Abstract:
We establish a new inequality of Hilbert type for a finite double number of nonnegative sequences of real numbers and some interrelated results, which are inverse and general forms of Pachpatte's and Handley's results. An integral version and some interrelated results are also obtained. These results provide some new estimates on such types of inequalities.

Abstract:
We establish some multiple integral Hilbert-Pachpatte-type inequalities. As applications, we get some inverse forms of Pachpatte's inequalities which were established in 1998.

Abstract:
In this paper some necessary and sufficient conditions are given for the Hilbert’s type operators to be bounded on the Herz spaces. The corresponding new operator norm inequalities are obtained.

Abstract:
In this paper, some new generalizations of Hilbert-Pachpatte type inequalities are given by introducing some parameters $ r_i,iin{1,2,ldots,n}$.

Abstract:
The main aim of this paper is to establish two new multidimensional integral inequalities similar to the integral analogue of the well known Hilbert's inequality by using elementary analysis.

Abstract:
In this paper, by using hardy inequality, we establish some new integral inequalities of Hardy-Hilbert type with general kernel. As applications, equivalent forms and some particular results are built; the corresponding to the double series inequalities are given.reverse forms are considered also.

Abstract:
We establish several operator extensions of the Chebyshev inequality. The main version deals with the Hadamard product of Hilbert space operators. More precisely, we prove that if $\mathscr{A}$ is a $C^*$-algebra, $T$ is a compact Hausdorff space equipped with a Radon measure $\mu$, $\alpha: T\rightarrow [0, +\infty)$ is a measurable function and $(A_t)_{t\in T}, (B_t)_{t\in T}$ are suitable continuous fields of operators in ${\mathscr A}$ having the synchronous Hadamard property, then \begin{align*} \int_{T} \alpha(s) d\mu(s)\int_{T}\alpha(t)(A_t\circ B_t) d\mu(t)\geq\left(\int_{T}\alpha(t) A_t d\mu(t)\right)\circ\left(\int_{T}\alpha(s) B_s d\mu(s)\right). \end{align*} We apply states on $C^*$-algebras to obtain some versions related to synchronous functions. We also present some Chebyshev type inequalities involving the singular values of positive $n\times n$ matrices. Several applications are given as well.