S. N. Bernstein Type Estimations in the Mean on the Curves in a Complex Plane

Abstract

The present paper discusses in the metricLpS. N. Bernstein type inequalities of the most general kind on very general accessible classes of curves in a complex plane. The obtained estimations, generally speaking, are not improvable.

Weighted polynomial inequalities in the complex plane

“…We [2,3] extended some of these results to the case of complex polynomials either on a quasismooth arc or on a domain bounded by the quasismooth curve. Our approach was influenced by the results on Markov-Bernstein-type inequalities for polynomials over a Jordan curve in the complex plane C that were proved first for some piecewise smooth curves by Szegő and Zygmund [30,Section 6] and, in more general form as well as for a larger class of curves, by Andrashko [1] and Mamedkhanov [20] (see also [22]). …”

On approximation by trigonometric polynomials in Orlicz spaces

“…Note that similar inequalities on different curves in the space L p (p 1) were investigated in [20][21][22][23] …”

Weighted L p Bernstein-type inequalities on a quasismooth curve in the complex plane

“…The starting point of our analysis are the results of Mastroianni and Totik [14] as well as Mamedkhanov and Dadashova [13] that extend a classical L p Bernstein inequality to the case of weighted inequalities for trigonometric polynomials and complex algebraic polynomials over a Jordan curve in the complex plane C. We describe these results below.…”

“…A complete survey of the results on the extension of inequality (1.2) to the case of more general L can be found in [16], [17], [2], [11][12][13].…”