Combined dynamic Gruss inequalities on time scales

TL;DR

This paper extends Gruss inequalities within the framework of time scales calculus by introducing combined dynamic derivatives and integrals, unifying continuous and discrete cases.

Contribution

It develops a more general form of the Gruss inequality using combined dynamic derivatives and integrals on time scales, bridging continuous and discrete inequalities.

Findings

Derived a generalized Gruss inequality on time scales.

Unified continuous and discrete Gruss inequalities.

Provided special cases for real and integer time scales.

Abstract

We prove a more general version of the Gruss inequality by using the recent theory of combined dynamic derivatives on time scales and the more general notions of diamond-alpha derivative and integral. For the particular case when alpha = 1, one gets a delta-integral Gruss inequality on time scales; for alpha = 0 a nabla-integral Gruss inequality. If we further restrict ourselves by fixing the time scale to the real (or integer) numbers, then the standard continuous (discrete) inequalities are obtained.