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/ (2n~(1/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.
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