In the present paper, we establish an equivalent form related to a Hilbert-type integral inequality with non-homogeneous kernel and best possible constant factor. We also consider case of homogeneous as well certain operator expressions.

By estimating a weight function, a reverse Hardy-Hilbert-type integral inequality with a best constant factor is obtained. As an application, some equivalent forms and some particular results have been established.

In this paper we introduce a new extension of Hilbert's integral inequality with a best constant factor involving the hypergeometric function. The equivalent form and some examples will be given.

By introducing some parameters and the weight coefficient, one proves a new extension of the Hilbert-type inequality with a best constant factor. The reverse, some equivalent forms and a number of new particular cases are considered.

In this paper, we give a new Hardy-Hilbert’s integral inequality with some parameters and a best constant factor. It includes an overwhelming majority of results of many papers.

In this paper, we obtain two new half-discrete Hilbert inequalities for three variables. The obtained inequalities are with the best constant factor. Moreover, we give their equivalent forms.

We give the rst approximation algorithm for the node-weighted Steiner tree problem. Its performance guarantee is within a constant factor of the best possible unless ~ P NP. Our algorithm generalizes to handle other network design problems.

This paper gives a new multiple extension of Hilbert’s integral inequality with a best constant factor, by introducing a parameter λ and the Γ function. Some particular results are obtained.