Abstract
In this paper, we apply (p,q)-calculus to establish some new Chebyshev-type integral inequalities for synchronous functions. In particular, we generalize results of quantum Chebyshev-type integral inequalities by using (p,q)-integral. By taking p=1 and q→1, our results reduce to classical results on Chebyshev-type inequalities for synchronous functions. Furthermore, we consider their relevance with other related known results.
Highlights
Let f, g : [ a, b] → R be integrable functions and φ, φ : [ a, b] → (0, ∞) be integrable functions
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Motivated by the results mentioned above, by using the four parameters of deformation ( p1, q1 ) and ( p2, q2 ), we propose generalizing and extending some new Chebyshev inequalities in q-integral to ( p, q)-integral
Summary
Let f , g : [ a, b] → R be integrable functions and φ, φ : [ a, b] → (0, ∞) be integrable functions. Aslam et al obtained quantum Ostrowski inequalities for the q-differentiable convex function. Yang [22] obtained some new Chebyshev-type quantum integral inequalities on finite intervals. H. Kalsoom et al [30] obtained ( p, q)-estimates of Hermite–Hadamard-type inequalities for coordinated convex and quasi-convex functions. Motivated by the results mentioned above, by using the four parameters of deformation ( p1 , q1 ) and ( p2 , q2 ), we propose generalizing and extending some new Chebyshev inequalities in q-integral to ( p, q)-integral. We obtain their relevance with other related known results. We hope that the ideas and techniques presented in this paper will inspire interested readers working in this field
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