In this paper, we mainly focus on the fractional GJMS operators P2γ which are defined on the conformal infinity of a Poincaré-Einstein manifold. We derive two comparison inequalities of the fractional Yamabe constants associated to the fractional GJMS operators. One is between P1 and P2γ
Nearby in the stack
for
γ∈(1/2,1)
, and the other is between
P2
and
P2γ
for
γ∈(1,2)
. They both imply the rigidity theorems by characterizing the equalities. Together with the result in WZ1, we partially provide some evidence for the monotonicity of the fractional Yamabe constants.