Permanental inequalities for totally positive matrices

We characterize ratios of permanents of (generalized) submatrices which are bounded on the set of all totally positive matrices. This provides a permanental analog of results of Fallat, Gekhtman, and Johnson [{ Adv.\ Appl.\ Math.} { 30} no.\ 3, (2003) pp.\ 442--470] concerning ratios of matrix minors. We also extend work of Drake, Gerrish, and the first author [{ Electron.\ J.\ Combin.,} { 11} no.\ 1, (2004) Note 6] by characterizing the differences of monomials in Z[x1,1,x1,2,...,xn,n] which evaluate positively on the set of all totally positive n × n matrices.