In this poster, I will briefly review renormalization group monotones (F-functions) of three dimensional quantum field theories and present our work on this topic. In our work (arXiv:2112.08715), we consider holographic F-functions in a top-down AdS/CFT setup involving flavored ABJM theory on a Euclidean 3-sphere. For quenched flavor, the holographic dual is type IIA supergravity with probe D6-branes. The flavor degrees of freedom are given a mass that drives an RG flow whose IR endpoint is pure ABJM theory. At non-zero mass, we find that the theory on the 3-sphere exhibits a quantum phase transition at a critical value of the sphere radius. The transition corresponds to a topology change in the D6-brane embeddings whose dual interpretation is the meson-melting transition. We perform the holographic computation of the free energy on the 3-sphere and we use it to construct various candidate F-functions. We find that while the F-functions of the flavored ABJM theory have the correct UV and IR limits, they are not monotonic. We surmise that the non-monotonicity is related to the presence of the phase transition.