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Using higher order delta derivatives on time scales, we demonstrated a few dynamic inequalities of the Opial type in this paper. Our findings expanded upon and generalised earlier findings in the literature. Furthermore, we give the discrete and continuous inequalities as special cases. At the end of this paper, we apply our results to study the behaviour of the solution of an initial value problem. In selecting the best ways to solve dynamic inequalities, symmetry is crucial. [ABSTRACT FROM AUTHOR]

We established some new α -conformable dynamic inequalities of Hardy–Knopp type. Some new generalizations of dynamic inequalities of α -conformable Hardy type in two variables on time scales are established. Furthermore, we investigated Hardy's inequality for several functions of α -conformable calculus. Our results are proved by using two-dimensional dynamic Jensen's inequality and Fubini's theorem on time scales. When α = 1 , then we obtain some well-known time-scale inequalities due to Hardy. As special cases, we derived Hardy's inequality for T = R , T = Z and T = h Z . Symmetry plays an essential role in determining the correct methods to solve dynamic inequalities. [ABSTRACT FROM AUTHOR]

Several inverse integral inequalities were proved in 2004 by Yong. It is our aim in this paper to extend these inequalities to time scales. Furthermore, we also apply our inequalities to discrete and continuous calculus to obtain some new inequalities as special cases. Our results are proved using some algebraic inequalities, inverse Hölder's inequality and inverse Jensen's inequality on time scales. Symmetry plays an essential role in determining the correct methods to solve dynamic inequalities. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

As a new usage of Leibniz integral rule on time scales, we proved some new extensions of dynamic Gronwall–Pachpatte-type inequalities on time scales. Our results extend some existing results in the literature. Some integral and discrete inequalities are obtained as special cases of the main results. The inequalities proved here can be used in the analysis as handy tools to study the stability, boundedness, existence, uniqueness and oscillation behavior for some kinds of partial dynamic equations on time scales. Symmetry plays an essential role in determining the correct methods to solve dynamic inequalities. [ABSTRACT FROM AUTHOR]

We established several novel inequalities of Gronwall–Pachpatte type on time scales. Our results can be used as handy tools to study the qualitative and quantitative properties of the solutions of the initial boundary value problem for a partial delay dynamic equation. The Leibniz integral rule on time scales has been used in the technique of our proof. Symmetry plays an essential role in determining the correct methods to solve dynamic inequalities. [ABSTRACT FROM AUTHOR]

In this article, with the help of Leibniz integral rule on time scales, we prove some new dynamic inequalities of Gronwall–Bellman–Pachpatte-type on time scales. These inequalities can be used as handy tools to study the qualitative and quantitative properties of solutions of the initial boundary value problem for partial delay dynamic equation. [ABSTRACT FROM AUTHOR]

Building on the work of Josip Pečarić in 2013 and 1982 and on the work of Srivastava in 2017. We prove some new α -conformable dynamic inequalities of Steffensen-type on time scales. In the case when α = 1 , we obtain some well-known time scale inequalities due to Steffensen inequalities. For some specific time scales, we further show some relevant inequalities as special cases: α -conformable integral inequalities and α -conformable discrete inequalities. Symmetry plays an essential role in determining the correct methods to solve dynamic inequalities. [ABSTRACT FROM AUTHOR]

Our work is based on the multiple inequalities illustrated in 1967 by E. K. Godunova and V. I. Levin, in 1990 by Hwang and Yang and in 1993 by B. G. Pachpatte. With the help of the dynamic Jensen and Hölder inequality, we generalize a number of those inequalities to a general time scale. In addition to these generalizations, some integral and discrete inequalities will be obtained as special cases of our results. [ABSTRACT FROM AUTHOR]