On a multiple Hilbert-type integral inequality involving the upper limit functions

By applying the weight functions, the idea of introducing parameters and the technique of real analysis, a new multiple Hilbert-type integral inequality involving the upper limit functions is given. The constant factor related to the gamma function is proved to be the best possible in a condition. A corollary about the case of the nonhomogeneous kernel and some particular inequalities are obtained.

In this paper, following the idea of [21], by using the weight functions, the way of introducing parameters and the technique of real analysis, a new multiple Hilbert-type integral inequality with the kernel 1 (x 1 +···+x n ) λ (λ > 0) involving the upper limit functions is given. The constant factor related to the gamma function is proved to be the best possible in a condition. A corollary about the case of the nonhomogeneous kernel and some particular inequalities are obtained.

Lemma 3
For n ∈ N\{1}, defining the following weight functions: we have In particular, for n i=1 1 Proof For j = i, setting u j = x j x i in (7), we have Then by Lemma 9.15 and (9.1.19) (cf. [2], p. 341-342), we obtain (8). The lemma is proved.

Lemma 4
We have the following inequality: Proof By (6) and Hölder's integral inequality (cf. [39]), we obtain If (11) takes the form of an equality, then there exist constants C i , C k (i = k) such that they are not all zero and Then by (8) and (11), we have (10). The lemma is proved.

Main results and a corollary
Theorem 1 We have the following inequality: In particular, for n i=1 1 and the following inequality: Proof By (4) and (5), we have Then by (12), we have (13). The theorem is proved. (14) is the best possible.
If there exists a positive constant M(M ≤ 1 (14) is valid when replacing 1 In view of Lemma 9.1.4 (9.1.5) in [2], we find Hence, we have For ε → 0 + , we find which yields that the constant factor M = 1 (14) is the best possible. The theorem is proved. (14), we have For f 1 (t) = t λ-2 f ( 1 t ), we find Then, replacing back x (resp. f (x)) by x 1 (resp. f 1 (x 1 )), we have then we have the following inequality with the nonhomogeneous kernel: where the constant factor 1 (15) is the best possible.
Remark 2 (i) For n = 2, (14) reduces to (cf. [40]) and (15) reduces to the following new inequality: and (15) reduces to The constant factors in the above inequalities are the best possible.

Conclusions
In this paper, following the idea of [21], by the use of the weight functions, the way of introducing parameters and the technique of real analysis, a new multiple Hilbert-type integral inequality with the kernel 1 (x 1 +···+x n ) λ (λ > 0) involving the upper limit functions is given in Theorem 1. In a condition, the best possible constant factor related to the gamma function and a few parameters is proved in Theorem 2. A corollary about the case of nonhomogeneous kernel and some particular inequalities are obtained in Corollary 1 and Remark 2. The lemmas and theorems provide an extensive account of this type of inequalities.