We study fixed points with N scalar fields in 4 − ε dimensions to leading order in ε using a bottom-up approach. We do so by analyzing O(N) invariants of the quartic coupling λ$_{ijkl}$ that describes such CFTs. In particular, we show that λ$_{iijj}$ 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 ε, there are no bosonic QED fixed points with N < 183 flavors.