Abstract
In this paper, we extend the Minkowski integral inequality on time scales by \( \varDelta \varDelta \)-integral and the inverse Minkowski inequality over the set of constant sign functions. We get similar results by replacing \( \varDelta \varDelta \)-integral with \( \varDelta \nabla \), \( \nabla \varDelta \) and \( \nabla \nabla \)-integrals. For application, we give another proof of the Hardy inequality on time scales according to the Minkowski inequality.
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References
Hilger, S.: Ein Maettenkalkul mit Anwendung auf Zentrmsmannigfaltingkeiten. Ph.D. Thesis, Univarsi. Wurzburg (1988)
Baric, J., et al.: Jensen Inequalities on Time Scales, Volume 9 of Monographs in Inequalities. ELEMENT, Zagreb (2015). Theory and Applications
Rehak, P.: Hardy inequality on time scales and its application to half-linear dynamic equations. J. Inequal. Appl. 5, 495–507 (2005). https://doi.org/10.1155/JIA.2005.495
Bohner, M., Peterson, A.: Dynamic Equations on Time Scales. An Introduction with Applications. Birkhäuser, Boston (2001)
Georgiev, S.G.: Fractional Dynamic Calculus and Fractional Dynamic Equations on Time Scales. Springer International Publishing Switzerland (2018). https://doi.org/10.1007/978-3-319-73954-0
Agarwal, R.P., O’Regan, D., Saker, S.H.: Hardy type inequalities on time scales. J. Springer International Publishing Switzerland (2016). https://doi.org/10.1007/978-3-319-44299-0
Bohner, M., Guseinov, G.S.H.: Multiple integration on time scales. Dyn. Syst. Appl. 14(3–4), 579–606 (2005)
Bohner, M., Guseinov, G.S.H.: Double integral calculus of variations on time scales. Comput. Math. Appl. 54, 45–57 (2007)
Liu, W., Chen, Q.N.W.: Ostrowski type inequalities on time scales for double integrals. Acta Appl. Math. 110, 477–497 (2010). https://doi.org/10.1007/s10440-009-9456-y
Benaissa, B.: On the reverse Minkowski’s integral inequality. Kragujevac. J. Math. 46(3), 407–416 (2022)
Acknowledgements
We thank the referees for their remarks and comments which allow us to correct and improve this document. The author wishes to thank the DGRSDT of Algeria for the support of this research.
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Benaissa, B. More on Minkowski and Hardy integral inequality on time scales. Ricerche mat 72, 853–866 (2023). https://doi.org/10.1007/s11587-021-00595-z
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DOI: https://doi.org/10.1007/s11587-021-00595-z