Some Companions of Gruss Inequality in Inner Product Spaces

Combined dynamic Gruss inequalities on time scales

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.

Chebyshev-Grüss-type inequalities via discrete oscillations

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…

Inegalităţi de tip Chebyshev-Grüss pentru operatorii Bernstein-Euler-Jacobi

The classical form of Grüss' inequality was first published by G. Grüss in 1935 and gives an estimate of the difference between the integral of the product and the product of the integrals of two functions. After that many variants of this inequality appeared in the literature. The aim of this paper is to consider some Chebyshev-Grüss-type inequalities and apply them to Bernstein-Euler-Jacobi (BEJ) operators of first and second kind. First and second moments of the operators are used to explain the situation.

Advances in Inequalities of the Schwarz, Gruss and Bessel Type in Inner Product Spaces

The main aim of this monograph is to survey some recent results obtained by the author related to reverses of the Schwarz, triangle and Bessel inequalities. Some Gruss' type inequalities for orthonormal families of vectors in real or complex inner product spaces are presented as well. Generalizations of the Boas-Bellman, Bombieri, Selberg, Heilbronn and Pecaric inequalities for finite sequences of vectors that are not necessarily orthogonal are also provided. Two extensions of the celebrated Ostrowski's inequalities for sequences or real numbers and the generalization of Wagner's inequality in…

Gruss inequality for some types of positive linear maps

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…

内积空间中 Gruss 型离散不等式再探

Gruss Type Discrete Inequalities in Inner Product Spaces, Revisited

本文重新探讨内积空间中的 Gruss 型离散不等式，得到尖锐结果并应用于凸函数的 Jensen 不等式。

Landau and Gruss type inequalities for inner product type integral transformers in norm ideals

For a probability measure $μ$ and for square integrable fields $(\mathscr{A}_t)$ and $(\mathscr{B}_t)$ ($t\inΩ$) of commuting normal operators we prove Landau type inequality \llu\int_Ω\mathscr{A}_tX\mathscr{B}_tdμ(t)- \int_Ω\mathscr{A}_t\,dμ(t)X \int_Ω\mathscr{B}_t\,dμ(t) \rru \le \llu \sqrt{\,\int_Ω|\mathscr{A}_t|^2\dt-|\int_Ω\mathscr{A}_t\dt|^2}X \sqrt{\,\int_Ω|\mathscr{B}_t|^2 \dt-|\int_Ω\mathscr{B}_t\dt|^2} \rru for all $X\in\mathcalb{B}(\mathcal{H})$ and for all unitarily invariant norms $\lluo\cdot\rruo$. For Schatten $p$-norms similar inequalities are given for arbitrary double square …

On Pompeiu-Chebyshev functional and its generalization

In this work, a generalization of Chebyshev functional is presented. New inequalities of Gruss type via Pompeiu's mean value theorem are established. Improvements of some old inequalities are proved. A generalization of pre-Gruss inequality is elaborated. Some remarks to further generalization of Chebyshev functional are presented. As applications, bounds for the reverse of CBS inequality are deduced. Hardy type inequalities on bounded real interval [a,b] under some other circumstances are introduced. Other related ramified inequalities for differentiable functions are also given.

A refinement of the companion of Ostrowski inequality for functions of bounded variation and applications

In this paper we establish a refinement of the companion of Ostrowski inequality for functions of bounded variation. Applications for the trapezoid inequality, the mid-point inequality, and to probability density functions are also given.

Some Companions of Ostrowski type inequality for s-convex and s-concave functions with applications

In this paper, we obtain some companions of Ostrowski type inequality for absolutely continuous functions whose second derivatives absolute value are s-convex and s-concave.

A companion of Ostrowski like inequality for mappings whose second derivatives belong to $L^{\infty}$ spaces and applications

A companion of Ostrowski like inequality for mappings whose second derivatives belong to $L^{\infty}$ spaces is established. Applications to composite quadrature rules, and to probability density functions are also given.

Some Companions of Ostrowski type inequality for functions whose second derivatives are convex and concave with applications

In this paper, we obtain some companions of Ostrowski type inequality for absolutely continuous functions whose second derivatives absolute value are convex and concave.Finally, we gave some applications for special means.

绝对连续函数的 Ostrowski 不等式的一些伴随不等式及其应用

Some Companions of Ostrowski's Inequality for Absolutely Continuous Functions and Applications

本文给出绝对连续函数的 Ostrowski 不等式伴随形式，并应用于复合求积规则与概率密度函数。

Some Bombieri Type Inequalities in Inner Product

Companion results to the Bombieri generalisation of Bessel's inequality in inner product spaces are given.

On the Signed Small Ball Inequality

This paper is a companion to our prior paper arXiv:0705.4619 on the `Small Ball Inequality in All Dimensions.' In it, we address a more restrictive inequality, and obtain a non-trivial, explicit bound, using a single essential estimate from our prior paper. The prior bound was not explicit and much more involved.