# Discrepancy Theorems via One-Sided Bounds for Potentials

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

## Abstract

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
and
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.

## Keywords

Conformal Mapping Jordan Curve Discrepancy Theorem Analytic Jordan Curve Outer Bound
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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