Some new error bounds for Hermite--Hadamard inequality and numerical integration formulas in quantum multiplicative calculus with computational analysis

Abstract

  This study presents a novel $q$-multiplicative calculus by merging the concepts of quantum and multiplicative calculus. This framework is crucial for phenomena that require discrete scaling and multiplicative differentiation, such as in biology, fractals, quantum mechanics, and finance. We present the $q$-multiplicative derivative, $q$-multiplicative integral, and their fundamental properties with their proofs. These definitions aid in the presentation of new versions of Hermite Hadamard and midpoint inequalities.  The generalized version of the Hermite-Hadamard inequality for $q$-multiplicative calculus has been established. In addition, a graphical and numerical analysis verifies the validity and effectiveness of the newly defined inequalities. We provide a detailed overview of the newly formed results and discussed the behavior of these inequalities based on various set parameters, numerically and graphically. Based on its applicability and effectiveness to complex systems, this work will greatly attract future research and exploration in this field.