From Wikipedia, Hilbert's inequality states that $$ \left|\sum _{{r\neq s}}{\dfrac {u_{{r}}\overline {u_{{s}}}}{r-s}}\right|\leq \pi \displaystyle \sum _{{r}}|u_{{r}}|^{2}\, ,$$ for any sequence $u_1$, $u_2$, $\dots$ of complex numbers.
I wonder whether the inequality is still valid for bilateral series and for $\sum _{{r\neq s}}{\dfrac {u_{{r}}\overline {v_{{s}}}}{r-s}}$, with $v_s\neq u_r$. I think yes for both questions. For example, I think that $$ \left|\sum _{{r\neq s}}{\dfrac {u_{{r}}\overline {v_{{s}}}}{r-s}}\right|\leq \pi \displaystyle \sqrt{\sum _{{r}}|u_{{r}}|^{2} \sum _{{r}}|v_{{r}}|^{2}} \, ,$$ but I can't some references for this.
Thanks in advance!