Asymptotic Uniqueness of Best Rational Approximants to Complex Cauchy Transforms in L2 of the Circle

For all n large enough, we show uniqueness of a critical point in best rational approximation of degree n, in the L2-sense on the unit circle, to functions of the form f(z) = ∫ dμ(t ) z − t + r (z), dμ= μ̇dω[a,b], with r a rational function and μ̇ a complex-valued Dini-continuous function on a real segment [a, b] ⊂ (−1,1) which does not vanish, and whose argument is of bounded variation. Here ω[a,b] stands the normalized arcsine distribution on [a, b].