  We study fixed points with N scalar fields in $4 - \varepsilon$ dimensions to
leading order in $\varepsilon$ using a bottom-up approach. We do so by
analyzing O(N) invariants of the quartic coupling $\lambda_{ijkl}$ that
describes such CFTs. In particular, we show that $\lambda_{iijj}$ and
$\lambda_{ijkl}^2$ are restricted to a specific domain, refining a result by
Rychkov and Stergiou. We also study averages of one-loop anomalous dimensions
of composite operators without gradients. In many cases, we are able to show
that the O(N) fixed point maximizes such averages. In the final part of this
work, we generalize our results to theories with N complex scalars and to
bosonic QED. In particular we show that to leading order in $\varepsilon$,
there are no bosonic QED fixed points with N < 183 flavors.
