Unified inequalities of the $\mathfrak{q}$-Trapezium-Jensen-Mercer type that incorporate majorization theory with applications

The objective of this paper is to explore novel unified continuous and discrete versions of the Trapezium-Jensen-Mercer (TJM) inequality, incorporating the concept of convex mapping within the framework of $\mathfrak{q}$-calculus, and utilizing majorized tuples as a tool. To accomplish this goal, we establish two fundamental lemmas that utilize the ${}_{{\varsigma }_1}\mathfrak{q}$ and ${}^{{\varsigma }_2}\mathfrak{q}$ differentiability of mappings, which are critical in obtaining new left and right side estimations of the midpoint $\mathfrak{q}$-TJM inequality in conjunction with convex mappings. Our findings are significant in a way that they unify and improve upon existing results. We provide evidence of the validity and comprehensibility of our outcomes by presenting various applications to means, numerical examples, and graphical illustrations.