On properties of inner product type integral transformers

In this paper, we give characterizations of certain properties of inner product type integral transformers. We first consider unitarily invariant norms and operator valued functions. We then give results on norm inequalities for inner product type integral transformers in terms of Landau inequality, Grüss inequality. Lastly, we explore some of the applications in quantum theory.

Two Alternative Proofs of the Grüss Inequality

The classical Grüs inequality has spurred a range of improvements, generalizations, and extensions. In this article, we provide new functional bounds that ultimately lead to two elementary proofs of the inequality that might be of interest. Our results are motivated by the extreme cases where the equality is reached, namely step functions of equal support. Our first proof is based on the standard Cauchy-Schwarz inequality and a simple bound on the variance of a function. Its simplicity would be of particular interest to those who are new to the study of functional inequalities. Our second proof utilizes non-intuitive and novel bounds on functionals defined on L∞(0, 1). As a result, we provide a detailed and new insight into the nature of the Grüss inequality.

A refinement of Grüss inequality for the complex integral

Assume that f and g are continuous on γ, γ ⊂ 𝔺 is a piecewise smooth path parametrized by z (t), t ∈ [a, b] from z (a) = u to z (b) = w with w ≠ u and the complex Čebyšev functional is defined by{{\cal D}_\gamma}\left({f,g} \right): = {1 \over {w - u}}\int_\gamma {f\left(z \right)} g\left(z \right)dz - {1 \over {w - u}}\int_\gamma {f\left(z \right)} dz{1 \over {w - u}}\int_\gamma {g\left(z \right)} dz.In this paper we establish some Grüss type inequalities for 𝒟 (f, g) under some complex boundedness conditions for the functions f and g.

Some Refinements of Ostrowski’s Inequality and an Extension to a 2-Inner Product Space

The purpose of this paper is to prove certain refinements of Ostrowski’s inequality in an inner product space. We study extensions of Ostrowski type inequalities in a 2-inner product space. Finally, some applications which are related to the Chebyshev function and the Grüss inequality are presented.