Inequalities based on a generalization of concavity

The concept of concavity is generalized to functions, $y$, satisfying $nth$ order differential inequalities, $(-1)^{n-k}y^{(n)}(t)\ge 0, 0\le t\le 1$, 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, $y$, below, is constructed, and then is employed to obtain that $y(t)\ge ||y||/4^{m}, 1/4\le t\le 3/4$, 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.