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Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

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

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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Inequalities based on a generalization of concavity
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by Paul W. Eloe and Johnny Henderson PDF
Proc. Amer. Math. Soc. 125 (1997), 2103-2107 Request permission

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
  • MR Author ID: 63110
  • ORCID: 0000-0002-6590-9931
  • Email: eloe@saber.udayton.edu
  • Johnny Henderson
  • Affiliation: Department of Mathematics, 218 Parker Hall, Auburn University, Alabama 36849-5310
  • MR Author ID: 84195
  • ORCID: 0000-0001-7288-5168
  • Email: hendej2@mail.auburn.edu
  • Received by editor(s): July 12, 1995
  • Received by editor(s) in revised form: February 6, 1996
  • Communicated by: Hal L. Smith
  • © Copyright 1997 American Mathematical Society
  • 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