[Paper Review] A new generalization of Ostrowski type inequality on time scales

Wenjun Liu, Quốc Anh Ngô+1 more|arXiv (Cornell University)|Apr 28, 2008

Nonlinear Differential Equations Analysis19 references19 citations

TL;DR

This paper introduces a new parameterized generalization of Ostrowski-type inequalities on time scales, unifying continuous and discrete versions by incorporating a parameter λ. It extends existing results by deriving bounds for the deviation of a function's value from its integral mean, with key results including unified trapezoid, midpoint, Simpson, and averaged midpoint-trapezoid inequalities on time scales as special cases when λ takes specific values.

ABSTRACT

In this paper we first extend a generalization of Ostrowski type inequality on time scales for functions whose derivatives are bounded and then unify corresponding continuous and discrete versions. We also point out some particular integral type inequalities on time scales as special cases.

Motivation & Objective

• To generalize Ostrowski-type inequalities on time scales by introducing a parameter λ to unify continuous and discrete settings.
• To extend existing Ostrowski-type results for functions with bounded Δ-derivatives on time scales.
• To derive unified integral inequalities that reduce to known results (e.g., trapezoid, midpoint, Simpson) under specific parameter choices.
• To provide a comprehensive framework encompassing various integral inequalities on time scales through a single generalized inequality.

Proposed method

• Introduce a parameter λ ∈ [0,1] to construct a generalized mean point on time scales, replacing the standard midpoint in classical Ostrowski inequalities.
• Use the Δ-derivative and the graininess functions μ(t) and ν(t) to define the time scale analogs of classical integral bounds.
• Apply the generalized integral representation involving h₂ functions h₂(t,a) and h₂(t,b), which are time scale analogs of (t−a)² and (b−t)².
• Derive the main inequality by bounding the difference between f(t) and the weighted average of f(σ(s)) over [a,b]Δ, using the supremum of |fΔ(t)|.
• Specialize the general inequality to λ = 1 (trapezoid), λ = 1/3 (Simpson), λ = 1/2 (averaged midpoint-trapezoid), and λ = 0 (midpoint) to recover known inequalities.
• Verify sharpness and consistency with existing results, including Theorem 1.2 from Bohner and Matthews (2001).

Experimental results

Research questions

• RQ1 How can Ostrowski-type inequalities on time scales be generalized using a parameter λ to unify continuous and discrete versions?
• RQ2 What are the time scale analogs of classical integral inequalities such as trapezoid, midpoint, and Simpson rules under this generalization?
• RQ3 Can the generalized inequality recover known Ostrowski-type results as special cases when λ takes specific values?
• RQ4 What is the role of the h₂ function and Δ-derivative in bounding the error between a function’s value and its integral mean on time scales?

Key findings

• The generalized inequality provides a unified framework for Ostrowski-type bounds on time scales, with the error term expressed via h₂ functions and the Δ-derivative's supremum norm.
• For λ = 1 and t = (a+b)/2, the trapezoid inequality on time scales is recovered, with error bounded by M/(b−a) times the sum of h₂(a, (a+b)/2) and h₂(b, (a+b)/2).
• For λ = 1/3 and t = (a+b)/2, the Simpson inequality on time scales is obtained, with error bounded by a sum of four h₂ terms evaluated at specific points.
• For λ = 1/2 and t = (a+b)/2, the averaged midpoint-trapezoid inequality on time scales is derived, showing a convex combination of midpoint and trapezoid rules.
• For λ = 0, the inequality reduces to the original Ostrowski-type inequality on time scales (Theorem 1.2), confirming consistency with prior work.
• The generalization is sharp, as the constant factors in the error bound cannot be improved, preserving the optimality of classical results.