 and (Equation18
(18)
(18) ), we get:
Combining the last three estimates, in case of
, for
, we have:
(39)
(39) Then, for
, from (Equation38
(38)
(38) –Equation39
(39)
(39) ), we obtain:
and, consequently, for
, from (Equation23
(23)
(23) ), we have:
(40)
(40) Therefore, combining (Equation29
(29)
(29) ) and (Equation40
(40)
(40) ), for any
, p>1, we obtain:
(41)
(41)
(42)
(42) (B) Now, we begin to estimate the
.
The Cauchy integral representation for the region gives:
Then, replacing the variable
and according to (Equation18
(18)
(18) ), we obtain:
(43)
(43) Combining estimates (Equation19
(19)
(19) )–(Equation23
(23)
(23) ), (Equation41
(41)
(41) ), (Equation42
(42)
(42) ) and (Equation43
(43)
(43) ), for any
, p>1,
, we get:
(44)
(44) where
Therefore, the proof of Theorem 2.1 is completed.
Now, we begin the proof of Theorem 2.2. Note that the proof of Theorem 2.2 is carried out similarly to the proof of Theorem 2.1, supplementing it with appropriate estimates for the given case.
2. Let for some
.
2.1. Let . Since
, for arbitrary
such that
and
, we have
. If
, applying Lemmas 5.1 and 5.2 to (Equation18
(18)
(18) ), we get:
or
(45)
(45) Further,
(46)
(46) and
(47)
(47) For
we have
and
. Therefore,
, by Lemma 5.1 and
, for
, by Lemmas 5.1 and 5.2. Then, for
, applying (Equation18
(18)
(18) ), we get:
(48)
(48) Combining (Equation45
(45)
(45) –Equation48
(48)
(48) ), for
and
, we get:
(49)
(49) If
, then
Therefore, for
, we get:
(50)
(50) and, for any
, from (Equation23
(23)
(23) ), (Equation49
(49)
(49) ) and (Equation50
(50)
(50) ), we obtain:
(51)
(51)
(52)
(52) 2.2. If
, for
, such that
, according to Lemma 5.1, we have:
For
holds and, therefore,
, by Lemma 5.1. Then, for
, such that
, applying Lemma 5.2, we get:
For
and each
,
holds. Therefore, from Lemma 5.1 and applying (Equation18
(18)
(18) ), we get:
Therefore, case of
for
, we have:
(53)
(53) If
, then
, from Lemma 5.2 and from (Equation18
(18)
(18) ), we get:
Combining the last tree estimates, in case of
for
, we have:
(54)
(54) Then, for
, from (Equation53
(53)
(53) ) and (Equation54
(54)
(54) ), we obtain:
and, consequently, for
from (Equation23
(23)
(23) ), we have:
(55)
(55) Therefore, combining (Equation52
(52)
(52) ) and (Equation55
(55)
(55) ), for any
, p>1, we get:
(56)
(56) where
.
In this case for the estimate , according to (Equation43
(43)
(43) ), we obtain:
(57)
(57) Then combining estimates (Equation19
(19)
(19) )–(Equation23
(23)
(23) ), (Equation41
(41)
(41) ), (Equation42
(42)
(42) ) and (Equation56
(56)
(56) ), we get:
(58)
(58) where for any
The proof of Theorem 2.2 is completed.
Proof
Proof of Theorem 2.3
Now let's start evaluations of . Proceeding in the same way as at the beginning of the proof of Theorems 2.1 and 2.2 for estimates (Equation20
(20)
(20) ) and (Equation21
(21)
(21) ), we have:
(59)
(59) where
So, for any k = 1, 2, 3 we will estimate the
for case
.
Let , for some
.
1. Let . Applying Lemma 5.2 and (Equation18
(18)
(18) ), we get:
(60)
(60)
(61)
(61) For
, we have
. Therefore,
, by Lemma 5.1 and applying (Equation18
(18)
(18) ), we get:
(62)
(62) Combining (Equation60
(60)
(60) –Equation62
(62)
(62) ), for
and
, we get:
(63)
(63) From (Equation59
(59)
(59) )–(Equation63
(63)
(63) ), we obtain:
(64)
(64) 2. If
, analogously we have:
Therefore, in case of
for
, we have:
(65)
(65) Combining estimates (Equation19
(19)
(19) )–(Equation59
(59)
(59) ), (Equation65
(65)
(65) ), we get:
and, we complete the proof of Theorem 2.3.
