On the behavior of algebraic polynomials in regions with piecewise smooth boundary without cusps

Abstract

We continue studying the estimation of Bernstein-Walsh type for algebraic polynomials in regions with piecewise smooth boundary.

On some inequalities for derivatives of algebraic polynomials in unbounded regions with angles

“…Analogous results of (7)-type for |𝑃 𝑛 (𝑧)|, different weight function ℎ, unbounded region Ω were obtained in [17, p.418-428], [5], [6], [7], [8], [9], [10], [11], [15], [22] and others.…”

“…To give a similar estimation to (5) for the 𝐴 𝑝 (ℎ, 𝐺) −norm, first of all we will give the following definition.…”

Polynomial inequalities in Lavrentiev regions with interior and exterior zero angles in the weighted Lebesgue space

“…Results analogous to (1.2) for some different norms and unbounded regions were obtained in [22], [2][3][4][5][6][7] and others.…”

Polynomial Inequalities in Regions Bounded by Piecewise Asymptotically Conformal Curve with Nonzero Angles in the Bergman Space

“…P n ∞ ≤ c 1 λ n (G, h, p) P n p , (1.3) where c 1 = c 1 (G, h, p) > 0 is a constant independent of n and P n , and λ n (G, h, p) → ∞, n → ∞, depending on the geometrical properties of region G, weight function h and of p. The estimate of (1.3)type for some (G, p, h) was investigated in [27, pp.122-133], [17], [26,Sect.5.3], [32], [15], [2]- [8] (see, also, references therein) and others. Further, analogous of (1.3) for some regions and the weight function h(z) were obtained: in [8] for p > 1 and for regions bounded by piecewise Dini-smooth boundary without cusps; in [11] for p > 0 and for regions bounded by quasiconformal curve; in [7] for p > 1 and for regions bounded by piecewise smooth curve without cusps; in [10] for p > 0 and for regions bounded by asymptotically conformal curve; in [16] for p > 0 and for regions bounded by piesewise smooth curves with interior (zero or nonzero) angles, in [12] for p > 0 and for regions bounded by piecewise asymptotically conformal curve having cusps and others.…”