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.

By means of the weight functions, the technique of real analysis and Hermite-Hadamard's inequality, a more accurate half-discrete Hardy-Hilbert-type inequality related to the kernel of logarithmic function and a best possible constant factor is given. Moreover, the equivalent forms, the operator expressions, the reverses and some particular cases are also considered.

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 prove some integral inequalities involving the Laplace transform. These are sharper than some known generalizations of the Hilbert integral inequality including a recent result of Yang.

*Correspondence: [email protected]; [email protected] 2Department of Mathematics, Guangdong University of Education, Guangzhou, Guangdong 510303, P.R. China Full list of author information is available at the end of the article Abstract By using the way of weight functions and Hadamard’s inequality, a half-discrete Hilbert-type inequality similar to Mulholland’s inequality with a best constant...

In a generalization of the classical Hilbert inequality by Hardy, Littlewood and Pólya, the best constant for an inequality is determined provided that the generating function for the corresponding matrix satisfies certain monotonicity condition. In this paper, we determine the best constant for a class of inequalities when the monotonicity condition is no longer satisfied. Mathematics subject ...