On a New Half-Discrete Hilbert-Type Inequality Involving the Variable Upper Limit Integral and Partial Sums

: In this paper we establish a new half-discrete Hilbert-type inequality involving the variable upper limit integral and partial sums. As applications, an inequality obtained from the special case of the half-discrete Hilbert-type inequality is further investigated; moreover, the equivalent conditions of the best possible constant factor related to several parameters are proved.


Introduction
In [2], an extension of (1) was established by introducing parameters is the beta function.
Recently, Yang and Wu et al. [28,29] gave a reverse half-discrete Hardy-Hilbert's inequality and an extended Hardy-Hilbert's inequality. For these inequalities, the equivalent statements of the best possible constant factor related to several parameters were also discussed therein.
Following the way of [2,4,21], the aim of this paper is to establish a new half-discrete Hilbert-type inequality involving the variable upper limit integral and partial sums via the kernel . With regard to the obtained inequality, the equivalent conditions of the best possible constant factor related to several parameters are also proved.

Some Lemmas
In what follows, we suppose that is a positive real function and In particular, for then the Euler-Maclaurin summation formulas hold: are the Bernoulli functions. ). 0 ( :   (7), (8), (9), and (10), we obtain Namely, g was previously defined as a function. We obtain that for ] 6 , and then we obtain , we obtain the following weight coefficient:   (12) and (14), we get inequality (13). The proof of Lemma 3 is complete.  (13), ) (x f and n a respectively with ) (x F and n A , in view of (5), we have

Proof. Integrating by parts, in view of
and then we obtain inequality (16).

In view of
where ) (  is the gamma function.

Remark 2. Putting
. Thus, we can reduce (20) to the following: By (22) and the decreasingness property of series, we obtain Then, in virtue of the above results, we have , we reduce (20) to the following: We obtain , , 0 and then we have (23)) is the best possible, then by (21), the unified best possible constant factor must be We observe that (24) (21), we have the following half-discrete Hilbert-type inequality with the best possible constant factor 1:

Conclusions
In this paper, by means of the weight coefficients, the idea of introduced parameters, the Euler-Maclaurin summation formula and Abel's summation by parts formula, a new half-discrete Hilbert-type inequality involving the variable upper limit integral and partial sums is given in Theorem 1. As an application, an inequality obtained from the special case of the half-discrete Hilbert-type inequality is investigated in Theorem 2, we obtained the equivalent conditions of the best possible constant factor related to several parameters. The lemmas and theorems proved in this paper reveal some new and interesting properties of this type of inequalities.