Erdos-Turan type theorems on quasiconformal curves and arcs

The theorems of Erdos and Turin mentioned in the title are concerned with t he distribution of zeros of a monic polynomial with known uniform norm alon g the unit interval or the unit disk. Recently, Blatt and Grothmann (Const, Approx. 7 (1991), 19-47). Grothmann ("Interpolation Points and Zeros of Po lynomials in Approximation Theory." Habilitationsschrift. katholische Unive rsitat Eichstatt, 1992), and Andrievskii and Blatt (J. Approx, Theory 88 (1 977), 109-134) established corresponding results for polynomials. considere d on a system of sufficiently smooth Jordan curves and arcs or piecewise sm ooth curves and arcs. We extend some of these results to polynomials with k nown uniform norm along an arbitrary quasiconformal curve: or are. As appli cations. estimates for the distribution of the zeros of best uniform approx imants. values of orthogonal polynomials, and zeros of Bieberbach polynomia ls and their derivatives are obtained. We also give a negative answer to on e conjecture of Eiermann :and Stahl ("Zeros of orthogonal polynomials on re gular N-gons." in Lecture Notes in Math. 1574 (1994). 187-189). (C) 1999 Ac ademic Press.