Abstract
In this paper, we introduce a novel formulation of dynamic Hardy-type inequalities on a time scale, motivated by a recently established convexity approach in the Haar measure. The classical Hardy inequality is refined so that the classical Lebesgue-measure constant is replaced by the sharp constant 1. We obtain time-scale analogues on finite intervals with best constants, and, for nonincreasing and nondecreasing functions, reversed inequalities with explicit weights described by incomplete β-functions. To establish our results, we employ two distinct time scales and apply the chain rule, together with the substitution rule, the derivative of inverse functions, and Fubini's theorem for delta integration. Our approach generalizes classical integral inequalities in the continuous setting, while yielding fundamentally new inequalities in the discrete setting. Furthermore, we explore the application of our results in the quantum case, demonstrating their broader relevance.
Recommended Citation
M. Bohner et al., "A New Formulation of Hardy-type Dynamic Inequalities on Time Scales," Aims Mathematics, vol. 10, no. 12, pp. 29627 - 29649, AIMS Press, Jan 2025.
The definitive version is available at https://doi.org/10.3934/math.20251302
Department(s)
Mathematics and Statistics
Publication Status
Open Access
Keywords and Phrases
chain rule on time scales; Hardy-type inequalities; Jensen's inequality
International Standard Serial Number (ISSN)
2473-6988
Document Type
Article - Journal
Document Version
Final Version
File Type
text
Language(s)
English
Rights
© 2026 The Authors, All rights reserved.
Creative Commons Licensing
This work is licensed under a Creative Commons Attribution 4.0 License.
Publication Date
01 Jan 2025
