Abstract
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.
Highlights
Hilbert-type inequality involving the variable upper limit integral and partial sums is given in Theorem 1
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
Summary
Hardy–Hilbert’s inequality with the best possible constant factor sin(ππ/p) reads as follows ([1], Theorem 315):. Hong and Wen [21] showed the equivalent statements of the extensions of (1) with the best possible constant factor related to several parameters. 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 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. Regard to the obtained inequality, the equivalent conditions of the best possible constant factor related to several parameters are proved
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