Erdős–Turán Type Theorems on Quasiconformal Curves and Arcs

Abstract

The theorems of Erdős and Turán mentioned in the title are concerned with the distribution of zeros of a monic polynomial with known uniform norm along the unit interval or the unit disk. Recently, Blatt and Grothmann ( Const. Approx. 7(1991), 19–47), Grothmann (“Interpolation Points and Zeros of Polynomials in Approximation Theory,” Habilitationsschrift, Katholische Universität Eichstätt, 1992), and Andrievskii and Blatt ( J. Approx. Theory 88(1977), 109–134) established corresponding results for polynomials, considered on a system of sufficiently smooth Jordan curves and arcs or piecewise smooth curves and arcs. We extend some of these results to polynomials with known uniform norm along an arbitrary quasiconformal curve or arc. As applications, estimates for the distribution of the zeros of best uniform approximants, values of orthogonal polynomials, and zeros of Bieberbach polynomials and their derivatives are obtained. We also give a negative answer to one conjecture of Eiermann and Stahl (“Zeros of orthogonal polynomials on regular N-gons,” in Lecture Notes in Math. 1574(1994), 187–189).