Mean value theorem for quantum integral operator with application to sharp Ostrowski inequality

TL;DR

This paper establishes a quantum calculus version of the mean value theorem, corrects a previous Ostrowski inequality, and introduces a sharp new inequality along with a midpoint inequality.

Contribution

It provides a corrected and sharp Ostrowski inequality in quantum calculus, along with a mean value theorem and a midpoint inequality.

Findings

Disproved a previous Ostrowski inequality in quantum calculus

Derived a correct and sharp Ostrowski inequality

Established a quantum mean value theorem and a midpoint inequality

Abstract

We derive a version of Lagrange's mean value theorem for quantum calculus. We disprove a version of Ostrowski inequality for quantum calculus appearing in the literature. We derive a correct statement and prove that our new inequality is sharp. We also derive a midpoint inequality.