Analysis of Quantum Multiplicative Calculus and Related Inequalities

This article investigates the exact meaning of a quantum derivative result and the corresponding definition of a quantum definite integral in multiplicative calculus from a geometrical viewpoint. After this critical analysis, we give an accurate definition of the q-multiplicative definite integral and the corresponding derivative result. Additionally, an example pertaining to q-multiplicative definite integrals is presented, and rigorous analysis to prove several fundamental results is provided. In addition, two other concepts are defined: the left q-multiplicative derivative and definite integral and the right q-multiplicative derivative and definite integral. Finally, several q-multiplicative Hermite–Hadamard-type inequalities are constructed, and related examples are shown to support our recent findings.