Equivalent Property of a Hilbert-Type Integral Inequality Related to the Beta Function in the Whole Plane

where the constant factor π/ sin(π/p) is the best possible. For p = q = 2, (1) reduces to the well-known Hilbert’s integral inequality. By using the weight functions, some extensions of (1) were given by [2, 3]. A few Hilbert-type inequalities with the homogenous and nonhomogenous kernels were provided by [4–7]. In 2017, Hong [8] also gave two equivalent statements betweenHilbert-type inequalities with the general homogenous kernel and parameters. Some other kinds of Hilbert-type inequalities were obtained by [9–16]. In 2007, Yang [17] gave a Hilbert-type integral inequality in the whole plane as follows:

In this paper, by means of the technique of real analysis and the weight functions, a few equivalent statements of a Hilbert-type integral inequality with the nonhomogeneous kernel in the whole plane similar to (2) are obtained.The constant factor related to the beta function is proved to be the best possible.As applications, the case of the homogeneous kernel, the operator expressions, and a few corollaries are considered.
The lemma is proved.
For  1 = , we have the following.

Lemma 3.
If there exists a constant , such that for any nonnegative measurable functions () and () in R, the following inequality holds true, then we have  ≥   ()(> 0).

Main Results and Some Corollaries
Theorem 4. If  is a constant, then the following statements (i), (ii), and (iii) are equivalent: (i) For any nonnegative measurable function () in R, we have the following inequality: (ii) For any nonnegative measurable functions () and () in R, we have the following inequality: (iii)  1 = , and  ≥   ()(> 0).