Abstract
In this paper, we prove some new characterizations of weighted functions for dynamic inequalities of Hardy’s type involving monotonic functions on a time scale \(\mathbb {T}\) in different spaces \(L^{p}(\mathbb {T})\) and \(L^{q}( \mathbb {T})\) when \(0<p<q<\infty \) and \(p\le 1\). The main results will be proved by employing the reverse Hölder inequality, integration by parts, and the Fubini theorem on time scales. The main contribution in this paper is the new proof in the case when \(p<1\), which has not been considered before on time scales. Moreover, the results unify and extend continuous and discrete systems under one theory.
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Hardy, G.H.: Note on a theorem of Hilbert. Math. Z. 6, 314–317 (1920)
Hardy, G.H.: Notes on some points in the integral calculus (LX). An inequality between integrals. Messenger Math. 54, 150–156 (1925)
Copson, E.T.: Note on series of positive terms. J. Lond. Math. Soc. 1, 49–51 (1928)
Hardy, G.H.: Notes on some points in the integral calculus (LXIV). Messenger Math. 57, 12–16 (1928)
Heinig, H.P.: Weighted norm inequalities for certain integral operators II. Proc. Am. Math. Soc. 95(3), 387–395 (1985)
Heinig, H.P., Maligranda, L.: Weighted inequalities for monotone and concave functions. Stud. Math. 116, 133–165 (1995)
Leindler, L.: Generalization of inequalities of Hardy and Littlewood. Acta Sci. Math. (Szeged) 31, 285–297 (1970)
Sinnamon, G.J.: Weighted Hardy and Opial-type inequalities. J. Math. Anal. Appl. 160, 434–445 (1991)
Stepanov, V.D.: Boundedness of linear integral operators on a class of monotone functions. Siber. Math. J. 32, 540–542 (1991)
Kufner, A., Maligranda, L., Persson, L.-E.: The Hardy Inequality. About its History and Some Related Results. Vydavatelský Servis, Plzeň (2007)
Kufner, A., Persson, L.-E.: Weighted Inequalities of Hardy Type. World Scientific, Singapore (2003)
Opic, B., Kufner, A.: Hardy-Type Inequalities. Pitman Research Notes in Mathematics Series. Longman Scientific and Technical, Harlow (1990)
Bennett, G., Grosse-Erdmann, K.-G.: Weighted Hardy inequalities for decreasing sequences and functions. Math. Ann. 334(3), 489–531 (2006)
Agarwal, R.P., O’Regan, D., Saker, S.H.: Hardy Type Inequalities on Time Scales. Springer, New York (2016)
Saker, S.H., Mahmoud, R.R., Peterson, A.: Weighted Hardy-type inequalities on time scales with applications. Mediterr. J. Math. 13(2), 585–606 (2016)
Bohner, M., Saker, S.H.: Gehring inequalities on time scales. J. Comput. Anal. Appl. 28(1), 11–23 (2020)
Bohner, M., Peterson, A.: Dynamic Equations on Time Scales: An Introduction with Applications. Birkhäuser, Boston (2001)
Bohner, M., Peterson, A. (eds.): Advances in Dynamic Equations on Time Scales. Birkhäuser, Boston (2003)
Bibi, R., Bohner, M., Pečarić, J., Varošanec, S.: Minkowski and Beckenbach–Dresher inequalities and functionals on time scales. J. Math. Inequ. 7, 299–312 (2013)
Barić, J., Bibi, R., Bohner, M., Nosheen, A., Pečarić, J.: Jensen Inequalities on Time Scales. Theory and Applications. Element, Zagreb (2015)
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Saker, S.H., Saied, A.I. & Anderson, D.R. Some New Characterizations of Weights in Dynamic Inequalities Involving Monotonic Functions. Qual. Theory Dyn. Syst. 20, 49 (2021). https://doi.org/10.1007/s12346-021-00489-3
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DOI: https://doi.org/10.1007/s12346-021-00489-3