On Further Analogs of Hilbert’s Inequality

By introducing the function | ln x − ln y | / ( x + y + | x − y | ), we establish new inequalities similar to Hilbert’s type inequality for integrals. As applications, we give its equivalent form as well.

Under the same condition of (1.2), we have Hardy-Hilbert's type inequality (see [1, Theorems 341 and 342]): where the constant factors 4 and π 2 are both the best possible.Recently, Li et al. [9] obtained the following result.
Theorem 1.1.If f, g are real functions such that 0 ) In this paper, we give a further analogs of Hilbert's type inequality and its applications.

Main results and applications
where the constant factor 4 is the best possible.
Assume that the constant factor 4 in (2.1) is not the best possible, then there exists a positive number K with K < 4 and a > 0; we have (2.12) By (2.10) and for b→0 + , we have that is

.16)
This contradicts the hypothesis.Hence the constant factor 4 in (2.1) is the best possible.
If the constant 16 in (2.17) is not the best possible, by (2.20), we may get a contradiction that the constant factor in (2.1) is not the best possible.This completes the proof.