We discuss renormalisation-group improvement of the effective potentia
l both in general and in the context of O(N) scalar phi4 and the Stand
ard Model. In the latter case we find that absolute stability of the e
lectroweak vacuum implies that m(H) greater-than-or-equal-to 1.95m(t)
- 189 GeV, for alpha3(M(Z)) = 0.11. We point out that the lower bound
on m(H) decreases if alpha3(M(Z)) is increased.