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This paper introduces and establishes new Hermite‐Hadamard type inequalities specifically tailored for m$$ m $$‐preinvex functions within the (p,q)$$ \left(p,q\right) $$‐calculus framework. These newly developed inequalities come with accompanying left‐right estimates, which enhance their practical utility. The primary objective of this research is to investigate the properties of (p,q)$$ \left(p,q\right) $$‐differentiable m$$ m $$‐preinvex functions and derive inequalities that extend and generalize existing results in the domain of integral inequalities. The techniques employed in this study hold broader implications, finding relevance in various fields where symmetry is paramount. The findings presented in this paper make a significant contribution to the field of analytic inequalities, offering valuable insights into the behavior and characteristics of m$$ m $$‐preinvex functions. Moreover, the established results demonstrate the wider applicability and generalization of analogous findings from prior literature. The techniques and inequalities introduced herein pave the way for further exploration and research in the realm of integral inequalities. [ABSTRACT FROM AUTHOR]

A new (p , q) -integral identity involving left and right post quantum derivatives, by using three times (p , q) -differentiable functions is established and then this identity is used to derive several new post-quantum Ostrowski type integral inequalities for three times (p , q) -differentiable functions. These results are generalizations of corresponding results in the area of integral inequalities. [ABSTRACT FROM AUTHOR]

A new generalized (p,q)-integral identity is derived. Using this new identity as an auxiliary result, we derive new parametric generalizations of certain integral inequalities using the class of s-preinvex functions. We discuss several new and known special cases of the obtained results. This shows that our results are quite unifying. To demonstrate the significance of the main results, we also present some interesting applications.

In this paper, we prove some new Simpson's type inequalities for partial (p,q)-differentiable convex functions of two variables in the context of (p,q)-calculus. We also show that the findings in this paper are generalizations of comparable findings in the literature.

The main objective of this paper is to derive some new post quantum analogues of Dragomir–Agarwal and Iyengar type integral inequalities essentially by using the strongly φ-preinvexity and strongly quasi φ-preinvexity properties of the mappings, respectively. We also discuss several new special cases which show that the results obtained are quite unifying. In order to illustrate the efficiency of our main results, some applications regarding (p,q)-differentiable mappings that are in absolute value bounded are given.

This paper deals with Lupaş post quantum Bernstein operators over arbitrary closed and bounded interval constructed with the help of Lupaş post quantum Bernstein bases. Due to the property that these bases are scale invariant and translation invariant, the derived results on arbitrary intervals are important from computational point of view. Approximation properties of Lupaş post quantum Bernstein operators on arbitrary compact intervals based on Korovkin type theorem are studied. More general situation along all possible cases have been discussed favouring convergence of sequence of Lupaş post quantum Bernstein operators to any continuous function defined on compact interval. Rate of convergence by modulus of continuity and functions of Lipschitz class are computed. Graphical analysis has been presented with the help of MATLAB to demonstrate approximation of continuous functions by Lupaş post quantum Bernstein operators on different compact intervals.

A new generalized (p; q)-integral identity is derived. Using this new identity as an auxiliary result, we derive new parametric generalizations of certain integral inequalities using the class of spreinvex functions. We discuss several new and known special cases of the obtained results. This shows that our results are quite unifying. To demonstrate the significance of the main results, we also present some interesting applications. [ABSTRACT FROM AUTHOR]

The quantum calculus has emerged as a connection between mathematics and physics. It has wide applications, particularly in quantum mechanics, analytic number theory, combinatorial analysis, operation theory etc. The q-calculus, which serves as a powerful tool to model quantum computing, has drawn attention of many researchers in the field of special functions and as a result the q-analogues of certain special functions, especially hypergeometric function, 1-variable Hermite polynomials, Appell polynomials etc., are introduced and studied. In this paper, we introduce the 2-variable q-Hermite polynomials by means of generating function. Also, its certain properties like series definition, recurrence relations, q-differential equation and summation formulas are established. The operational definition and some integral representations of these polynomials are obtained. Some examples are also considerd to show the efficacy of the proposed method. Some concluding remarks are given. At the end of this paper, the graphical representations of these polynomials of certain degrees with specified values of q are given.

A new (p,q)-integral identity involving left and right post quantum derivatives, by using three times (p,q)-differentiable functions is established and then this identity is used to derive several new post-quantum Ostrowski type integral inequalities for three times (p,q)-differentiable functions. These results are generalizations of corresponding results in the area of integral inequalities.

In this work, first, we consider novel parameterized identities for the left and right part of the (p,q)-analogue of Hermite–Hadamard inequality. Second, using these new parameterized identities, we give new parameterized (p,q)-trapezoid and parameterized (p,q)-midpoint type integral inequalities via η-quasiconvex function. By changing values of parameter μ∈[0,1], some new special cases from the main results are obtained and some known results are recaptured as well. Finally, at the end, an application to special means is given as well. This new research has the potential to establish new boundaries in comparative literature and some well-known implications. From an application perspective, the proposed research on the η-quasiconvex function has interesting results that illustrate the applicability and superiority of the results obtained.

In this paper, we establish some new Hermite–Hadamard type inequalities for preinvex functions and left-right estimates of newly established inequalities for p,q-differentiable preinvex functions in the context of p,q-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 work, first, we consider novel parameterized identities for the left and right part of the (p , q) -analogue of Hermite–Hadamard inequality. Second, using these new parameterized identities, we give new parameterized (p , q) -trapezoid and parameterized (p , q) -midpoint type integral inequalities via η -quasiconvex function. By changing values of parameter μ ∈ [ 0 , 1 ] , some new special cases from the main results are obtained and some known results are recaptured as well. Finally, at the end, an application to special means is given as well. This new research has the potential to establish new boundaries in comparative literature and some well-known implications. From an application perspective, the proposed research on the η -quasiconvex function has interesting results that illustrate the applicability and superiority of the results obtained. [ABSTRACT FROM AUTHOR]

In this paper, we establish some new Hermite–Hadamard type inequalities for preinvex functions and left-right estimates of newly established inequalities for p , q -differentiable preinvex functions in the context of p , q -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. [ABSTRACT FROM AUTHOR]