An Ostrowski-Gruss type inequality on time scales

In this paper we derive a new inequality of Ostrowski-Gruss type on time scales and thus unify corresponding continuous and discrete versions. We also apply our result to the quantum calculus case.

💡 Research Summary

The paper “An Ostrowski‑Grüss type inequality on time scales” presents a unified inequality that simultaneously generalizes the classical Ostrowski and Grüss inequalities within the framework of time‑scale calculus. Time‑scale calculus, introduced by Hilger, provides a single theory that encompasses both continuous differential calculus (when the time scale 𝕋 = ℝ) and discrete difference calculus (when 𝕋 = ℤ), as well as more exotic scales such as the quantum (q‑) calculus.

The authors begin by recalling the two classical results. The Ostrowski inequality bounds the deviation of a function value from its average over an interval in terms of the supremum of its derivative; the Grüss inequality bounds the difference between the average of a product and the product of averages in terms of the suprema of the derivatives of the two functions. Both inequalities have been proved separately for continuous and discrete settings, but no single statement has covered all cases.

In Section 2 the paper reviews the necessary notions of time‑scale calculus: rd‑continuity, the delta derivative Δ, the delta integral ∫_a^b f(t)Δt, and the mean‑value theorems adapted to arbitrary closed bounded time scales. The authors define the key quantity

M_f = sup_{t∈