  In this paper we prove a class of second order Caffarelli-Kohn-Nirenberg
inequalities which contains the sharp second order uncertainty principle
recently established by Cazacu, Flynn and Lam \cite{CFL2020} as a special case.
We also show the sharpness of our inequalities for several classes of
parameters. Finally, we prove two stability versions of the sharp second order
uncertainty principle of Cazacu, Flynn and Lam by showing that the difference
of both sides of the inequality controls the distance to the set of extremal
functions both in $L^2$ norm of function and in $L^2$ norm of gradient of
functions.
