Inequalities based on a generalization of concavity.

*(English)*Zbl 0868.34008Summary: 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.

##### MSC:

34A40 | Differential inequalities involving functions of a single real variable |

34B27 | Green’s functions for ordinary differential equations |

##### Keywords:

differential inequalities; concavity; \(n\)th order differential inequalities; homogeneous two-point boundary inequalities; related Green’s function; cone theoretic fixed point theorems
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\textit{P. W. Eloe} and \textit{J. Henderson}, Proc. Am. Math. Soc. 125, No. 7, 2103--2107 (1997; Zbl 0868.34008)

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