Some New Midpoint and Trapezoidal-Type Inequalities for General Convex Functions in q -Calculus

Dafang Zhao, Ghazala Gulshan, Muhammad Aamir Ali and Kamsing Nonlaopon
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Dafang Zhao: School of Mathematics and Statistics, Hubei Normal University, Huangshi 435002, China
Ghazala Gulshan: Department of Mathematics, Faculty of Science, Mirpur University of Science and Technology (MUST), Mirpur 10250, Pakistan
Muhammad Aamir Ali: Jiangsu Key Laboratory for NSLSCS, School of Mathematical Sciences, Nanjing Normal University, Nanjing 210023, China
Kamsing Nonlaopon: Department of Mathematics, Faculty of Science, Khon Kaen University, Khon Kaen 40002, Thailand

Mathematics, 2022, vol. 10, issue 3, 1-14

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.

Keywords: midpoint inequalities; trapezoid inequalities; quantum calculus; ( ? , m )-convex functions (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2022
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