Assume that $\,I\subseteq\mathbb{ R}\,$ is an interval and $\,\beta:\,I\rightarrow\,I\,$ a strictly increasing and continuous function with a single fixed point $\,s_0\in I\,$, satisfying $\,(s_0-t)(\beta(t)-t)\leq 0\,$ for all $\,t\in I$, where the equality occurs only when $\,t=s_0$. Hamza et al. considered the general quantum operator, $\,D_{ \beta}[f](t):=\displaystyle\frac{ f\big(\beta(t)\big)-f(t)}{ \beta(t)-t}\,$ when $\,t\neq s_0\,$ 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 Gr\"uss type inequalities for its inverse operator, the $\beta$-integral. Furthermore, we introduced the concept of $\,\beta$-Riemann-Stieltjes integral and obtained Gr\"uss type inequalities associated with it.
Categories: math.GM