Formation of Versions of Some Dynamic Inequalities Unified on Time Scale Calculus

Abstract

The aim of this paper is to present some comprehensive and extended versions of classical inequalities such as Radon's Inequality, Bergström's Inequality, the weighted power mean inequality, Schlömilch's Inequality and Nesbitt's Inequality on time scale calculus. In time scale calculus, results are unified and extended. The theory of time scale calculus is applied to unify discrete and continuous analysis and to combine them in one comprehensive form. This hybrid theory is also widely applied on dynamic inequalities. The study of dynamic inequalities has received a lot of attention in the literature and has become a major field in pure and applied mathematics.

Assembling classical and dynamic inequalities accumulated on calculus of time scales

“…This directly yields (16). Similarly, we can prove the Case (2) by applying reverse Rogers-Hölder's inequality.…”

“…In order to conclude our main results, now we present an extension of dynamic Radon's inequality by using Kantorovich's ratio and the time scale ∆-Riemann-Liouville type fractional integral. Extensions of dynamic Radon's inequality are also proved in [14,15,16].…”

Analogy of Classical and Dynamic Inequalities Merging on Time Scales

“…In this research article, we have presented some dynamic inequalities on diamond-α calculus, which is the linear combination of the delta and nabla integrals. Some generalizations and applications of Radon's inequality, Bergström's inequality, Nesbitt's inequality and other dynamic inequalities on time scales are also given in [17,18].…”

A refinement of the Bergström inequality

“…Bergström inequality has stimulated several mathematicians' interest, and various extensions, refinements, and proofs of the inequality have been provided. We refer to [7,9,14,17,28,29] and the references given therein.…”