Some results on quantum Hahn integral inequalities

Abstract

Abstract In this paper the quantum Hahn difference operator and the quantum Hahn integral operator are defined via the quantum shift operator Φqθ(t)=qt+(1−q)θ $_{\theta }\varPhi _{q}(t)=qt+(1-q)\theta $, t∈[a,b] $t\in [a,b]$, θ=ω/(1−q)+a $\theta = \omega /(1-q)+a$, 0<q<1 $0< q<1$, ω≥0 $\omega \ge 0$. 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.

Keywords

• Hahn difference operator
• Hahn integral operator
• Hahn difference inequalities
• Hermite–Hadamard quantum Hahn integral inequality
• Pólya–Szegö type fractional Hahn integral inequalities
• Grüss–C̆ebyšev type fractional Hahn integral inequality