[Paper Review] Gruss-type inequality by mean of a fractional integral

J. Vanterler da C. Sousa, D. S. Oliveira+1 more|arXiv (Cornell University)|Apr 28, 2017

Mathematical Inequalities and Applications4 citations

TL;DR

This paper generalizes Gruss-type integral inequalities using Katugampola's fractional integral, which unifies six existing fractional integrals. By applying this operator to bounded functions, the authors derive new inequalities that extend classical Gruss bounds, with tightness proven via Cauchy-Schwarz and AM-GM-type estimates, yielding sharper error bounds in fractional calculus contexts.

ABSTRACT

In this paper, using a fractional integral as proposed by Katugampola we establish a generalization of integral inequalities of Gruss-type. We prove two theorems associated with these inequalities and then immediately we enunciate and prove others inequalities associated with these fractional operator.

Motivation & Objective

• To extend classical Gruss-type inequalities to the framework of fractional calculus using Katugampola's generalized fractional integral.
• To unify six known fractional integrals (Riemann-Liouville, Hadamard, Erdélyi-Kober, etc.) under a single operator for broader applicability.
• To establish new fractional Gruss-type inequalities with improved error bounds through operator-specific estimates.
• To provide a foundation for generalizing other classical inequalities (e.g., Hermite-Hadamard) in fractional settings.

Proposed method

• Utilizes Katugampola's left-sided fractional integral operator defined via a parameterized kernel involving $ x^\rho $, $ \tau^\rho $, and the gamma function.
• Operates within the space $ X_c^p(a,b) $, ensuring integrability and convergence for the fractional integral under $ L^p $-type norms.
• Applies the inequality $ (f(t) - mg(t))(Mg(t) - f(t)) \geq 0 $ to derive bounds on $ {}^\rho\mathcal{I}_{\eta,\kappa}^{\alpha,\beta}(fg)(x) $.
• Employs the Cauchy-Schwarz and AM-GM inequalities on fractional integral forms to derive upper bounds on $ \left({}^\rho\mathcal{I}_{\eta,\kappa}^{\alpha,\beta}f^2\right)\left({}^\rho\mathcal{I}_{\eta,\kappa}^{\alpha,\beta}g^2\right) $.
• Derives a key bound: $ \left({}^\rho\mathcal{I}_{\eta,\kappa}^{\alpha,\beta}f^2\right)\left({}^\rho\mathcal{I}_{\eta,\kappa}^{\alpha,\beta}g^2\right) \leq \frac{(M+m)^2}{4} \left({}^\rho\mathcal{I}_{\eta,\kappa}^{\alpha,\beta}fg\right)^2 $.
• Validates the results through algebraic manipulation and limit cases, recovering known Riemann-Liouville bounds as special cases.

Experimental results

Research questions

• RQ1 Can Gruss-type inequalities be generalized using Katugampola's fractional integral operator?
• RQ2 How do the bounds of the classical Gruss inequality transform under fractional integration?
• RQ3 What is the tightest possible error bound for the fractional Gruss inequality under bounded function assumptions?
• RQ4 Can the fractional Gruss inequality be derived using Cauchy-Schwarz and AM-GM estimates on the fractional integral form?
• RQ5 What are the limiting cases of the new inequality when reducing to known fractional integrals like Riemann-Liouville or Hadamard?

Key findings

• The paper establishes a new fractional Gruss-type inequality: $ \left({}^\rho\mathcal{I}_{\eta,\kappa}^{\alpha,\beta}f^2\right)\left({}^\rho\mathcal{I}_{\eta,\kappa}^{\alpha,\beta}g^2\right) \leq \frac{(M+m)^2}{4} \left({}^\rho\mathcal{I}_{\eta,\kappa}^{\alpha,\beta}fg\right)^2 $, which generalizes the classical result.
• The inequality achieves tighter bounds than classical versions by incorporating the fractional integral's kernel structure and parameter dependence.
• The bound $ \frac{(M+m)^2}{4} $ is preserved in the fractional form, indicating that the classical extremal behavior is maintained under the generalized operator.
• A second inequality is derived: $ \sqrt{\left({}^\rho\mathcal{I}_{\eta,\kappa}^{\alpha,\beta}f^2\right)\left({}^\rho\mathcal{I}_{\eta,\kappa}^{\alpha,\beta}g^2\right)} - \left({}^\rho\mathcal{I}_{\eta,\kappa}^{\alpha,\beta}fg\right) \leq \frac{(\sqrt{M} - \sqrt{m})^2}{2\sqrt{mM}} \left({}^\rho\mathcal{I}_{\eta,\kappa}^{\alpha,\beta}fg\right) $, providing a relative error estimate.
• The results reduce to the classical Gruss inequality when $ \rho \to 1 $, $ \beta = \alpha $, $ \kappa = 0 $, $ \eta = 0 $, confirming consistency with known theory.
• The framework allows for future generalization of Hermite-Hadamard and Hermite-Hadamard-Fejér inequalities in fractional settings.