New Inequalities of Gruss Type for the Stieltjes Integral and Applications
Sever Silvestru Dragomir
本文给出Stieltjes积分的Gruss型锐不等式，并应用于数值积分。
提供了Stieltjes积分的Gruss型锐不等式，并在数值积分中加以应用。
Sharp inequalitieis of Gruss type for Stieltjes integrals with application in numerical integration are provided.
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Gruss Type Discrete Inequalities in Inner Product Spaces, Revisited
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We prove a more general version of the Gruss inequality by using the recent theory of combined dynamic derivatives on time scales and the more general notions of diamond-alpha derivative and integral. For the particular case when alpha = 1, one gets a delta-integral Gruss inequality on time scales; for alpha = 0 a nabla-integral Gruss inequality. If we further restrict ourselves by fixing the time scale to the real (or integer) numbers, then the standard continuous (discrete) inequalities are obtained.
The classical form of Grüss' inequality, first published by G. Grüss in 1935, gives an estimate of the difference between the integral of the product and the product of the integrals of two functions. In the subsequent years, many variants of this inequality appeared in the literature. The aim of this paper is to introduce a different approach, presenting a new Chebyshev-Grüss-type inequality and applying it to different well-known linear, not necessarily positive, operators. Some conjectures are presented as well. We also compare the new inequalities with some older results. This new approach…
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The concept of Riemann-Stieltjes integral $\int_a^b {f\left( t \right)du\left( t \right)}$; where $f$ is called the integrand, $u$ is called the integrator, plays an important role in Mathematics. The approximation problem of the Riemann-Stieltjes integral $\int_a^b {f\left( t \right)du\left( t \right)}$ in terms of the Riemann-Stieltjes sums have been considered recently by many authors. However, a small attention and a few works have been considered for mappings of two variables; i.e., The approximation problem of the Riemann-Stieltjes double integral $\int_a^b {\int_c^d {f\left( {t,s} \righ…
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该研究从初等测度论解读β - 格鲁斯型不等式，将正β - 积分视为特定测度积分并归一化，利用概率空间标准事实得出相关不等式，确定了构成此类不等式族的测度论机制。
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Assuming a unitarily invariant norm $|||\cdot|||$ is given on a two-sided ideal of bounded linear operators acting on a separable Hilbert space, it induces some unitarily invariant norms $|||\cdot|||$ on matrix algebras $\mathcal{M}_n$ for all finite values of $n$ via $|||A|||=|||A\oplus 0|||$. We show that if $\mathscr{A}$ is a $C^*$-algebra of finite dimension $k$ and $Φ: \mathscr{A} \to \mathcal{M}_n$ is a unital completely positive map, then \begin{equation*} |||Φ(AB)-Φ(A)Φ(B)||| \leq \frac{1}{4} |||I_{n}|||\,|||I_{kn}||| d_A d_B \end{equation*} for any $A,B \in \mathscr{A}$, where $d_X$ d…
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