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arXiv · 2410.12838

Gr\"uss inequalities for the $\beta-$integral associated with the general quantum operator

J. L. Cardoso·N. Haque·A. Macedo
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Abstract

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.

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BibTeXRIS

J. L. Cardoso, N. Haque, A. Macedo. 2024-10-07. Gr\"uss inequalities for the $\beta-$integral associated with the general quantum operator. https://arxiv.org/abs/2410.12838

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