Abstract
In this work, two generalized quantum integral identities are proved by using some parameters. By utilizing these equalities, we present several parameterized quantum inequalities for convex mappings. These quantum inequalities generalize many of the important inequalities that exist in the literature, such as quantum trapezoid inequalities, quantum Simpson’s inequalities, and quantum Newton’s inequalities. We also give some new midpoint-type inequalities as special cases. The results in this work naturally generalize the results for the Riemann integral.
Highlights
Preliminaries of q-Calculus and Some InequalitiesWe first present some known definitions and related inequalities in q-calculus
Introduction omasSimpson has developed crucial methods for the numerical integration and estimation of definite integrals considered as Simpson’s rule during 1710–1761
In this work, using quantum integrals, we developed a new extension of quantum trapezoid, quantum Simpson’s, and quantum Newton’s type estimations for quantum differentiable convex functions
Summary
We first present some known definitions and related inequalities in q-calculus. Jackson [30] defined the q-integral of a given function F from 0 to κ2 as follows:. He defined the q-integral of a given function over the interval [κ1, κ2] as follows:. In [11, 21], the authors proved quantum Hermite–Hadamard-type inequalities and their estimations by using the notions of the qκ1-derivative and qκ1-integral. We consider the mapping F: [κ1, κ2] ⟶ R. en, the qκ2 -definite integral on [κ1, κ2] is defined by. We consider the mapping F: [κ1, κ2] ⟶ R. en, the qκ1-derivative of F at x ∈ [κ1, κ2] is defined by the following expression: Theorem 2 (see [32]). E present paper aims to generalize the results proved in [13, 33, 34]. e key benefit of our paper is that it includes multiple inequalities at the same time, such as Simpson’s inequalities, Newton’s inequalities, midpoint inequalities, and trapezoidal inequities
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