The Diamond Integral on Time Scales

TL;DR

This paper introduces a new, more general diamond integral on time scales, extending previous work, and establishes fundamental properties including a mean value theorem and classical inequalities.

Contribution

It presents a novel, generalized diamond integral on time scales, improving upon the earlier diamond-alpha integral with new theoretical results.

Findings

Proves a mean value theorem for the diamond integral

Establishes Holder's, Cauchy-Schwarz's, and Minkowski's inequalities for the new integral

Extends the theoretical framework of integrals on time scales

Abstract

We define a more general type of integral on time scales. The new diamond integral is a refined version of the diamond-alpha integral introduced in 2006 by Sheng et al. A mean value theorem for the diamond integral is proved, as well as versions of Holder's, Cauchy-Schwarz's and Minkowski's inequalities.