Grüss inequalities for the $\beta -$integral associated with the general quantum operator

Abstract

Assume that $I\subseteq \mathbb{R}$ is an interval and $\beta :I\to I$ a strictly increasing and continuous function with a single fixed point $s_0\in I$, satisfying $(s_0-t)(\beta (t)-t)\le 0$ for all $t\in I$, where the equality occurs only when $t=s_0$. Hamza et al. considered the general quantum operator, $D_{\beta }[f](t):=\frac{f(\beta (t))-f(t)}{\beta (t)-t}$ when $t\ne s_0$ and $D_{\beta }[f](s_0):=f^{\prime }(s_0)$ when $t=s_0$. It generalizes the Jackson $q$-derivative operator $D_q$ as well as the Hahn (quantum derivative) operator, $D_{q,\omega }$. We obtained Grüss type inequalities for its inverse operator, the $\beta$-integral. Furthermore, we introduced the concept of $\beta$-Riemann-Stieltjes integral and obtained Grüss type inequalities associated with it.