Summary: The concept of concavity is generalized to functions \(y\) satisfying \(n\)th order differential inequalities, \((-1)^{n-k}y^{(n)}(t)\geq 0, 0\leq t\leq 1\), and homogeneous two-point boundary conditions, \(y(0)=\ldots =y^{(k-1)}(0)=0, y(1)=\ldots =y^{(n-k-1)}(1)=0\), for some \(k\in \{ 1,\ldots ,n-1\}\). A piecewise polynomial, which bounds the function \(y\) from below, is constructed, and then is employed to obtain that \(y(t)\geq ||y||/4^{m}, 1/4\leq t\leq 3/4\), where \(m=\max\{ k, n-k\}\) and \(||\cdot ||\) 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.