Inequalities based on a generalization of concavity

The concept of concavity is generalized to functions, y, satisfying nth order differential inequalities, (-1)(n-k)y((n))(t) greater than or equal to 0, 0 less than or equal to t less than or equal to 1, and homogeneous two-point boundary conditions, y(0) = ... = y((k-1))(0) = 0, y(1) = ... = y((n-k-1))(1) = 0, for some k is an element of {1,..., n - 1}. A piecewise polynomial, which bounds the function, y, below, is constructed, and then is employed to obtain that y(t) greater than or equal to \\y\\/4(m), 1/4 less than or equal to t less than or equal to 3/4, where m = max{k, n - k} and \\.\\ denotes the supremum norm. An analogous inequality for a related Green's function is also obtained. These inequalities are useful in applications of certain cone theoretic fixed point theorems.