  Black hole no-hair theorems are proven using inequalities that govern the
radial dependence of spherically symmetric configurations of matter fields. In
this paper, we analyze the analogous inequalities for geometries dual to
renormalization group flows via the AdS/CFT correspondence. These inequalities
give much useful information about the qualitative properties of such flows.
For Poincare invariant flows, we show that generic flows of relevant or
irrelevant operators lead to singular geometries. For the case of irrelevant
operators, this leads to an apparent conflict with Polchinski's decoupling
theorem, and we offer two possible resolutions to this problem.
