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

J. L. Cardoso, N. Haque, A. Macedo

Abstract

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.

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

Abstract

Assume that is an interval and a strictly increasing and continuous function with a single fixed point , satisfying for all , where the equality occurs only when . Hamza et al. considered the general quantum operator, when and when . It generalizes the Jackson -derivative operator as well as the Hahn (quantum derivative) operator, . We obtained Grüss type inequalities for its inverse operator, the -integral. Furthermore, we introduced the concept of -Riemann-Stieltjes integral and obtained Grüss type inequalities associated with it.

Paper Structure

This paper contains 18 sections, 17 theorems, 112 equations.

Table of Contents

  1. Introduction
  2. The $\beta$-difference operator and the $\beta$-integral
  3. The $\beta$-difference operator
  4. The $\beta$-integral
  5. Properties of the $\beta$-derivative and of the $\beta$-integral
  6. The fundamental theorem of $\,\beta$-calculus
  7. The $\,\beta$-integration by parts formula
  8. Conditions for the monotony property of the $\,\beta$-integral
  9. The spaces $\,\mathscr{L}_{\beta}^p[a,b]\,$ and $\,L_{\beta}^p[a,b]$
  10. The space $\,\mathscr{L}_{\beta}^p[a,b]$
  11. The space $L_{\beta}^p[a,b]$
  12. $\beta$-Grüss-type inequality
  13. $\beta$-Grüss type inequalities in the context of the Riemann-Stieltjes $\beta$-integral
  14. The Riemann-Stieltjes $\beta$-integral
  15. $\beta$-Grüss type inequality associated with the Riemann-Stieltjes $\beta$-integral
  16. ...and 3 more sections

Key Result

Theorem 3.1

Let $\,f,\,g\,:\,[a,b]\to\mathbb{R}\,$ be absolutely $\,\beta$-integrable functions on $\,[a,b]\,$, and suppose that there exist real numbers $\,m\,$, $\,n\,$, $\,M\,$ and $\,N\,$ such that If $\,a<s_0<b\,$, then and the constant $\,\frac{1}{4}\,$ is the best possible.

Theorems & Definitions (41)

  • Theorem 3.1
  • proof
  • Definition 3.2
  • Proposition 3.3
  • proof
  • Lemma 3.4: $\beta$-Korkine identity
  • proof
  • Proposition 3.5
  • Proposition 3.6
  • proof
  • ...and 31 more