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Open Access Research Article

Weighted Hardy-type inequalities with Diamond-alpha derivative on time scales

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pp. 1835–1847Vol. 29Issue 6June 2026DOI: 10.47974/JIM-2319XML
Received:
01 Nov 2024
Published Online:
30 Jun 2026
Article type:
Research Article
Language:
EN
Article no.:
JIM-2319
Pages:
1835–1847

Abstract

We establish new weighted Hardy-type inequalities involving the Diamond-alpha derivative within time scales calculus, unifying discrete and continuous settings. By introducing weight functions, we generalize classical Hardy-type inequalities and derive conditions ensuring the sharpness of the constants. The results extend known inequalities by providing optimal bounds and covering various time scales, such as real numbers, integers, and q-calculus. Applications to dynamic equations and illustrative examples are presented, demonstrating the interaction between weighted functions and dynamic derivatives. These findings deepen the understanding of weighted inequalities on time scales, offering new avenues for generalization and application.

Keywords

Subject Classifications

26E7026D1526D1033A40

References

[1] G. H. Hardy, J. E. Littlewood, G. Pólya, and D. E. Littlewood, Inequalities. Cambridge, U.K.: Cambridge Univ. Press (1952).
[2] R. P. Agarwal, D. O’Regan, and S. H. Saker, Hardy Type Inequalities on Time Scales. Cham, Switzerland: Springer (2016). 
[3] N. Levinson, “Generalization of an inequality of Hardy,” Duke Math. J., vol. 31, no. 3, pp. 389–394 (1964).
[4] W. Cheung, Z. Hanjš, and J. Pečarić, “Some Hardy-type inequalities,” J. Math. Anal. Appl., vol. 250, no. 2, pp. 621–634 (2000).
[5] E. T. Copson, “Some integral inequalities,” Proceedings of the Royal Society of Edinburgh Section A: Mathematics, vol. 75, pp. 157–164 (1976), doi: 10.1017/S0308210500017868.
[6] G. S. Yang and D. Y. Hwang, “Generalizations of some reverse integral inequalities,” J. Math. Anal. Appl., vol. 233, no. 1, pp. 193–204 (1999).
[7] P. Řehák, “Hardy inequality on time scales and its application to half-linear dynamic equations,” J. Inequal. Appl., vol. 2005, no. 5, pp. 942–973 (2005).
[8] S. H. Saker and D. O’Regan, “Littlewood inequalities on time scales,” Bull. Malays. Math. Sci. Soc., vol. 39, no. 2, pp. 527–543 (2016).
[9] S. H. Saker, D. O’Regan, and R. P. Agarwal, “Littlewood and Bennett inequalities on time scales,” Mediterr. J. Math., vol. 12, no. 3, pp. 605–619 (2015).
[10] C. P. Selvan, A. Shrivastava, S. Ramaswamy, and K. Shipra, “Applying control theory in engineering using dynamical systems methods,” J. Interdiscip. Math., vol. 29, no. 3, pp. 627–635 (2026). 
[11] G. Vaidya, S. M. U. Iqbal, M. I. Khan, and K. Gupta, “Analyzing non-linear dynamical systems and chaos using machine learning in physical and biological systems,” J. Interdiscip. Math., vol. 29, no. 3, pp. 663–670 (2026).
[12] M. Bohner and A. C. Peterson, Dynamic Equations on Time Scales: An Introduction with Applications. New York, USA: Springer (2001).
[13] M. Bohner and A. C. Peterson, Advances in Dynamic Equations on Time Scales. Boston, MA, USA: Springer (2002).
[14] A. Hamiaz, W. Abuelela, S. H. Saker, and D. Baleanu, “Some new dynamic inequalities with several functions of Hardy type on time scales,” J. Inequal. Appl., vol. 2021, no. 3 (2021).

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