Discrepancy Theorems via One-Sided Bounds for Potentials

  • Vladimir V. Andrievskii
  • Hans-Peter Blatt
Part of the Springer Monographs in Mathematics book series (SMM)


In Chapter 2 we obtained discrepancy estimates for the zero distribution of a polynomial p in connection with the equilibrium measure µ L of a Jordan curve or arc L. The basic quantities involved have been the two terms
where z 0 ∈ int L is fixed if L is a curve. In Section 2.3 we have outlined that it is possible to restrict the essential quantities to the outer bounds
in the case of a Jordan arc. If L is a curve, we replace δ p,L by the smaller inner bound
where L r is the level line of the conformal mapping φ(z) of int L onto D normalized by φ(z 0) = 0, φ'(z 0) > 0, as in (1.4.5). Then the discrepancy estimates can be formulated in terms of ε p,L (r) + δ p,L (r). In this chapter we shall discuss this approach carefully for general signed measures.


Conformal Mapping Jordan Curve Discrepancy Theorem Analytic Jordan Curve Outer Bound 
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Copyright information

© Springer Science+Business Media New York 2002

Authors and Affiliations

  • Vladimir V. Andrievskii
    • 1
  • Hans-Peter Blatt
    • 2
  1. 1.Department of Mathematics and Computer ScienceKent State UniversityKentUSA
  2. 2.Mathematisch-Geographische FakultätKatholische Universität EichstättEichstättGermany

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