Sharp unifying generalizations of Opial's inequality

Opial's inequality and its ramifications play an important role in the theory of differential and difference equations. A sharp unifying generalization of Opial's inequality is presented that contains both its continuous and discrete version. This generalization based on distribution funct...

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Bibliographic Details
Published in:arXiv.org 2023-12
Main Author: Klaassen, Chris A J
Format: Article
Language:English
Subjects:
Derivatives
Difference equations
Differential equations
Distribution functions
Weighting functions
Online Access:Get full text
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Summary:Opial's inequality and its ramifications play an important role in the theory of differential and difference equations. A sharp unifying generalization of Opial's inequality is presented that contains both its continuous and discrete version. This generalization based on distribution functions is extended to the case of derivatives of arbitrary order. This extension optimizes and improves the constant as given in the literature. The special case of derivatives of second order is studied in more detail. Two closely related Opial inequalities with a weight function are presented as well. The associated Wirtinger inequality is studied briefly.
ISSN:2331-8422
DOI:10.48550/arxiv.2312.05150