  We establish three families of Sobolev trace inequalities of orders two and
four in the unit ball under higher order moments constraint, and are able to
construct \emph{smooth} test functions to show all such inequalities are
\emph{almost optimal}. Some distinct feature in \emph{almost sharpness}
examples between the fourth order and second order Sobolev trace inequalities
is discovered. This has been neglected in higher order Sobolev inequality case
in \cite{Hang}. As a byproduct, the method of our construction can be used to
show the sharpness of the generalized Lebedev-Milin inequality under
constraints.
