Bernstein–Nikol’skii-Type Inequalities for Algebraic Polynomials from the Bergman Space in Domains of the Complex Plane

2021 ◽  
Author(s):  
F. G. Abdullayev ◽  
C. D. Gün
Keyword(s):  
Bergman Space ◽  
Complex Plane ◽  
Algebraic Polynomials

Approximation in weighted generalized grand smirnov classes

2017 ◽  
Vol 54 (4) ◽  
pp. 471-488 ◽  
Author(s):  
Daniyal M. Israfilov ◽  
Ahmet Testici
Keyword(s):  
Complex Plane ◽  
Approximation Theory ◽  
Connected Domain ◽  
Modulus Of Smoothness ◽  
Algebraic Polynomials ◽  
Lipschitz Classes ◽  
Carleson Curve ◽  
Smirnov Classes ◽  
Direct And Inverse Theorems ◽  
Inverse Theorems

Let G be a finite simple connected domain in the complex plane C, bounded by a Carleson curve Γ := ∂G. In this work the direct and inverse theorems of approximation theory by the algebraic polynomials in the weighted generalized grand Smirnov classes εp),θ(G,ω) and , 1 < p < ∞, in the term of the rth, r = 1, 2,..., mean modulus of smoothness are proved. As a corollary the constructive characterizations of the weighted generalized grand Lipschitz classes are obtained.

scholarly journals On inequalities of Kolmogorov type for the functions being analytic in the disk

2012 ◽  
Vol 20 ◽  
pp. 82
Author(s):  
S.B. Vakarchuk ◽  
M.B. Vakarchuk
Keyword(s):  
Unit Disk ◽  
Bergman Space ◽  
Complex Plane ◽  
Sharp Inequality ◽  
Kolmogorov Type

Sharp inequality of Kolmogorov type is obtained in the Bergman space $B_2$ for functions being analytic in the unit disk. The application of this inequality to problems of the theory of approximation in the complex plane is presented too.

scholarly journals Generalizations of some Zygmund-type integral inequalities for polar derivatives of a complex polynomial

2021 ◽  
Vol 110 (124) ◽  
pp. 57-69
Author(s):  
Abdullah Mir
Keyword(s):  
Unit Disk ◽  
Complex Plane ◽  
Integral Inequalities ◽  
Complex Polynomial ◽  
Algebraic Polynomials ◽  
Polar Derivative ◽  
Type Integral ◽  
Polar Derivatives ◽  
Derivatives Of

We prove some results for algebraic polynomials in the complex plane that relate the L-norm of the polar derivative of a complex polynomial and the polynomial under some conditions. The obtained results include several interesting generalizations of some Zygmund-type integral inequalities for polynomials and derive polar derivative analogues of some classical Bernsteintype inequalities for the sup-norms on the unit disk as well.

Evaluation operators on the Bergman space

1995 ◽  
Vol 117 (3) ◽  
pp. 513-523 ◽  
Author(s):  
Kehe Zhu
Keyword(s):  
Bergman Space ◽  
Complex Plane ◽  
Analytic Functions ◽  
Unit Disc ◽  
Open Unit ◽  
Open Unit Disc ◽  
Space Of Analytic Functions ◽  
Area Measure ◽  
Image Position

Let D be the open unit disc in the complex plane C and let dA be the normalized area measure on D. The Bergman space is the space of analytic functions f in D such that

scholarly journals Inequalities concerning the rate of growth of polynomials involving the polar derivative

2020 ◽  
Vol 74 (1) ◽  
pp. 67
Author(s):  
Abdullah Mir ◽  
Adil Hussain Malik
Keyword(s):  
Complex Plane ◽  
Algebraic Polynomials ◽  
Polar Derivative ◽  
Rate Of Growth

This paper contains some results for algebraic polynomials in the complex plane involving the polar derivative that are inspired by some classical results of Bernstein. Obtained results yield the polar derivative analogues of some inequalities giving estimates for the growth of derivative of lacunary polynomials.

A system of functions biorthogonal with the derivatives of Chebyshev second-kind polynomials of a complex variable

2020 ◽  
Vol 17 (2) ◽  
pp. 256-277
Author(s):  
Ol'ga Veselovska ◽  
Veronika Dostoina
Keyword(s):  
Complex Plane ◽  
Complex Variable ◽  
Analytic Functions ◽  
Combinatorial Identities ◽  
Independent Interest ◽  
Closed Curves ◽  
In Series ◽  
Derivatives Of

For the derivatives of Chebyshev second-kind polynomials of a complex vafiable, a system of functions biorthogonal with them on closed curves of the complex plane is constructed. Properties of these functions and the conditions of expansion of analytic functions in series in polynomials under consideration are established. The examples of such expansions are given. In addition, we obtain some combinatorial identities of independent interest.

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