A generalization of concavity for finite differences

The concept of concavity is generalized to discrete functions, u, satisfying the nth order difference inequality, (-1)(n-k)Delta(n)u(m) greater than or equal to 0, m = 0, 1,..., N and the homogeneous boundary conditions, u(0) = ... = u(k-1) = 0, u(N+k+1) = ... = u(N+n) = 0 for some k is an element of {1,..., n-1}. A piecewise polynomial is constructed which bounds u below. The piecewise polynomial is employed to obtain a positive lower bound on u(m) for m = k,..., N + k, where the lower bound is proportional to the supremum of u. An analogous bound is obtained for a related Green's function. (C) 1998 Elsevier Science Ltd. All rights reserved.