  Some known constraints on Renormalization Group flow take the form of
inequalities: in even dimensions they refer to the coefficient $a$ of the Weyl
anomaly, while in odd dimensions to the sphere free energy $F$. In recent work
arXiv:1409.1937 it was suggested that the $a$- and $F$-theorems may be viewed
as special cases of a Generalized $F$-Theorem valid in continuous dimension.
This conjecture states that, for any RG flow from one conformal fixed point to
another, $\tilde F_{\rm UV} > \tilde F_{\rm IR}$, where $\tilde F=\sin (\pi
d/2)\log Z_{S^d}$. Here we provide additional evidence in favor of the
Generalized $F$-Theorem. We show that it holds in conformal perturbation
theory, i.e. for RG flows produced by weakly relevant operators. We also study
a specific example of the Wilson-Fisher $O(N)$ model and define this CFT on the
sphere $S^{4-\epsilon}$, paying careful attention to the beta functions for the
coefficients of curvature terms. This allows us to develop the $\epsilon$
expansion of $\tilde F$ up to order $\epsilon^5$. Pade extrapolation of this
series to $d=3$ gives results that are around $2-3\%$ below the free field
values for small $N$. We also study RG flows which include an anisotropic
perturbation breaking the $O(N)$ symmetry; we again find that the results are
consistent with $\tilde F_{\rm UV} > \tilde F_{\rm IR}$.
