[Paper Review] New sharp inequalities of Ostrowski and generalized trapezoid type for the Riemann-Stieltjes integrals and applications

Mohammad W. Alomari|arXiv (Cornell University)|Aug 7, 2014

Mathematical Inequalities and Applications21 references9 citations

TL;DR

This paper establishes new sharp weighted inequalities of Ostrowski and generalized trapezoid types for Riemann–Stieltjes integrals under various smoothness and variation conditions on the integrand and integrator. The key contribution is a unified error bound for a general quadrature rule, expressed as $\left| R(f,u,I_n,\xi) \right| \leq \left[\frac{1}{2} + \left|\frac{1}{2} - \alpha\right|\right] \left[u(b)-u(a)\right] \cdot \bigvee_a^b(f)$, which generalizes and sharpens existing results for numerical integration of RS-integrals.

ABSTRACT

In this paper, new sharp weighted generalizations of Ostrowski and generalized trapezoid type inequalities for the Riemann--Stieltjes integrals are proved. Several related inequalities are deduced and investigated. New Simpson's type inequalities for $\mathcal{RS}$--integral are pointed out. Finally, as application; an error estimation of a general quadrature rule for $\mathcal{RS}$--integral via Ostrowski--generalized trapezoid quadrature formula is given.

Motivation & Objective

• To derive new sharp weighted generalizations of Ostrowski and generalized trapezoid inequalities for Riemann–Stieltjes integrals.
• To unify and extend existing error bounds for quadrature rules under diverse smoothness and bounded variation assumptions.
• To provide a comprehensive error estimation framework for general quadrature rules applied to Riemann–Stieltjes integrals.
• To deduce new Simpson-type inequalities and apply the results to practical numerical integration scenarios.

Proposed method

• Derives new weighted Ostrowski-type inequalities using Hölder continuity, bounded variation, and Lipschitz conditions on the integrand and integrator.
• Applies the generalized trapezoid rule with a parameter $\alpha$ to partitioned intervals $[x_i, x_{i+1}]$ to model the Riemann–Stieltjes sum $S(f,u,I_n,\xi)$.
• Uses the generalized triangle inequality and variation norms to bound the remainder term $R(f,u,I_n,\xi)$ over the full interval $[a,b]$.
• Establishes the key error bound $\left| R \right| \leq \left[\frac{1}{2} + \left|\frac{1}{2} - \alpha\right|\right] \left[u(b)-u(a)\right] \cdot \bigvee_a^b(f)$ via summation over subintervals.
• Leverages known results from Dragomir, Kikianty, and others as foundational references to build sharper inequalities.
• Validates the results through case analysis under various regularity assumptions, including Hölder, Lipschitz, and monotonicity conditions.

Experimental results

Research questions

• RQ1 What are the sharpest possible error bounds for the Ostrowski-generalized trapezoid quadrature rule applied to Riemann–Stieltjes integrals?
• RQ2 How can weighted inequalities be derived for the Riemann–Stieltjes integral under Hölder, bounded variation, and Lipschitz conditions?
• RQ3 Can the error bound be uniformly expressed in terms of total variation of the integrand and the total variation of the integrator?
• RQ4 What is the role of the parameter $\alpha$ in minimizing the error bound across different quadrature schemes?
• RQ5 How do the new inequalities compare with existing ones in terms of sharpness and generality?

Key findings

• The remainder term $R(f,u,I_n,\xi)$ in the general quadrature rule satisfies $\left| R \right| \leq \left[\frac{1}{2} + \left|\frac{1}{2} - \alpha\right|\right] \left[u(b)-u(a)\right] \cdot \bigvee_a^b(f)$, which is sharp and independent of the partition mesh.
• The bound is uniform across all $\alpha \in [0,1]$, with the worst case at $\alpha = 0$ or $\alpha = 1$, yielding a factor of 1, and the best at $\alpha = 0.5$, yielding a factor of 0.5.
• The result generalizes and improves upon earlier bounds from Dragomir, Kikianty, and others by incorporating weighted norms and variation structures.
• The method allows for error estimation under multiple regularity assumptions, including Hölder continuity, bounded variation, and Lipschitz conditions.
• New Simpson-type inequalities for the Riemann–Stieltjes integral are deduced as special cases of the main framework.
• The analysis confirms that the total variation of the integrand $\bigvee_a^b(f)$ is a key determinant of the error magnitude, while the integrator's variation $u(b)-u(a)$ scales the bound linearly.