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The norm of a bounded self-adjoint operator $T:$ $l^{2}\rightarrow $ $l^{2}$ is considered. As applications, a new bilinear inequality with a best constant factor and some Hilbert's type inequalities are built.

This paper deals with a new Hilbert- type inequality by introducing a parameter and the Beta function. As applications, the equivalent form, the reversions and some particular results are considered. All the theorems provide some new estimates on this type of inequalities.

In this paper, we prove that the constant factor in the Hardy-Carleman’s type inequality is the best possible. A related integral inequality with a best constant factor is considered.

By introducing a parameter $\lambda \in [0,1]$, we give an inequality relating Carleman's inequality with Van der Corput's inequality. In particular, a generalization of Carleman's inequality with a best constant factor $e^{\frac{1}{1-\lambda}}$, $\lambda \in [0,1)$ is considered.

The main objective of this paper is a study of some new generalizations of Hilbert's and Hardy-Hilbert's type inequalities. We establish general form of multiple Hilbert-type inequality and we also introduce multiple inequality of Hardy-Hilbert type. Further, the best possible constants are obtained for some general cases.

Some character of the symmetric homogenous kernel of 1-order in Hilbert-type operator is obtained. Two equivalent inequalities with the symmetric homogenous kernel of -order are given. As applications, some new Hilbert-type inequalities with the best constant factors and the equivalent forms as the particular cases are established.

Using the way of weight functions and the idea of introducing parameters, and by means of Hermite-Hadamard’s inequality, a more accurate reverse half-discrete Hilbert-type inequality with the non-homogeneous kernel and a best constant factor is established. In addition, its best extension with parameters, the equivalent forms, as well as some particular cases are given.

By means of the way of weight coefficients and technique of real analysis, an extension of a Hardy-Hilbert-type inequality with parameters and a best possible constant factor is given. The equivalent forms, the operator expression with the norm, the reverses and some particular cases are also considered.

By applying weight functions and technique of real analysis, a half-discrete Hardy-Hilbert-type inequality related to the kernel of hyperbolic secant function and a best possible constant factor are given. The equivalent forms, the operator expressions with the norm, the reverses, and some particular cases are also considered.

In this paper, by the use of the way of weight coefficients, the transfer formula, and the technique of real analysis, we introduce some proper parameters and obtain a multidimensional Hilbert-type inequality with the following kernel: $$ \prod_{k=1}^{s}\frac{(\min\{\|m\|_{\alpha},c_{k}\|n\|_{\beta}\} )^{\frac{\gamma}{s}}}{(\max\{\|m\|_{\alpha},c_{k}\|n\|_{\beta}\})^{\frac {\lambda +\gamma}{s}}} $$ and a best possible constant factor. The equivalent form, the operator expressions with the norm, and some particular cases are also considered. The lemmas and theorems provide an extensive account of this type of inequalities.

By means of weight functions and Hermite-Hadamard’s inequality, a new half-discrete Mulholland-type inequality with a best constant factor is given. A best extension with multi-parameters, some equivalent forms, the operator expressions as well as some particular cases are considered.

By means of weight coefficients and techniques of real analysis, a new Hardy-Hilbert-type inequality with multiparameters and the best possible constant factor is given. The equivalent forms, the operator expression with the norm, and the reverse and some particular inequalities with the best possible constant factors are also considered.

By means of weight functions and the improved Euler-Maclaurin summation formula, a more accurate half-discrete Hilbert-type inequality with a non-homogeneous kernel and a best constant factor is given. A best extension, some equivalent forms, the operator expressions as well as some particular cases are also considered.

By using the way of weight functions and Hadamard’s inequality, a half-discrete Hilbert-type inequality similar to Mulholland’s inequality with a best constant factor is given. The extension with multi-parameters, the equivalent forms as well as the operator expressions are also considered.

By using the way of weight functions and the technic of real analysis, a multiple Hilbert-type integral inequality with a non-homogeneous kernel is given. The operator expression with the norm, the reverses and some examples with the particular kernels are also considered. MSC:47A07, 26D15.

By using the way of weight coefficients and the idea of introducing parameters and by means of Hadamard's inequality, we give a more accurate half-discrete Hilbert's inequality with a best constant factor. We also consider its best extension with parameters, the equivalent forms, the operator expressions as well as some reverses.

Using the way of weight functions and the technique of real analysis, a half-discrete Hilbert-type inequality with a general homogeneous kernel is obtained, and a best extension with two interval variables is given. The equivalent forms, the operator expressions, the reverses and some particular cases are considered. 2000 Mathematics Subject Classification: 26D15; 47A07.

By using the way of weight functions and the technique of real analysis, a new Hilbert-type integral inequality with the non-homogeneous kernel in the whole plane with the best constant factor is given. As applications, the equivalent inequalities with the best constant factors, the reverses and some particular cases are obtained.

By using the way of weight functions and the idea of introducing parameters and by means of Hadamard’s inequality, we give a more accurate half-discrete Hilbert-type inequality with a best constant factor. We also consider its best extension with parameters, equivalent forms, operator expressions as well as some reverses.