Proofs
Proofs of Theorems 2.5 and 2.6
From (Equation41(41)
(41) ), (Equation42
(42)
(42) ) and (Equation44
(44)
(44) ), we get:
(66)
(66) where for any
Taking into account estimates for
from (Equation10
(10)
(10) ), (Equation11
(11)
(11) ) respectively and combining with (Equation66
(66)
(66) ) prove the needed estimates.
Analogously for the case of
,
, from (Equation56
(56)
(56) ), (Equation57
(57)
(57) ) and (Equation66
(66)
(66) ), we get:
where for any
,
Therefore,
(67)
(67) for
and
(68)
(68) for
. Taking into account estimates for
from (Equation10
(10)
(10) ), (Equation11
(11)
(11) ), respectively, and combining with (Equation66
(66)
(66) ) gives the necessary estimates.
Proofs
Proofs of Theorems 2.7 and 2.8
According to the Theorems 2.1, 2.3, 2.5 and estimates (Equation20(20)
(20) ), (Equation21
(21)
(21) ), (Equation41
(41)
(41) ), (Equation42
(42)
(42) ), (Equation44
(44)
(44) ), for
,
, m = 2 and p>1 from (Equation19
(19)
(19) ), we have:
Substituting estimates for
, j = 1, 2,
and
from Theorems 2.1, 2.3 and 2.5 respectively, we get:
Now, according to Theorems 2.2, 2.4, 2.6 and estimates (Equation20
(20)
(20) ), (Equation21
(21)
(21) ), (Equation58
(58)
(58) ), (Equation57
(57)
(57) ), (Equation67
(67)
(67) ), (Equation68
(68)
(68) ), for the case
,
,
, m = 2 and p>1, we have:
if
and
if
. Therefore, the proof of Theorems 2.7 and 2.8 is completed.
In conclusion, note that in proofs everywhere there is a quantity . Let us show that
holds for all
. For the points
, we have:
. Now, let
. Denote by
the point such that
, and point
,such that
, and for
, we have:
. Then, according to Lemma 5.1, we obtain
.
Proofs
Proofs of Theorems 3.1 and 3.2
First of all, we give two theorems that we will use in this case, and after then we give an estimate for ,
, for each
.
Theorem A
[Citation15, Th. 2.4]
Assume that for some
and
defined as in (Equation1
(1)
(1) ). Then, for any p>1 there exists a constant
such that the following is fulfilled:
(69)
(69)
Theorem B
[Citation11, Th.1.5]
Assume that for some
and
and
defined as in (Equation1
(1)
(1) ). Then, for any p>1 there exists a constant
such that the following is fulfilled:
(70)
(70) where
is defined as in (Equation8
(8)
(8) ).
Now, we will give an estimate for ,
, for each
and for the regions
and
, respectively. Let
and
is an arbitrary fixed point. By the Cauchy integral formulas for mth derivatives, we have:
Then, applying (Equation3
(3)
(3) ), we get:
Now, applying Theorems thmA and thmBand using Corollary 5.1, we get:
Since
is arbitrary, we complete the proof of Theorems 3.1 and 3.2.
Proof
Proof of Remark
Sharpness of the inequalities (Equation16(16)
(16) ) and (Equation17
(17)
(17) ) can be argued as follows. These inequalities can be interpreted as a combination of the well-known sharp Markov inequalities
with inequalities (Equation69
(69)
(69) ) and (Equation70
(70)
(70) ), respectively. And the sharpness of the last inequalities can be verified to the following examples: For the polynomial
,
,
,
and any
there exist
,
such that
Really, if
, then,
. (a)
; (b)
.
Obviously,
So,
On the other hand, according to [Citation27, p. 236], we have:
and
Therefore,
Acknowledgements
The authors would like to thank the referees for their remarks and useful advice, which helped the authors to significantly improve the work.
