In this article, we prove some new fractional dynamic inequalities on time scales via conformable calculus. By using chain rule and Hölder’s inequality on timescales we establish the main results. When $\alpha =1$ we obtain some well-known time-scale inequalities due to Hardy, Copson, Bennett and Leindler inequalities.

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 $\alpha =1$.

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 , 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.

In this paper, some fractional Leindler and Hardy-type inequalities and their reversed will be proved by using integration by parts and Hölder inequality on conformable fractional calculus. As a special case, some classical integral inequalities will be obtained. Symmetrical properties play an essential role in determining the correct methods to solve inequalities. The new fractional inequalities in special cases yield some recent relevance, which also provide new estimates on inequalities of these type.

Through the paper, we present several inequalities involving C-monotonic functions with $C\ge 1,$ 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 $\mathbb{T}=\mathbb{R},$ we obtain the continuous analouges of inequalities that had previously been proved in the literature. When $\mathbb{T}=\mathbb{N}$, 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.

In this paper, we will generate some fractional Liendler type inequalities by using integration by parts, chain rule and Hölder inequality on conformable fractional calculus. As a special case, we obtain some formulas of Liendler type inequalities at \(\alpha=1\).