Abstract
Our purpose in this paper is to use \(\psi-\)Riemann-Liouville fractional integral operator which is the fractional integral of any function with respect to another increasing function to establish some new fractional integral inequalities of Hermite-Hadamard, involving concave functions. Using the concave functions, we establish some new fractional integral inequalities related to the Hermite-Hadamard type inequalities via \(\psi-\)Riemann-Liouville fractional integral operator.
Highlights
The classical calculus of derivatives and integrals which involves integer orders is extended with fractional orders that belong to the real numbers
Several distinct kinds of fractional integrals and derivatives operators have been realized, such as the Liouville, Riemann-Liouville, Katugampola, Weyl types, Hadamard and some other types which can be found in Kilbas et al, [1]
This section is dedicated for some basic definitions and properties of fractional integrals used to obtain and discuss our new results
Summary
The classical calculus of derivatives and integrals which involves integer orders is extended with fractional orders that belong to the real numbers. In 2010, Dahmani [15], studied the Hermite-Hadamard type inequalities for concave functions by means of Riemann–Liouville fractional integral. In 2013 [16], gave the Hermite-Hadamard type inequalities for convex function using Riemann–Liouville fractional integral. Liu et al, [19] in 2016, introduced some inequalities of Hermite-Hadamard type for MT-convex functions using classical integrals and Riemann-Liouville fractional integrals. In 2017, Agarwal et al, [20], obtained some Hermite-Hadamard type inequalities for convex functions via (k, s)−Riemann-Liouville fractional integrals. The main objective of this paper is to establish some new fractional integral Hermite-Hadamard inequalities for concave functions by using ψ−Riemann-Liouville fractional integral operator. We introduce some new fractional integral inequalities related to the Hermite-Hadamard inequalities via ψ−Riemann-Liouville fractional integral operator.
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