A Comprehensive Review of the Hermite–Hadamard Inequality Pertaining to Fractional Integral Operators

Muhammad Tariq, Sotiris K. Ntouyas () and Asif Ali Shaikh
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Muhammad Tariq: Department of Basic Sciences and Related Studies, Mehran University of Engineering and Technology, Jamshoro 76062, Pakistan
Sotiris K. Ntouyas: Department of Mathematics, University of Ioannina, 451 10 Ioannina, Greece
Asif Ali Shaikh: Department of Basic Sciences and Related Studies, Mehran University of Engineering and Technology, Jamshoro 76062, Pakistan

Mathematics, 2023, vol. 11, issue 8, 1-106

Abstract: In the frame of fractional calculus, the term convexity is primarily utilized to address several challenges in both pure and applied research. The main focus and objective of this review paper is to present Hermite–Hadamard (H-H)-type inequalities involving a variety of classes of convexities pertaining to fractional integral operators. Included in the various classes of convexities are classical convex functions, m -convex functions, r -convex functions, ( α , m ) -convex functions, ( α , m ) -geometrically convex functions, harmonically convex functions, harmonically symmetric functions, harmonically ( θ , m ) -convex functions, m -harmonic harmonically convex functions, ( s , r ) -convex functions, arithmetic–geometric convex functions, logarithmically convex functions, ( α , m ) -logarithmically convex functions, geometric–arithmetically s -convex functions, s -convex functions, Godunova–Levin-convex functions, differentiable ϕ -convex functions, M T -convex functions, ( s , m ) -convex functions, p -convex functions, h -convex functions, σ -convex functions, exponential-convex functions, exponential-type convex functions, refined exponential-type convex functions, n -polynomial convex functions, σ , s -convex functions, modified ( p , h ) -convex functions, co-ordinated-convex functions, relative-convex functions, quasi-convex functions, ( α , h − m ) − p -convex functions, and preinvex functions. Included in the fractional integral operators are Riemann–Liouville (R-L) fractional integral, Katugampola fractional integral, k -R-L fractional integral, ( k , s ) -R-L fractional integral, Caputo-Fabrizio (C-F) fractional integral, R-L fractional integrals of a function with respect to another function, Hadamard fractional integral, and Raina fractional integral operator.

Keywords: Hermite–Hadamard inequality; convex function; Riemann–Liouville fractional integral; Katugampola fractional integral; Hadamard fractional integral (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2023
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