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In this work, we prove several new (γ,a) -nabla Bennett and Leindler dynamic inequalities on time scales. The results proved here generalize some known dynamic inequalities on time scales, unify and extend some continuous inequalities and their corresponding discrete analogues. Our results will be proved by using integration by parts, chain rule and Hölder inequality for the (γ,a)-nabla-fractional derivative on time scales.

In this paper, we employ some algebraic equations due to Hardy and Littlewood to establish some conditions on weights in dynamic inequalities of Hardy and Copson type. For illustrations, we derive some dynamic inequalities of Wirtinger, Copson and Hardy types and formulate the classical integral and discrete inequalities with sharp constants as particular cases. The results improve some results obtained in the literature.

We prove new Hardy–Copson-type (γ,a)-nabla fractional dynamic inequalities on time scales. Our results are proven by using Keller’s chain rule, the integration by parts formula, and the dynamic Hölder inequality on time scales. When γ=1, then we obtain some well-known time-scale inequalities due to Hardy. As special cases, we obtain new continuous and discrete inequalities. Symmetry plays an essential role in determining the correct methods to solve dynamic inequalities.

In this important work, we discuss some novel Hilbert-type dynamic inequalities on time scales. The inequalities investigated here generalize several known dynamic inequalities on time scales and unify and extend some continuous inequalities and their corresponding discrete analogues. Our results will be proved by using some algebraic inequalities, Hölder inequality, and Jensen’s inequality on time scales.

In the present paper, some new generalizations of dynamic inequalities of Gronwall–Bellman–Pachpatte-type on time scales are established. Some integral and discrete Gronwall–Bellman–Pachpatte-type inequalities that are given as special cases of main results are original. The main results are proved by using the dynamic Leibniz integral rule on time scales. To highlight our research advantages, several implementations of these findings are presented. Symmetry plays an essential role in determining the correct methods to solve dynamic inequalities.

In this paper, we consider generalized conformable integrals to establish new Opial-type inequalities. The obtained results mainly depend on Hölder’s inequality, some algebraic inequalities, and a simple consequence of Keller’s chain rule on time scales. Our obtained results unify and extend some continuous and discrete inequalities. In the special case α = 1 , our results cover some well-known inequalities of Opial-type on time scales.

We prove some new dynamic inequalities of the Gronwall–Bellman–Pachpatte type on time scales. Our results can be used in analyses as useful tools for some types of partial dynamic equations on time scales and in their applications in environmental phenomena and physical and engineering sciences that are described by partial differential equations.

This study demonstrates several novel dynamic inequalities of the Hardy and Littlewood types on time scales. As special cases, our studies include Hardy’s integral inequalities and Hardy and Littlewood’s discrete inequalities. The research findings are demonstrated using algebraic inequalities, Hölder’s inequality, and the chain rule on time scales.

In this article, we present some novel dynamic Hilbert-type inequalities within the framework of time scales T. We achieve this by utilizing Hölder’s inequality, the chain rule, and the mean inequality. As specific instances of our findings (when T=N and T=R), we obtain the discrete and continuous analogues of previously established inequalities. Additionally, we derive other inequalities for different time scales, such as T=qN0 for q>1, which, to the best of the authors’ knowledge, is a largely novel conclusion.

In the present article, we prove some new generalizations of dynamic inequalities of Hardy-type by utilizing diamond-α dynamic integrals on time scales. Furthermore, new generalizations of dynamic inequalities of Hardy-type in two variables on time scales are proved. Moreover, we present Hardy inequalities for several functions by using the diamond-α dynamic integrals on time scales. The results are proved by using the dynamic Jensen inequality and the Fubini theorem on time scales. Our main results extend existing results of the integral and discrete Hardy-type inequalities. Symmetry plays an essential role in determining the correct methods to solve dynamic inequalities.