Multiplicative view point analysis of Hahn calculus and their applications to inequality theory

Hahn multiplicative calculus is the generalization of quantum multiplicative ( $\mathit{q}$-multiplicative) calculus. In this manuscript, we defined novel definitions for derivative and definite integral called left Hahn multiplicative derivative and definite integral in the Hahn multiplicative calculus. In addition, we derived fundamental results for this newly defined integral. Furthermore, we constructed left Hahn multiplicative Hermite-Hadamard inequalities. Additionally, we defined new definitions for the derivative and definite integral in the Hahn calculus, which enabled us to define further definitions in Hahn multiplicative calculus called the right Hahn multiplicative derivative and definite integral. Moreover, we construct the power rule of the newly defined definite integral in the Hahn calculus, which assisted us in deriving the right Hahn multiplicative Hermite-Hadamard inequalities. Finally, we gave an application of the newly established Hermite-Hadamard inequalities through an example wherein it could be seen that these inequalities were crucial for finding the lower and upper bounds of the range of those functions whose Hahn multiplicative definite integrals were very difficult to find.