On further strengthened Hardy-Hilbert's inequality

Abstract

We obtain an inequality for the weight coefficient ω(q,n) (q>1, 1/q+1/q=1, n∈ℕ) in the form ω(q,n)=:∑m=1∞(1/(m+n))(n/m)1/q<π/sin(π/p)−1/(2n1/p+(2/a)n−1/q) where 0<a<147/45, as n≥3; 0<a<(1−C)/(2C−1), as n=1,2, and C is an Euler constant. We show a generalization and improvement of Hilbert's inequalities. The results of the paper by Yang and Debnath are improved.