  Fixed points of scalar field theories with quartic interactions in
$d=4-\varepsilon$ dimensions are considered in full generality. For such
theories it is known that there exists a scalar function $A$ of the couplings
through which the leading-order beta-function can be expressed as a gradient.
It is here proved that the fixed-point value of $A$ is bounded from below by a
simple expression linear in the dimension of the vector order parameter, $N$.
Saturation of the bound requires a marginal deformation, and is shown to arise
when fixed points with the same global symmetry coincide in coupling space.
Several general results about scalar CFTs are discussed, and a review of known
fixed points is given.
