Coordination of Classical and Dynamic Inequalities Complying on Time Scales

Abstract

In this research article, we present extensions of some classical inequalities such as Schweitzer, Pólya–Szegö, Kantorovich and Greub–Rheinboldt inequalities of fractional calculus on time scales. To investigate generalizations of such types of classical inequalities, we use the time scales Riemann–Liouville type fractional integrals. We explore dynamic inequalities on delta calculus and their symmetric nabla versions. A time scale is an arbitrary nonempty closed subset of the real numbers. The theory of time scales is applied to combine results in one comprehensive form. The calculus of time scales unifies and extends continuous versions and their discrete and quantum analogues. By using the calculus of time scales, results are presented in more general form. This hybrid theory is also widely applied on dynamic inequalities.

Specht's ratio and logarithmic mean in time scale dynamic inequalities and their retrospective variants

“…The time scales calculus is studied as delta calculus, nabla calculus, and diamond-α calculus. This hybrid theory is also widely applied on dynamic inequalities, see [1,6,12,13,14,15]. The basic work on dynamic inequalities is done by Ravi Agarwal, George Anastassiou, Martin Bohner, Allan Peterson, Donal O'Regan, Samir Saker, and many other authors.…”

“…Without loss of generality, we may suppose that b a |w(x)| α x = 1. Therefore, by inequality(14), we have inequality(15).…”