We study the properties of a logconcavity operator on a symmetric, unimodal subset of finite sequences. In doing so we are able to prove that there is a large unbounded region in this subset that is ∞-logconcave. This problem was motivated by the conjecture of Boros and Moll in  that the binomial coefficients are ∞-logconcave.
David Uminsky and Karen Yeats. Unbounded regions of infinitely logconcave sequences. The Electronic Journal of Combinatorics 14 (2007), #R72.