Diamond-$\alpha$ Jensen's Inequality on Time Scales

TL;DR

This paper introduces the diamond-$ ext{alpha}$ derivative on time scales, establishing a generalized Jensen's inequality and related integral inequalities, thereby advancing the mathematical framework for dynamic derivatives on time scales.

Contribution

It develops basic properties of diamond-$ ext{alpha}$ derivatives and proves a generalized Jensen's inequality on time scales, including corollaries like Hölder's and Minkowski's inequalities.

Findings

Established properties of diamond-$ ext{alpha}$ derivatives.

Proved a generalized Jensen's inequality on time scales.

Derived corollaries such as Hölder's and Minkowski's inequalities.

Abstract

The theory and applications of dynamic derivatives on time scales has recently received considerable attention. The primary purpose of this paper is to give basic properties of diamond-$\alpha$ derivatives which are a linear combination of delta and nabla dynamic derivatives on time scales. We prove a generalized version of Jensen's inequality on time scales via the diamond-$\alpha$ integral and present some corollaries, including H\"{o}lder's and Minkowski's diamond-$\alpha$ integral inequalities.