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Fixed points of scalar field theories with quartic interactions in$d=4-\varepsilon$ dimensions are considered in full generality. For suchtheories it is known that there exists a scalar function $A$ of the couplingsthrough 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 asimple expression linear in the dimension of the vector order parameter, $N$.Saturation of the bound requires a marginal deformation, and is shown to arisewhen fixed points with the same global symmetry coincide in coupling space.Several general results about scalar CFTs are discussed, and a review of knownfixed points is given.

General Properties of Multiscalar RG Flows in $d=4-\varepsilon$