In this paper, we first establish two right-quantum integral equalities involving a right-quantum derivative and a parameter $m\in \left[0,1\right]$. Then, we prove modified versions of Simpson’s and Newton’s type inequalities using established equalities for right-quantum differentiable $\left(\alpha ,m\right)$-convex functions. The newly developed inequalities are also proven to be expansions of comparable inequalities found in the literature.

The main goal of the current study is to establish some new Milne’s rule type inequalities for single-time differentiable convex functions in the setting of quantum calculus. For this, we establish a quantum integral identity and then we prove some new inequalities of Milne’s rule type for quantum differentiable convex functions. These inequalities are very important in Open-Newton’s Cotes formulas because, with the help of these inequalities, we can find the bounds of Milne’s rule for differentiable convex functions in classical or quantum calculus. The method adopted in this work to prove these inequalities are very easy and less conditional compared to some existing results. Finally, we give some mathematical examples to show the validity of newly established inequalities.

In this paper, we establish a new integral identity involving differentiable functions, and then we use the newly established identity to prove some Ostrowski–Mercer-type inequalities for differentiable convex functions. It is also demonstrated that the newly established inequalities are generalizations of some of the Ostrowski inequalities established inside the literature. There are also some applications to the special means of real numbers given.

In this work, we prove a new $(p,q)$-integral identity involving a $(p,q)$-derivative and $(p,q)$-integral. The newly established identity is then used to show some new Simpson’s formula type inequalities for $(p,q)$-differentiable convex functions. Finally, the newly discovered results are shown to be refinements of comparable results in the literature. Analytic inequalities of this type, as well as the techniques used to solve them, have applications in a variety of fields where symmetry is important.

In this paper, we establish some new Hermite–Hadamard type inequalities for preinvex functions and left-right estimates of newly established inequalities for $\left(p,q\right)$-differentiable preinvex functions in the context of $\left(p,q\right)$-calculus. We also show that the results established in this paper are generalizations of comparable results in the literature of integral inequalities. Analytic inequalities of this nature and especially the techniques involved have applications in various areas in which symmetry plays a prominent role.

In this study, first we establish a $\left(p,q\right)$-integral identity involving the second $\left(p,q\right)$-derivative, and then, we use this result to prove some new midpoint-type inequalities for twice-$\left(p,q\right)$-differentiable convex functions. It is also shown that the newly established results are the refinements of the comparable results in the literature.