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In this paper, we present the preliminaries of ( p , q ) -calculus for functions of two variables. Furthermore, we prove some new Hermite-Hadamard integral-type inequalities for convex functions on coordinates over [ a , b ] × [ c , d ] by using the ( p , q ) -calculus of the functions of two variables. Furthermore, we establish an identity for the right-hand side of the Hermite-Hadamard-type inequalities on coordinates that is proven by using the ( p , q ) -calculus of the functions of two variables. Finally, we use the new identity to prove some trapezoidal-type inequalities with the assumptions of convexity and quasi-convexity on coordinates of the absolute values of the partial derivatives defined in the ( p , q ) -calculus of the functions of two variables.

Post-quantum integral inequalities involving the Riemann–Liouville fractional integral have a significant role in understanding and modeling systems with nonlocal interactions; anomalous diffusion and memory effects make them indispensable for addressing modern challenges in applied mathematics and physics. In this paper, the authors introduced a right fractional ( p , q ) -integral operator of Riemann–Liouville type, including the q -shifting operator, in parallel to the left fractional ( p , q ) -integral operator of Riemann–Liouville type introduced by Neang et al. This is done to investigate the post-quantum multiparameter fundamental identity of continuous functions on finite intervals and to derive some new and existing estimates of various inequalities, such as Bullen type, Simpson type, midpoint type, trapezoid type, etc. In order to verify the accuracy of the findings, graphical and numerical analysis and an application to special means are provided.

The Hermite–Hadamard inequality was first considered for convex functions and has been studied extensively. Recently, many extensions were given with the use of general convex functions. In this paper we present some variants of the Hermite–Hadamard inequality for general convex functions in the context of q-calculus. From our theorems, we deduce some recent results in the topic.

Recently, Sofonea (Gen. Math. 16:47-54, 2008 ) considered some relations in the context of quantum calculus associated with the q -derivative operator D q and divided difference. As applications of the post-quantum calculus known as the ( p , q ) -calculus, we derive several relations involving the ( p , q ) -derivative operator and divided differences.

Certain subclasses S H p , q m , ℓ [ Φ i , Ψ j ; α , A , B ] and S H ¯ p , q m , ℓ [ Φ i , Ψ j ; α , A , B ] of the function class S H ( m ) of multivalent harmonic functions associated with the ( p ,  q )-derivative operator and subordination function are introduced and investigated. Further, on the one hand, we obtain the necessary and sufficient conditions and the coefficient characterization for the class of ( p ,  q )-Sălăgean multivalent harmonic functions for the subclasses. On the other hand, we also provide the bound estimates of the coefficient a n and b n and distortion theorems and covering theorems as well as the closeness of convex combination for these harmonic function classes.

In this paper, we conduct a research on a new version of the (p,q)-Hermite–Hadamard inequality for convex functions in the framework of postquantum calculus. Moreover, we derive several estimates for (p,q)-midpoint and (p,q)-trapezoidal inequalities for special (p,q)-differentiable functions by using the notions of left and right (p,q)-derivatives. Our newly obtained inequalities are extensions of some existing inequalities in other studies. Lastly, we consider some mathematical examples for some (p,q)-functions to confirm the correctness of newly established results.

In this paper, the new concepts of (p,q)-difference operators are introduced. The properties of fractional (p,q)-calculus in the sense of a (p,q)-difference operator are introduced and developed.

The main goal of this paper is to establish some error bounds for Maclaurin’s formula which is three point quadrature formula using the notions of q-calculus. For this, we first prove a q-integral identity involving fist time q-differentiable functions. Then, by using the new established identity we find the error bounds for maclaurin’s formula by using the convexity of fist time q-differentiable functions. It is also shown that the newly established inequalities are extension of some existing inequalities inside the literature.

In this paper, we establish some new Milne’s type inequalities for the differentiable convex functions in quantum calculus (q-calculus). We prove q-integral identity first and then we prove some new Milne’s type inequalities for q-differentiable convex functions. These inequalities play an important role in Open-Newton’s Cotes formulas. Furthermore, we give the computational analysis of these inequalities for convex functions and prove that the bounds of this paper are better than the existing ones. Ultimately, we provide some mathematical examples to show the validity of newly establish inequalities in q-calculus.

Based on the fundamental concepts of ( p , q )-calculus, this paper introduces a key lemma that successfully generalizes the classical Anderson inequality to the ( p , q )-calculus framework. We further investigate the ( p , q )-Anderson-type inequality for s -Breckner convex functions, which extends the classical notion of convexity. Numerical experiments are provided to validate the theoretical results, demonstrating the correctness and applicability of the proposed inequalities. The findings not only enrich the theoretical system of ( p , q )-calculus but also offer new analytical tools for studying generalized convexity and related integral inequalities.