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.
In this study, we apply Hölder’s inequality, Jensen’s inequality, chain rule and the properties of convex functions and submultiplicative functions to develop an innovative category of dynamic Hardy-type inequalities on time scales delta calculus. A time scale, denoted by T, is any closed nonempty subset of R. In time scale calculus, results are unified and extended. As particular cases of our findings (when T=R), we have the continuous analogues of inequalities established in some the literature. Furthermore, we can find other inequalities in different time scales, such as T=N, which, to the best of the authors’ knowledge, is a largely novel conclusion.
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