On the Growth of Algebraic Polynomials in the Whole Complex Plane

Abstract

Abstract. In this paper, we study the estimation for algebraic polynomials in the bounded and unbounded regions bounded by piecewise Dini smooth curve having interior and exterior zero angles.

Uniform and Pointwise Polynomial Estimates in Regions with Interior and Exterior Cusps

“…In this work, we investigate similar problem [3] for 0 < ≤ 1 in bounded G and unbounded region  with piecewise Dini-smooth curve having interior and exterior angles (also cusps) for weight function h defined in (1.1) . Finally, combining obtained estimates for () n Pz on G and ,  we get the evaluation for () n Pz in whole complex plane, depending on the geometrical properties of the region G , weight function () hz and p.…”

Polynomial inequalities in regions with corners in theweighted Lebesgue spaces

“…Therefore, according to 2.3, we can calculate α in the right parts of estimations (6) and (7) for each case, respectively. Now, let's introduce "special" singular points on the curve L. It is well known that each quasicircle satisfies the condition b).…”

Polynomial Inequalities in Regions with Piecewise Asymptotically Conformal Curve in the Weighted Lebesgue Space

“…Further, analogous estimates of (2) for some regions and the weight function h(z) were obtained: in [2] (p > 1) and in [25] (p > 0, h ≡ h 0 ) for regions bounded by rectifiable quasiconformal curve having some general properties; in [4] (p > 1) for piecewise Dini-smooth curve with interior and exterior cusps; in [3] ( p > 1) for regions bounded by piecewise smooth curve with exterior cusps but without interior cusps; in [5] (p > 0) for regions bounded by piecewise rectifiable quasiconformal curve with cusps; in [6] (p > 0) for regions bounded by piecewise quasismooth (by Lavrentiev) curve with cusps. Now, let's give some definitions and notations.…”