Beurling regular variation, Bloom dichotomy, and the Gołąb–Schinzel functional equation

The class of 'self-neglecting' functions at the heart of Beurling slow variation is expanded by permitting a positive asymptotic limit function λ(t), in place of the usual limit 1, necessarily satisfying the following 'self-neglect' condition:(Formula presented.)known as the Goła{ogonek}b-Schinzel functional equation, a relative of the Cauchy equation (which is itself also central to Karamata regular variation). This equation, due independently to Aczél and Goła{ogonek}b, occurring in the study of one-parameter subgroups, is here accessory to the λ -Uniform Convergence Theorem (λ-UCT) for the recent, flow-motivated, 'Beurling regular variation'. Positive solutions, when continuous, are known to be λ(t) = 1 + at (below a new, 'flow', proof is given); a = 0 recovers the usual limit 1 for self-neglecting functions. The λ-UCT allows the inclusion of Karamata multiplicative regular variation in the Beurling theory of regular variation, with λ (t) = 1 + t being the relevant case here, and generalizes Bloom's theorem concerning self-neglecting functions.