Integral inequalities for some convex functions via generalized fractional integrals

Area of Science:

• Mathematical Analysis
• Fractional Calculus

Background:

• The Hermite-Hadamard inequality is a fundamental result in convex analysis.
• Fractional integrals generalize classical integrals, offering broader applications.
• Existing fractional integrals like Riemann-Liouville and Hadamard have limitations.

Purpose of the Study:

• To establish new Hermite-Hadamard type inequalities for s-convex and m-convex functions.
• To utilize the generalized Katugampola fractional integral as a unifying framework.
• To demonstrate the relationship between Katugampola and Riemann-Liouville fractional integrals.

Main Methods:

• Application of the Katugampola fractional integral.
• Derivation of inequalities for s-convex and m-convex functions.
• Analysis of the generalization properties of the Katugampola fractional integral.

Main Results:

• New Hermite-Hadamard type inequalities are obtained for s-convex and m-convex functions.
• The Katugampola fractional integral is shown to generalize existing fractional integral inequalities.
• A connection is established between the Katugampola fractional integral and the Riemann-Liouville fractional integral.

Conclusions:

• The Katugampola fractional integral provides a powerful tool for extending classical inequalities.
• This work unifies and advances the study of fractional integral inequalities.
• The findings contribute to the theory of convex functions and fractional calculus.