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In this article, we will demonstrate some Hardy’s inequalities by utilizing Hölder inequality, integration by parts, and chain rule of the conformable fractional calculus. When α=1\alpha =1, we can obtain some of classical Copson’s and Hardy’s inequalities. In the end, we will obtain an application of Hardy’s inequality utilizing the conformable fractional calculus.

The oscillation and asymptotic behavior of solutions of a general class of damped second-order differential equations with several sub-linear neutral terms is considered. New sufficient conditions are established to fulfill a part of the gap in the oscillation theory for the case of sub-linear neutral equations. Our main results improve and generalize some of those recently published in the literature. Several examples are given to support our results. [ABSTRACT FROM AUTHOR]

This research investigates innovative extensions of Hardy-type inequalities through the use of nabla Hölder's and nabla Jensen's inequalities, combined with the nabla chain rule and the characteristics of convex and submultiplicative functions. We extend these inequalities within a cohesive framework that integrates elements of both continuous and discrete calculus. Furthermore, our study revisits specific integral inequalities from the existing literature, showcasing the wide-ranging relevance of our results. [ABSTRACT FROM AUTHOR]

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 = q N 0 for q > 1 , which, to the best of the authors' knowledge, is a largely novel conclusion. [ABSTRACT FROM AUTHOR]

In this article, we establish some new generalized inequalities of the Hilbert-type on time scales' delta calculus, which can be considered similar to formulas for inequalities of Hilbert type. The major innovation point is to establish some dynamic inequalities of the Hilbert-type on time scales' delta calculus for delta differentiable functions of one variable and two variables. In this paper, we use the condition  a j (s j) = 0  and  a j (s j , z j) = a j (w j , n j) = 0 ,  ∀ j = 1 , 2 , ... , n . These inequalities will be proved by applying Hölder's inequality, the chain rule on time scales, and the mean inequality. As special cases of our results (when  T = N  and  T = R ), we obtain the discrete and continuous inequalities. Also, we can obtain other inequalities in different time scales, like  T = q Z − ,  q > 1 . [ABSTRACT FROM AUTHOR]

In this research, we aim to explore generalizations of Hardy-type inequalities using nabla Hölder's inequality, nabla Jensen's inequality, chain rule on nabla calculus and leveraging the properties of convex and submultiplicative functions. Nabla calculus on time scales provides a unified framework that unifies continuous and discrete calculus, making it a powerful tool for studying various mathematical problems on time scales. By utilizing this approach, we seek to extend Hardy-type inequalities beyond their classical continuous or discrete settings to a more general time scale domain. As specific instances of our discoveries, we have the integral inequalities previously established in the existing literature. [ABSTRACT FROM AUTHOR]

Through the paper, we present several inequalities involving C-monotonic functions with C ≥ 1 , on nabla calculus via time scales. It is known that dynamic inequalities generate many different inequalities in different calculus. The main results will be proved by applying the chain rule formula on nabla calculus. As a special case for our results, when T = R , we obtain the continuous analouges of inequalities that had previously been proved in the literature. When T = N , the results, to the best of the authors' knowledge, are essentially new. Symmetrical properties of C -monotonic functions are critical in determining the best way to solve inequalities. [ABSTRACT FROM AUTHOR]

In this paper, we derive some new fractional extensions of Hardy's type inequalities. The corresponding reverse relations are also obtained by using the conformable fractional calculus from which the classical integral inequalities are deduced as special cases at α = 1 . [ABSTRACT FROM AUTHOR]