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