Disclosure statement
No potential conflict of interest was reported by the author(s).
References
- Walsh JL. Interpolation and approximation by rational functions in the complex domain. Providence Rhode Island: AMS; 1960.
- Hille E, Szegö G, Tamarkin JD. On some generalization of a theorem of A.Markoff. Duke Math J. 1937;3:729–739.
- Abdullayev FG, Özkartepe P. On the growth of algebraic polynomials in the whole complex plane. J Korean Math Soc. 201552(4):699–725.
- Pommerenke Ch. Univalent functions. Göttingen: Vandenhoeck & Ruprecht; 1975.
- Belinskii PP. General properties of quasiconformal mappings. Novosibirsk: Nauka, Sib. otd.; 1974. Russian.
- Lehto O, Virtanen KI. Quasiconformal mapping in the plane. Berlin: Springer; 1973.
- Abdullayev FG. On the some properties on orthogonal polynomials over the regions of complex plane 1. Ukr Math J. 2000;52(12):1807–1817.
- Abdullayev FG, Özkartepe P. An analogue of the Bernstein–Walsh lemma in Jordan regions of the complex plane. J Inequal Appl. 20131(2013):1–7.
- Stylianopoulos N. Strong asymptotics for Bergman polynomials over domains with corners and applications. Constr Approx. 2012;38(1):59–100.
- Abdullayev FG, Özkartepe P. On the behavior of the algebraic polynomial in unbounded regions with piecewise dini-smooth boundary. Ukr Math J. 2014;66(5):645–665.
- Abdullayev FG, Özkartepe P, Gün CD. Uniform and pointwise polynomial inequalities in regions without cusps in the weighted Lebesgue space. Bull TICMI. 2014;18(1):146–167.
- Abdullayev FG, Gün CD, Ozkartepe NP. Inequalıtıes for algebraıc polynomıals ın regıons wıth exterıor cusps. J Nonlinear Funct Anal. 2015;3:1–32.
- Abdullayev FG, Özkartepe P. Uniform and pointwise polynomial inequalities in regions with cusps in the weighted Lebesgue space. Jaen J Approx. 2015;7(2):231–261.
- Abdullayev FG, Özkartepe NP. Polynomial inequalities in Lavrentiev regions with interior and exterior zero angles in the weighted Lebesgue space. Publ l'Inst Math (Beograd). 2016;100(114):209–227.
- Abdullayev FG, Özkartepe NP. Interference of the weight and boundary contour for algebraic polynomials in the weighted Lebesgue spaces. I. Ukr Math J. 201768(10):1574–1590. (Translated from Ukr Mat Zh. 2016;68(10):1365–1379.)
- Abdullayev FG. Polynomial inequalities in regions with corners in the weighted Lebesgue spaces. Filomat. 2017;31(18):5647–5670.
- Abdullayev FG, Imashkyzy M, Abdullayeva G. Bernstein–Walsh type inequalities in unbounded regions with piecewise asymptotically conformal curve in the weighted Lebesgue space. J Math Sci. 2018;234(1):35–48. DOI:10.1007/s10958-018-3979-6. (Translated from Ukr Mat Visnik. 2017;14(4):515–531.)
- Abdullayev FG, Tunc T, Abdullayev GA. Polynomial inequalities in quasidisks on weighted Bergman space. Ukr Math J. 2017;69(5):675–695.
- Abdullayev FG, Özkartepe NP. The uniform and pointwise estimates for polynomials on the weighted Lebesgue spaces in the general regions of complex plane. Hacet J Math Stat. 2019;48(1):87–101.
- Abdullayev FG, Özkartepe P, Tunç T. Uniform and pointwise estimates for algebraic polynomials in regions with interior and exterior zero angles. Filomat. 2019;33(2):403–413.
- Abdullayev FG, Gün CD. Bernstein–Walsh-type inequalities for derivatives of algebraic polynomials. Bull Korean Math Soc. 2022;59(1):45–72. DOI:10.4134/BKMS.b210023
- Abdullayev FG. Bernstein–Walsh-type inequalities for derivatives of algebraic polynomials in quasidiscs. OPEN Math. 2022;19(1):1847–1876.
