Let ω(θ) be a positive weight function on the unit circle of the complex plane. For a sequence of points {αk}_(k=1)~∞ included in a compact subset of the unit disk, we consider the orthogonal rational functions φ_n that are obtained by orthogonalization of the sequence {1, z/π_1, z~2/π_2,...} where π_k(z) = Π_(j = 1)~k(1 - α-bar_jz), with respect to the inner product = 1/(2π) ∫ from x = -π to x = π f(e~(iθ)) (g(e~(iθ)))-barω(θ) dθ. In this paper we discuss the behaviour of φ_n(t) for |t| = 1 and n → ∞ under certain conditions. The main condition on the weight is that it satisfies a Lipschitz-Dini condition and that it is bounded away from zero. This generalizes a theorem given by Szego in the polynomial case, that is when all α_k = 0.
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