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For all n large enough, we show uniqueness of a critical point in bestrational approximation of degree n, in the L^2-sense on the unit circle, tofunctions f, where f is a sum of a Cauchy transform of a complex measure \musupported on a real interval included in (-1,1), whose Radon-Nikodym derivativewith respect to the arcsine distribution on its support is Dini-continuous,non-vanishing and with and argument of bounded variation, and of a rationalfunction with no poles on the support of \mu.

Asymptotic Uniqueness of Best Rational Approximants to Complex Cauchy Transforms in ${L}^2$ of the Circle