Discrepancy Theorems via One-Sided Bounds for Potentials
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.
KeywordsConformal Mapping Jordan Curve Discrepancy Theorem Analytic Jordan Curve Outer Bound
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