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.

This study shows that a refinement of the Hilbert inequality for double series can be established by introducing a real function u x and a parameter λ. In particular, some sharp results of the classical Hilbert inequality are obtained by means of a sharpening of the Cauchy inequality. As applications, some refinements of both the Fejer-Riesz inequality and Hardy inequality inHp function are given.

In this paper, we obtain an extension of a multivariable integral inequality of Hilbert-Pachpatte type. By specializing the upper estimate functions in the hypothesis and the parameters, we obtain many special cases.

By using thefractal theory and the methods of weight function, a Hilbert-type fractal integral inequality and its equivalent form are given. Their constant factors are proved being the best possible, and their applications are discussed briefly.

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.