  To set the stage, I discuss the $\beta$-function of the massless O(N) model
in three dimensions, which can be calculated exactly in the large N limit.
Then, I consider SU(N) Yang-Mills theory in 2+1 space-time dimensions. Relating
the $\beta$-function to the expectation value of the action in lattice gauge
theory, and the latter to the trace of the energy-momentum tensor, I show that
$\frac{d \ln g^2/\mu}{d\ln \mu}=-1$ for all $g$ and all N in one particular
renormalization scheme. As a consequence, I find that the Yang-Mills
$\beta$-function in three dimensions must have the same sign for all finite and
positive bare coupling parameters in any renormalization scheme, and all
non-trivial infrared fixed points are unreachable in practice.
