The main objective of this study is to establish two important right q-integral equalities involving a right-quantum derivative with parameter $m\in [0,1]$. Then, utilizing these equalities, we derive some new variants for midpoint- and trapezoid-type inequalities for the right-quantum integral via differentiable $(\alpha ,m)$-convex functions. The fundamental benefit of these inequalities is that they may be transformed into q-midpoint- and q-trapezoid-type inequalities for convex functions, classical midpoint inequalities for convex functions and classical trapezoid-type inequalities for convex functions are transformed without having to prove each one independently. In addition, we present some applications of our results to special means of positive real numbers. It is expected that the ideas and techniques may stimulate further research in this field.

In this investigation, we first establish new quantum Hermite–Hadamard type integral inequalities for s-convex functions by utilizing newly defined $T_q$-integrals. Then, by using obtained inequality, we establish a new Hermite–Hadamard inequality for coordinated $\left(s_1,s_2\right)$-convex functions. The results obtained in this paper provide significant extensions of other related results given in the literature. Finally, some examples are given to illustrate the result obtained in this paper. These types of analytical inequalities, as well as solutions, apply to different areas where the concept of symmetry is important.

In this investigation, we established some new post quantum-Hermite-Hadamard inequalities for differentiable convex function with critical point by using the notion of 〖(p,q)〗^b-integeral .The perseverance of this article is to establish different results on the left hand side of 〖(p,q)〗^b- Hermite-Hadamard inequality for differentiable convex function along with critical a point. Some Special cases are obtain for different 〖(p,q)〗^b- Hermite Hadamard inequalies at critical point c for some specific values of q.

In this study, we obtained some new post quantum-Hermite-Hadamard inequalities for differentiable convex function with critical point by using generalized (p, q)- Hermite-Hadamard Inequality. The perseverance of this article is to establish different results on the left-hand side of (p,q)-Hermite-Hadamard inequality for differentiable convex function along with critical point. Special cases were obtained for different (p, q)-Hermite Hadamard inequalies with the critical point c for some special values of q.

This article is pedestal for the (p,q)-calculus connecting two concepts of (p,q)-derivatives and (p,q)-integrals. The purpose of this paper is to establish different type of identities for (p,q)-calculus. Some special cases of the (p,q)-midpoint, Simpson, Averaged midpoint trapezoid, and trapezoid type integral identities are also derived.