Derivation of Hermite–Hadamard-type inequalities via quasi-preinvex functions and strongly preinvex functions

Quantum calculus, similar to calculus without limits, is the same as ordinary “infinitesimal calculus”. Quantum Hermite–Hadamard-type inequalities, according to quantum calculus, recently discovered improvements within quantum Hermite–Hadamard-type inequalities. New results about the derivatives and integrals identities related to both [Formula: see text]-derivatives and [Formula: see text]-integrals will be obtained. This research work is motivated by this fact, so using properties of “generalized higher-order strongly preinvex” and “quasi-preinvex” functions, we drive innovative Quantum Hermite–Hadamard-type inequalities. As applications, new Hermite–Hadamard-type inequalities Hermite–Hadamard [Formula: see text]-integral and [Formula: see text]-integral-type inequalities will be obtained. These types of identities are applied to “preinvex functions”. The newly obtained important outcomes are present. The results of these new generalizations are used to assess a variety of mathematical difficulties. These new findings have a huge impact on integrated symmetrical functions and approximations, functions with a symmetric degree. These visions are encouraging new and significant achievements in a wide range of mathematics and engineering disciplines. The generalized strongly “preinvex functions” are the “quasi-preinvex function” studied using “elementary Quantum” Calculus methods.