  The F theorem states that, for a unitary three dimensional quantum field
theory, the F quantity defined in terms of the partition function on a three
sphere is positive, stationary at fixed point and decreases monotonically along
a renormalization group flow. We construct holographic renormalization group
flows corresponding to relevant deformations of three-dimensional conformal
field theories on spheres, working to quadratic order in the source. For these
renormalization group flows, the F quantity at the IR fixed point is always
less than F at the UV fixed point, but F increases along the RG flow for
deformations by operators of dimension $3/2 < \Delta < 5/2$. Therefore the
strongest version of the F theorem is in general violated.
