Post-Quantum Midpoint-Type Inequalities Associated with Twice-Differentiable Functions

Thanin Sitthiwirattham,Muhammad Aamir Ali,Chanon Promsakon,Ifra Bashir Sial,Ghulam Murtaza,Praveen Agarwal

doi:10.3390/axioms11020046

Abstract

In this study, first we establish a p,q-integral identity involving the second p,q-derivative, and then, we use this result to prove some new midpoint-type inequalities for twice-p,q-differentiable convex functions. It is also shown that the newly established results are the refinements of the comparable results in the literature.

Highlights

Twice-Differentiable Functions .In convex functions theory, the Hermite–Hadamard (HH) inequality is very important, which was discovered by C
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Physicists make up the majority of scientists who utilize q-calculus today

Summary

Introduction

For generalized quasi-convex functions, Nwaeze et al proved certain parameterized quantum integral inequalities in [24]. Budak et al [26], Ali et al [27], and Vivas-Cortez et al [28] developed new quantum Simpson and quantum Newton-type inequalities for convex and coordinated convex functions. Kunt et al [31] generalized the results of [19] and proved HH-type inequalities and their left estimates using the π1 D p,q -difference operator and ( p, q)π1 -integral. Generalized the results of [12] and proved the HH-type inequalities and their left estimates using the π2 D p,q -difference operator and ( p, q)π2 -integral. Inspired by the ongoing studies, we use the ( p, q)-integral to develop some new postquantum midpoint-type inequalities for ( p, q)-differentiable convex functions.

Quantum Derivatives and Integrals

Post-Quantum Derivatives and Integrals

Post-Quantum Midpoint-Type Inequalities

Concluding Remarks