Abstract
In this article, we use quantum integrals to derive Hermite–Hadamard inequalities for preinvex functions and demonstrate their validity with mathematical examples. We use the qϰ2-quantum integral to show midpoint and trapezoidal inequalities for qϰ2-differentiable preinvex functions. Furthermore, we demonstrate with an example that the previously proved Hermite–Hadamard-type inequality for preinvex functions via qϰ1-quantum integral is not valid for preinvex functions, and we present its proper form. We use qϰ1-quantum integrals to show midpoint inequalities for qϰ1-differentiable preinvex functions. It is also demonstrated that by considering the limit q→1− and ηϰ2,ϰ1=−ηϰ1,ϰ2=ϰ2−ϰ1 in the newly derived results, the newly proved findings can be turned into certain known results.
Highlights
In the theory of convexity, the Hermite–Hadamard inequality is a well-established inequality with many applications and geometrical interpretations
This inequality states that if a function φ : I ⊆ R → R is convex, for κ1, κ2 ∈ I with κ1 < κ2, we have the following: Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations
The main objective of this paper is to prove the Hermite–Hadamard inequality for qκ2 -integrals and find its left and right side estimates
Summary
Several research studies were recently carried out on the subject of q-analysis, beginning with Euler, due to a large need for mathematics that models quantum computing q-calculus, occurring for the interaction between physics and mathematics It has a wide range of applications in mathematics, including combinatorics, number theory, basic hypergeometric functions, orthogonal polynomials, and other disciplines, as well as mechanics, relativity theory, and quantum theory [22,23,24,25]. For generalized quasi-convex functions, Nwaeze et al proved certain parameterized quantum integral inequalities in [39]. We believe that our work’s viewpoint and methodology may stimulate additional study in this field
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