Abstract
In this paper, we establish a new (p,q)b-integral identity involving the first-order (p,q)b-derivative. Then, we use this result to prove some new (p,q)b-integral inequalities related to Hermite–Hadamard inequalities for (p,q)b-differentiable convex functions. Furthermore, our main results are used to study some special cases of various integral inequalities. The newly presented results are proven to be generalizations of some integral inequalities of already published results. Finally, some examples are given to illustrate the investigated results.
Highlights
In mathematics, the study of calculus without limits is called quantum calculus, and was first studied by Euler (1707–1783), introducing the number in the q-infinite series defined by Newton
In 2021, Li et al [46] presented a new generalization of qb-integral inequalities related to Hermite–Hadamard inequalities for qb-differentiable convex functions
Inspired by the above-mentioned literature, we propose establishing a new generalization of (p, q)b-integral inequalities related to Hermite–Hadamard inequalities for (p, q)b-differentiable convex functions to extend and generalize the results given in the above-mentioned literature
Summary
The study of calculus without limits is called quantum calculus (briefly called q-calculus), and was first studied by Euler (1707–1783), introducing the number in the q-infinite series defined by Newton ( called Newton’s infinite series). Q-calculus has had numerous applications in various disciplines of physics and mathematics; see [2–10] and the references cited therein for more details. Hermite–Hadamard inequalities have been studied by using q-calculus for convex functions by many researchers; see [12,19–25] and the references cited therein for more details. In 2021, Li et al [46] presented a new generalization of qb-integral inequalities related to Hermite–Hadamard inequalities for qb-differentiable convex functions. Inspired by the above-mentioned literature, we propose establishing a new generalization of (p, q)b-integral inequalities related to Hermite–Hadamard inequalities for (p, q)b-differentiable convex functions to extend and generalize the results given in the above-mentioned literature.
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