- Dzjadyk VK. Introduction to the theory of uniform approximation of functiıon by polynomials. Moskow: Nauka; 1977.
- Mamedhanov DI. Inequalities of S.M.Nikol'skii type for polynomials in the complex variable on curves. Sov Math Dokl. 1974;15:34–37.
- Bernstein SN. Sur la limitation des derivees des polnomes. C R Acad Sci Paris. 1930;190:338–341.
- Bernstein SN. On the best approximation of continuos functions by polynomials of given degree. Izd. Akad. Nauk SSSR I. 1952; II; 1954 (O nailuchshem problizhenii nepreryvnykh funktsii posredstrvom mnogochlenov dannoi stepeni). Sobraniye Sochinenii. 1912;I(4):11–10.
- Szegö G, Zygmund A. On certain mean values of polynomials. J Anal Math. 1953;3(1):225–244.
- Abdullayev FG. On some properties of the orthogonal polynomials over the region of the complex plane (Part III). Ukr Math J. 2001;53(12):1934–1948.
- Abdullayev FG. The properties of the orthogonal polynomials with weight having singularity on the boundary contour. J Comput Anal Appl. 2004;6(1):43–59.
- Abdullayev FG. On the interference of the weight boundary contour for orthogonal polynomials over the region. J Comput Anal Appl. 2004;6(1):31–42.
- Abdullayev FG, Gün CD. Bernstein–Nikolskii-type inequalities for algebraic polynomials in Bergman space in regions of complex plane. Ukr Math J. 202173(4):513–531. (Translated from Ukr Mat Zh. 2021 Apr;73(4):439–454.)
- Andrievskii VV. Weighted polynomial inequalities in the complex plan. J Approx Theory. 2012;164(9):1165–1183.
- Balcı S, Imashkyzy M, Abdullayev FG. Polynomial inequalities in regions with interior zero angles in the Bergman space. Ukr Math J. 2018;70(3):362–384.
- Benko D, Dragnev P, Totik V. Convexity of harmonic densities. Rev Mat Iberoamericana. 2012;28(4):1–14.
- Ditzian Z, Tikhonov S. Ul'yanov and Nikol'skii-type inequalities. J Approx Theory. 2005;133(1):100–133. DOI:10.1016/j.jat.2004.12.008 .
- Ditzian Z, Prymak A. Nikol'skii inequalities for Lorentz spaces. Rocky Mountain J Math. 2010;40(1):209–223.
- Milovanovic GV, Mitrinovic DS, Rassias ThM. Topics in polynomials: extremal problems, inequalities, zeros. Singapore: World Scientific; 1994.
- Nevai P, Totik V. Sharp Nikolskii inequalities with exponential weights. Anal Math. 1987;13(4):261–267.
- Nikol'skii SM. Approximation of function of several variable and imbeding theorems. New York: Springer; 1975.
- Pritsker I. Comparing norms of polynomials in one and several variables. J Math Anal Appl. 1997;216:685–695. DOI:10.1006/jmaa.1997.5699
- Lesley FD. Hölder continuity of conformal mappings at the boundary via the strip method. Indiana Univ Math J. 1982;31:341–354.
- Becker J, Pommerenke C. Uber die quasikonforme Fortsetzung schlichten Funktionen. Math Z. 1978;161:69–80.
- Pommerenke Ch. Boundary behaviour of conformal maps. Berlin: Springer; 1992.
- Ahlfors L. Lectures on quasiconformal mappings. Princeton (NJ): Van Nostrand; 1966.
- Warschawski SE. On differentiability at the boundary in conformal mapping. Proc Am Math Soc. 1961;12:614–620.
- Warschawski SE. On Hölder continuity at the boundary in conformal maps. J Math Mech. 1968;18:423–427.
- Abdullayev FG, Andrievskii VV. On the orthogonal polynomials in the domains with K-quasiconformal boundary. Izv Akad Nauk Azerb SSR Ser FTM. 1983;1:3–7.
- Andrievskii VV, Belyi VI, Dzyadyk VK. Conformal invariants in constructive theory of functions of complex plane. Atlanta: World Federation; 1995.