In this paper, a new inequality for the weight coefficient ω ( q , n ) in the form 1,\frac{1} {p} + \frac{1} {q} = 1,n \in N} \right) \hfill \\ \end{gathered} \] " display="block"> ω ( q , n ) : = ∑ m = 1 ∞ 1 m + n ( n m ) 1 / q                             < π sin ( π / p ) − 1 2 π 1 / p + n − 1 / q ( q > 1 , 1 p + 1 q = 1 , n ∈ N ) is proved. This is followed by a strengthened version ofthe Hardy-Hilbert inequality.