  We study $F$-functions in the context of field theories on $S^3$ using
gauge-gravity duality, with the radius of $S^3$ playing the role of RG scale.
We show that the on-shell action, evaluated over a set of holographic RG flow
solutions, can be used to define good $F$-functions, which decrease
monotonically along the RG flow from the UV to the IR for a wide range of
examples. If the operator perturbing the UV CFT has dimension $\Delta > 3/2$
these $F$-functions correspond to an appropriately renormalized free energy. If
instead the perturbing operator has dimension $\Delta < 3/2$ it is the quantum
effective potential, i.e. the Legendre transform of the free energy, which
gives rise to good $F$-functions. We check that these observations hold beyond
holography for the case of a free fermion on $S^3$ ($\Delta=2$) and the free
boson on $S^3$ ($\Delta=1$), resolving a long-standing problem regarding the
non-monotonicity of the free energy for the free massive scalar. We also show
that for a particular choice of entangling surface, we can define good
$F$-functions from an entanglement entropy, which coincide with certain
$F$-functions obtained from the on-shell action.
