Uniqueness theorems for Korenblum type spaces

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For a scale of spaces X of functions analytic in the unit disc, including the Korenblum space, and for some natural families E of uniqueness subsets for X, we describe minorants for (X,E), that is non-decreasing functions M:(0,1)→(0,∞) such that f∈ X, E, and |f(z)|≤ -M(|z|) on E imply f=0. We give an application of this result to approximation by simple fractions with restrictions on the coefficients.