Hermite-Hadamard type inequalities obtained via Riemann-Liouville fractional calculus (Q2921638)

In this paper, the authors establish new inequalities of Hermite-Hadamard type for functions \(f\in L^1[a, b], a \geq 0\) whose derivatives in absolute value are convex. These inequalities are generalizations of the Hermite-Hadamard inequality. Many mathematicians have devoted their efforts to generalise, refine, counter-part and extend the Hermite-Hadamard inequality for different classes of functions. The authors define the Hermite-Hadamard \(\alpha \)-gap function \(H_\alpha (x)\) which is a generalization of the function \(H(x)\), where NEWLINE\[NEWLINE H(x) = f(x) + \frac {(b-x)f(b)+(x-a)f(a)}{b-a} - \frac {2}{b-a} \int \limits ^b_a f(t)\,{\operatorname {d}}t NEWLINE\]NEWLINE and they prove some new inequalities for \(H_\alpha (x)\), where the bounds depend only on \(f'(x)\). The Riemann-Liouville fractional integrals are used in the definition of \(H_\alpha (x)\). The Hölder inequality, the Jensen integral inequality, the power mean inequality and integration by parts are used in the paper.NEWLINENEWLINEIn my opinion, the paper is very good. The authors use good mathematical ideas, much cleverness and mathematical skills. The paper also inspires other ways of research of Hermite-Hadamard-type inequalities (for example looking for new Hermite-Hadamard-type inequalities for functions of \(n\) variables on suitable convex sets). I recommend this paper to everyone interested in mathematics.