Some inequalities for interval-valued functions on time scales

TL;DR

This paper introduces new integral concepts for interval-valued functions on time scales, establishing fundamental properties and inequalities like Jensen's, Hölder's, and Minkowski's, with illustrative examples.

Contribution

It develops the interval Darboux delta and Riemann delta integrals for functions on time scales, extending integral inequalities to this setting.

Findings

Defined the $ID$ $ riangle$-integral and $IR$ $ riangle$-integral for interval-valued functions.

Proved Jensen's, Hölder's, and Minkowski's inequalities for the $IR$ $ riangle$-integral.

Provided examples illustrating the properties and inequalities.

Abstract

We introduce the interval Darboux delta integral (shortly, the $ID$ $\Delta$-integral) and the interval Riemann delta integral (shortly, the $IR$ $\Delta$-integral) for interval-valued functions on time scales. Fundamental properties of $ID$ and $IR$ $\Delta$-integrals and examples are given. Finally, we prove Jensen's, H\"{o}lder's and Minkowski's inequalities for the $IR$ $\Delta$-integral. Also, some examples are given to illustrate our theorems.