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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 

 

Inequalities based on a generalization
of concavity


Authors: Paul W. Eloe and Johnny Henderson
Journal: Proc. Amer. Math. Soc. 125 (1997), 2103-2107
MSC (1991): Primary 34A40; Secondary 34B27
DOI: https://doi.org/10.1090/S0002-9939-97-03800-8
MathSciNet review: 1376760
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Abstract: 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.


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Additional Information

Paul W. Eloe
Affiliation: Department of Mathematics, University of Dayton, Dayton, Ohio 45469-2316
Email: eloe@saber.udayton.edu

Johnny Henderson
Affiliation: Department of Mathematics, 218 Parker Hall, Auburn University, Alabama 36849-5310
Email: hendej2@mail.auburn.edu

DOI: https://doi.org/10.1090/S0002-9939-97-03800-8
Keywords: Differential inequalities
Received by editor(s): July 12, 1995
Received by editor(s) in revised form: February 6, 1996
Communicated by: Hal L. Smith
Article copyright: © Copyright 1997 American Mathematical Society