Inequalities for α-fractional differentiable functions

Area of Science:

• Fractional Calculus
• Mathematical Analysis
• Inequality Theory

Background:

• The Hermite-Hadamard inequality is a fundamental result in convex analysis.
• Fractional calculus extends the concept of differentiation and integration to non-integer orders.
• Conformable fractional integrals provide a novel framework for fractional calculus operations.

Purpose of the Study:

• To introduce a new identity for conformable fractional integrals.
• To establish novel Hermite-Hadamard type inequalities within the conformable fractional calculus framework.
• To explore applications of these inequalities in special means and numerical integration.

Main Methods:

• Development of a new integral identity tailored for conformable fractional integrals.
• Application of the established identity to derive new Hermite-Hadamard type inequalities.
• Utilizing the derived inequalities to analyze properties of special means and estimate errors in numerical methods.

Main Results:

• A novel identity for conformable fractional integrals is presented.
• Several new Hermite-Hadamard type inequalities are derived for conformable fractional integrals.
• Applications include establishing inequalities for special means and providing error bounds for the trapezoidal formula.

Conclusions:

• The study successfully extends Hermite-Hadamard inequalities to the domain of conformable fractional integrals.
• The derived inequalities and their applications offer valuable tools for further research in fractional analysis and numerical methods.
• This work contributes to the growing body of literature on fractional calculus and its applications.