  We have considered the corrections to the finite-size-scaling functions for a
general class of $O(N)$ $\sigma$-models with two-spin interactions in two
dimensions for $N=\infty$. We have computed the leading corrections finding
that they generically behave as $(f(z) \log L + g(z))/L^2$ where $z = m(L) L$
and $m(L)$ is a mass scale; $f(z)$ vanishes for Symanzik improved actions for
which the inverse propagator behaves as $q^2 + O(q^6)$ for small $q$, but not
for on-shell improved ones. We also discuss a model with four-spin interactions
which shows a much more complicated behaviour.
