Abstract
In this paper, using the notions of qκ2-quantum integral and qκ2-quantum derivative, we present some new identities that enable us to obtain new quantum Simpson’s and quantum Newton’s type inequalities for quantum differentiable convex functions. This paper, in particular, generalizes and expands previous findings in the field of quantum and classical integral inequalities obtained by various authors.
Highlights
Thomas Simpson developed crucial techniques for numerical integration and estimation of definite integrals, which became known as Simpson’s law during his lifetime (1710–1761)
Using Formula (2), from the definition of the function Λ(s), we find that
Using Formula (2), from the definition of the function ∆(s), we find that
Summary
We first present the definitions and some properties of quantum derivatives and quantum integrals. Tariboon and Ntouyas defined the following qκ1 -derivative and qκ1 -integral: Definition 1 ([13]). Alp et al proved quantum Hermite–Hadamard inequalities for qκ1 -integrals by utilizing the convex functions, as follows: Theorem 2 ([21]). The qκ2 -definite integral of mapping F : [κ1 , κ2 ] → R on [κ1 , κ2 ] is defined as:. Bermudo et al proved the corresponding quantum Hermite–Hadamard inequalities for the qκ2 -integral: Theorem 3 ([14]). For a convex mapping F : [κ1 , κ2 ] → R that is differentiable on [κ1 , κ2 ], the following inequality holds: κ1 + qκ.
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