Abstract
The main objective of this study is to establish two important right q-integral equalities involving a right-quantum derivative with parameter m∈[0,1]. Then, utilizing these equalities, we derive some new variants for midpoint- and trapezoid-type inequalities for the right-quantum integral via differentiable (α,m)-convex functions. The fundamental benefit of these inequalities is that they may be transformed into q-midpoint- and q-trapezoid-type inequalities for convex functions, classical midpoint inequalities for convex functions and classical trapezoid-type inequalities for convex functions are transformed without having to prove each one independently. In addition, we present some applications of our results to special means of positive real numbers. It is expected that the ideas and techniques may stimulate further research in this field.
Highlights
It is well known that modern investigation, directly or indirectly, involves the applications of convexity
Inspired by the ongoing studies, we derive some new inequalities of midpoint and trapezoid type inequalities for (α, m)-convex functions by utilizing quantum calculus
Under the conditions of Lemma 2, if |y2 Dq F| is (α, m)-convex function over [y1, y2 ], we find the following midpoint type inequality:
Summary
It is well known that modern investigation, directly or indirectly, involves the applications of convexity. Convex functions are powerful tools for proving a large class of inequalities. In [15], Alp et al proved the following version of the quantum. Hermite–Hadamard type for convex functions using the left-quantum integrals:. Bermudo et al [16] used the right-quantum integrals and proved the following variant of the Hermite–Hadamard type inequality for convex functions: y1 + qy. In [26,27,28,29], the authors used convexity and coordinated convexity to prove Simpson’s and Newton’s type inequalities via q-calculus. Inspired by the ongoing studies, we derive some new inequalities of midpoint and trapezoid type inequalities for (α, m)-convex functions by utilizing quantum calculus.
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