Grüss inequalities for the $β-$integral associated with the general quantum operator
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arXiv
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866914975536119808 |
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| author | Cardoso, J. L. Haque, N. Macedo, A. |
| author_facet | Cardoso, J. L. Haque, N. Macedo, A. |
| contents | Assume that $\,I\subseteq\mathbb{R}\,$ is an interval and $\,β:\,I\rightarrow\,I\,$ a strictly increasing and continuous function with a single fixed point $\,s_0\in I\,$, satisfying $\,(s_0-t)(β(t)-t)\leq 0\,$ for all $\,t\in I$, where the equality occurs only when $\,t=s_0$. Hamza et al. considered the general quantum operator, $\,D_β[f](t):=\displaystyle\frac{f\big(β(t)\big)-f(t)}{β(t)-t}\,$ when $\,t\neq s_0\,$ and $\,D_β[f](s_0):=f^{\prime}(s_0)\,$ when $\,t=s_0\,$. It generalizes the Jackson $\,q$-derivative operator $\,D_{q}\,$ as well as the Hahn (quantum derivative) operator, $\,D_{q,ω}$. We obtained Grüss type inequalities for its inverse operator, the $β$-integral. Furthermore, we introduced the concept of $\,β$-Riemann-Stieltjes integral and obtained Grüss type inequalities associated with it. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2410_12838 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Grüss inequalities for the $β-$integral associated with the general quantum operator Cardoso, J. L. Haque, N. Macedo, A. General Mathematics Primary 26D15, Secondary 26A42 Assume that $\,I\subseteq\mathbb{R}\,$ is an interval and $\,β:\,I\rightarrow\,I\,$ a strictly increasing and continuous function with a single fixed point $\,s_0\in I\,$, satisfying $\,(s_0-t)(β(t)-t)\leq 0\,$ for all $\,t\in I$, where the equality occurs only when $\,t=s_0$. Hamza et al. considered the general quantum operator, $\,D_β[f](t):=\displaystyle\frac{f\big(β(t)\big)-f(t)}{β(t)-t}\,$ when $\,t\neq s_0\,$ and $\,D_β[f](s_0):=f^{\prime}(s_0)\,$ when $\,t=s_0\,$. It generalizes the Jackson $\,q$-derivative operator $\,D_{q}\,$ as well as the Hahn (quantum derivative) operator, $\,D_{q,ω}$. We obtained Grüss type inequalities for its inverse operator, the $β$-integral. Furthermore, we introduced the concept of $\,β$-Riemann-Stieltjes integral and obtained Grüss type inequalities associated with it. |
| title | Grüss inequalities for the $β-$integral associated with the general quantum operator |
| topic | General Mathematics Primary 26D15, Secondary 26A42 |
| url | https://arxiv.org/abs/2410.12838 |