Squashing, mass, and holography for 3d sphere free energy

We consider the sphere free energy F(b; m(I)) in N = 6 ABJ(M) theory deformed by both three real masses m(I) and the squashing parameter b, which has been computed in terms of an N dimensional matrix model integral using supersymmetric localization. We show that setting m3 = ib-b-/12 relates F(b; m(I)) to the round sphere free energy, which implies infinite relations between m(I) and b derivatives of F(b; m(I)) evaluated at m(I) = 0 and b = 1. For N = 8 ABJ(M) theory, these relations fix all fourth order and some fifth order derivatives in terms of derivatives of m(1), m(2), which were previously computed to all orders in 1/N using the Fermi gas method. This allows us to compute partial derivative b4Fb=1 and partial derivative b5Fb=1 to all orders in 1/N, which we precisely match to a recent prediction to sub-leading order in 1/N from the holographically dual AdS(4) bulk theory.