A variant of Hilbert's inequality and the norm of the Hilbert Matrix on $K^p$

We prove the nontrivial variant $$\sum _{m,n=1}^{\mathrm{\infty }}(\frac{n}{m}{)}^{\frac{1}{q}-\frac{1}{p}}\frac{a_m{b}_n}{m+n-1}\le \frac{\pi }{\sin \frac{\pi }{p}}(\sum _{m=1}^{\mathrm{\infty }}{a}_m^p{)}^{\frac{1}{p}}(\sum _{n=1}^{\mathrm{\infty }}{b}_n^q{)}^{\frac{1}{q}}$$ of the well known Hilbert’s inequality. Then we use this to determine the exact value $\frac{\pi }{\sin \frac{\pi }{p}}$ of the norm of the Hilbert matrix as an operator acting on the Hardy-Littlewood space $K^p$. This space consists of all functions $f(z)={\sum }_{m=0}^{\mathrm{\infty }}{a}_m{z}^m$ analytic in the unit disc with $\mathrm{\parallel }f{\mathrm{\parallel }}_{K^p}^p={\sum }_{m=0}^{\mathrm{\infty }}(m+1{)}^{p-2}\mathrm{\mid }{a}_m{\mathrm{\mid }}^p<+\mathrm{\infty }$.