  The stability conditions of a renormalization group improved effective
potential have been discussed in the case of scalar QED and QCD with a
colorless scalar. We calculate the same potential in these models assuming the
existence of non-perturbative fixed points associated to a conformal phase. In
the case of scalar QED the barrier of instability found previously is barely
displaced as we approach the fixed point, and in the case of QCD with a
colorless scalar not only the barrier is changed but the local minimum of the
potential is also changed.
