A discrepancy theorem for dini-smooth arcs

Abstract

If L is a Dini-smooth arc in the complex plane we obtain estimates for the discrepancy between the equilibrium measure μ = μ_Lof L and an arbitrary positive Borel unit measure v supported on L based on one-sided bounds for the-difference of their logarithmic potentials along level lines of the conformal mapping of onto the exterior of unit disk with standard normalization at ∞.

Keywords:

• Discrepancy
• lograithmic potential
• equilibrium measure
• theorem of Erdos-turan

Classification Categories: AMS No:

• 30C10
• 30C15
• 41A10
• 41A50

^*Also at Institute for Applied Mathematics and Mechanics. R. Luxemburg Street 74,3401 14 Donetsk-114. Ukraine

^*Also at Institute for Applied Mathematics and Mechanics. R. Luxemburg Street 74,3401 14 Donetsk-114. Ukraine

Notes

^*Also at Institute for Applied Mathematics and Mechanics. R. Luxemburg Street 74,3401 14 Donetsk-114. Ukraine