Quantum Integral Inequalities in the Setting of Majorization Theory and Applications

In recent years, the theory of convex mappings has gained much more attention due to its massive utility in different fields of mathematics. It has been characterized by different approaches. In 1929, G. H. Hardy, J. E. Littlewood, and G. Polya established another characterization of convex mappings involving an ordering relationship defined over ${\mathbb{R}}^{\mathfrak{n}}$ known as majorization theory. Using this theory many inequalities have been obtained in the literature. In this paper, we study Hermite–Hadamard type inequalities using the Jensen–Mercer inequality in the frame of $\underset{\dot{}}{\mathrm{q}}$-calculus and majorized l-tuples. Firstly we derive $\underset{\dot{}}{\mathrm{q}}$-Hermite–Hadamard–Jensen–Mercer (H.H.J.M) type inequalities with the help of Mercer’s inequality and its weighted form. To obtain some new generalized (H.H.J.M)-type inequalities, we prove a generalized quantum identity for $\underset{\dot{}}{\mathrm{q}}$-differentiable mappings. Next, we obtain some estimation-type results; for this purpose, we consider $\underset{\dot{}}{\mathrm{q}}$-identity, fundamental inequalities and the convexity property of mappings. Later on, We offer some applications to special means that demonstrate the importance of our main results. With the help of numerical examples, we also check the validity of our main outcomes. Along with this, we present some graphical analyses of our main results so that readers may easily grasp the results of this paper.