Some results on quantum Hahn integral inequalities

Suphawat Asawasamrit,Jessada Tariboon,Chayapat Sudprasert,Sotiris K Ntouyas

doi:10.1186/s13660-019-2101-z

Abstract

In this paper the quantum Hahn difference operator and the quantum Hahn integral operator are defined via the quantum shift operator _{theta }varPhi _{q}(t)=qt+(1-q)theta , tin [a,b], theta = omega /(1-q)+a, 0< q<1, omega ge 0. Some new fractional integral inequalities are established by using the quantum Hahn integral for one and two functions bounded by quantum integrable functions. The Hermite–Hadamard type of ordinary and fractional quantum Hahn integral inequalities as well as the Pólya–Szegö type fractional Hahn integral inequalities and the Grüss–C̆ebyšev type fractional Hahn integral inequality are also presented.

Highlights

Introduction and preliminariesLet be f defined on an interval I ⊆R containing ω0 := ω 1–q TheHahn difference operatorDq,ω, introduced in [1], is defined as ⎧ Dq,ω f (t) =
Before going to state the definitions of fractional quantum Hahn calculus on an interval [a, b], we should introduce the θ -power function which is defined by k–1 (n – m)(θ0) = 1, (n – m)(θk) = n – θ Φqi (m), k ∈ N ∪ {∞}
In this paper we prove some new quantum Hahn fractional integral inequalities by using the quantum Hahn integral

Summary

Introduction

Before going to state the definitions of fractional quantum Hahn calculus on an interval [a, b], we should introduce the θ -power function which is defined by k–1 (n – m)(θ0) = 1, (n – m)(θk) = n – θ Φqi (m) , k ∈ N ∪ {∞}. Definition 3 The fractional quantum Hahn difference of Riemann–Liouville type of a function f : [a, b] → R of order α ≥ 0 is defined by (aD0q,ωf )(t) = f (t) and a Dαq,ω f (t) The fractional quantum Hahn integral of Riemann–Liouville type is defined by (aIq0,ωf )(t) = f (t) and a Iqα,ω f

Results

Conclusion