Inequalities for derivatives of algebraic polynomials in the complex plane

1967 ◽  
Vol 8 (4) ◽  
pp. 723-726 ◽  
Author(s):  
M. Ya. Zinger
Keyword(s):  
Complex Plane ◽  
Algebraic Polynomials ◽  
Derivatives Of

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.

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.

scholarly journals On the derivatives at the origin of entire harmonic functions

1979 ◽  
Vol 20 (2) ◽  
pp. 147-154 ◽  
Author(s):  
D. H. Armitage
Keyword(s):  
Entire Function ◽  
Euclidean Space ◽  
Complex Plane ◽  
Harmonic Functions ◽  
Image Position ◽  
Derivatives Of

If f is an entire function in the complex plane such thatwhere 0 ≤ α < 1, and all the derivatives of f at 0 are integers, then it is easy to show that f is a polynomial (see e.g. Straus [10]). The best possible result of this type was proved by Pólya [9]. The main aim of this paper is to prove two analogous results for harmonic functions defined in the whole of the Euclidean space Rn, where n ≥ 2 (i.e. entire harmonic functions).

Some Inequalities for Derivatives of Trigonometric and Algebraic Polynomials

1996 ◽  
Vol 30 (1-2) ◽  
pp. 79-92
Author(s):  
Theodore Kilgore ◽  
Michael Felten
Keyword(s):  
Algebraic Polynomials ◽  
Derivatives Of

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 Inequalities for the First and Second Derivatives of Algebraic Polynomials on an Ellipse

2021 ◽  
Author(s):  
Tatiana M. Nikiforova
Keyword(s):  
Special Form ◽  
Algebraic Polynomials ◽  
Second Derivatives ◽  
Derivatives Of

We prove a theorem on the preservation of inequalities between functions of a special form after differentiation on an ellipse. In particular, we obtain generalizations of the Duffin–Schaeffer inequality and the Vidensky inequality for the first and second derivatives of algebraic polynomials to an ellipse.

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