  Let Z be a strictly a-stable real Levy process (a>1) and X be a fluctuating
b-homogeneous additive functional of Z. We investigate the asymptotics of the
first passage-time of X above 1, and give a general upper bound. When Z has no
negative jumps, we prove that this bound is optimal and does not depend on the
homogeneity parameter b. This extends a result of Y. Isozaki and solves
partially a conjecture of Z. Shi.
