New integral inequalities in the class of functions $(h,m)$-convex

J. E. Nápoles ; P. M. Guzmán ; B. Bayraktar

Izvestiya of Saratov University. Mathematics. Mechanics. Informatics, Tome 24 (2024) no. 2, pp. 173-183

Résumé

In this article, we have defined new weighted integral operators. We formulated a lemma in which we obtained a generalized identity through these integral operators. Using this identity, we obtain some new generalized Simpson's type inequalities for $(h,m)$-convex functions. These results we obtained using the convexity property, the classical Hölder inequality, and its other form, the power mean inequality. The generality of our results lies in two fundamental points: on the one hand, the integral operator used and, on the other, the notion of convexity. The first, because the “weight” allows us to encompass many known integral operators (including the classic Riemann and Riemann – Liouville), and the second, because, under an adequate selection of the parameters, our notion of convexity contains several known notions of convexity. This allows us to show that many of the results reported in the literature are particular cases of ours.