Some dynamic Hardy-type inequalities with negative parameters on time scales nabla calculus

Elkhateeb S. Aly; Y. A. Madani; F. Gassem; A. I. Saied; H. M. Rezk; Wael W. Mohammed; Elkhateeb S. Aly; Y. A. Madani; F. Gassem; A. I. Saied; H. M. Rezk; Wael W. Mohammed

doi:10.3934/math.2024250

AIMS Mathematics

2024, Volume 9, Issue 2: 5147-5170. doi: 10.3934/math.2024250

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Some dynamic Hardy-type inequalities with negative parameters on time scales nabla calculus

1.
Department of Mathematics, College of Science, Jazan University, P.O. Box 114, Jazan 45142, Kingdom of Saudi Arabia
2.
Department of Mathematics, College of Science, University of Ha'il, Ha'il 2440, Saudi Arabia
3.
Department of Mathematics, Faculty of Science, Benha University, Benha, Egypt
4.
Department of Mathematics, Faculty of Science, Al-Azhar University, Nasr City 11884, Egypt
5.
Department of Mathematics, Faculty of Science, Mansoura University, Mansoura 35516, Egypt

Received: 13 December 2023 Revised: 08 January 2024 Accepted: 19 January 2024 Published: 25 January 2024
MSC : 26D10, 26D15, 34N05, 47B38, 39A12

In this paper, we establish some new dynamic Hardy-type inequalities with negative parameters on time scales nabla calculus by applying the reverse H ölder's inequality, integration by parts, and chain rule on time scales nabla calculus. As special cases of our results (when $ \mathbb{ T = R} $), we get the continuous analouges of inequalities proven by Benaissa and Sarikaya, and when $ \mathbb{T = N}_{0} $, the results to the best of the authors' knowledge are essentially new.
- Hardy-type inequalities with negative parameters,
- time scales,
- reverse Hölder's inequality,
- chain rule
Citation: Elkhateeb S. Aly, Y. A. Madani, F. Gassem, A. I. Saied, H. M. Rezk, Wael W. Mohammed. Some dynamic Hardy-type inequalities with negative parameters on time scales nabla calculus[J]. AIMS Mathematics, 2024, 9(2): 5147-5170. doi: 10.3934/math.2024250

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Abstract

In this paper, we establish some new dynamic Hardy-type inequalities with negative parameters on time scales nabla calculus by applying the reverse H ölder's inequality, integration by parts, and chain rule on time scales nabla calculus. As special cases of our results (when $ \mathbb{ T = R} $), we get the continuous analouges of inequalities proven by Benaissa and Sarikaya, and when $ \mathbb{T = N}_{0} $, the results to the best of the authors' knowledge are essentially new.

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