Polynomial Inequalities in Quasidisks on Weighted Bergman Spaces

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.…”