Considerable attention has been given to Hilbert inequalities and Hilbert-type inequalities and their various generalizations by several authors including Handley et al. 1 , Minzhe and Bicheng 2 , Minzhe 3 , Hu 4 , Jichang 5 , Bicheng 6 , and Zhao 7, 8 . In 1998, Pachpatte 9 gave some new integral inequalities similar to Hilbert inequality see 10, page 226 . In 2000, Zhao and Debnath 11 establi...

We obtain new proofs with improved constants of the Khintchine-type inequality with matrix coefficients in two cases. The first case is the Pisier and Lust-Piquard noncommutative Khintchine inequality for p = 1 , where we obtain the sharp lower bound of 1 √ 2 in the complex Gaussian case and for the sequence of functions {en}n=1 . The second case is Junge’s recent Khintchine-type inequality for...

In this paper, by introducing four parameters A,B,?,? and using the Euler-Maclaurin expansion for Riemann zeta function, we establish an inequality of a weight coefficient. Using inequality, derive generalizations Hilbert?s type inequality.

A finite Hilbert transformation associated with a polynomial is the analogue of a Hilbert transformation associated with an entire function which is a generalization of the classical Hilbert transformation. The weighted Hilbert inequality, which has applications in analytic number theory, is closely related to the finite Hilbert transformation associated with a polynomial. In this note, we stud...

In this paper, by using the Euler-Maclaurin expansion for Riemann-$zeta$ function, we establish an inequality of a weight coefficient. Using inequality, derive new reverse Hilbert's type inequality. As applications, equivalent form is obtained.