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A theorem of D. Hilbert on double series:

$$ \tag{* }
\sum _ {m = 1 } ^  \infty   \sum _ {n = 1 } ^  \infty  

\frac{a _ {n} b _ {m} }{n + m }
  < \ 

\frac \pi {\sin ( \pi /p) }

\left (
\sum _ {n = 1 } ^  \infty  
a _ {n}  ^ {p} \right )  ^ {1/p} \ 
\left (
\sum _ {m = 1 } ^  \infty  
b _ {m} ^ {q } \right ) ^ {1/q } ,
$$

where

$$ 
p  >  1,\ \ 
q  =  
\frac{p}{p - 1 }
 ,\ \ 
{
\frac{1}{p}
 } + {
\frac{1}{q }
 }  =  1,\ \ 
a _ {n} , b _ {m}  \geq   0,
$$

and the series on the right-hand side have finite positive sums. The constant  $  \pi / {\sin ( \pi / p ) } $
is precise, i.e. it cannot be decreased. The validity of (*) with  $  p= 2 $
was demonstrated by Hilbert, without the precise constant, in his course on integral equations. Its proof was published by H. Weyl [[#References|[1]]]. The precise constant was found by I. Schur [[#References|[2]]], while the inequality (*) for arbitrary  $  p > 1 $
was first quoted by G.H. Hardy and M. Riesz in 1925. There exist integral analogues and generalizations of (*), for example

$$ 
\int\limits _ { 0 } ^  \infty  
\int\limits _ { 0 } ^  \infty  
K  ^  \lambda  ( x, y) f ( x) g ( y)  dx  dy \leq 
$$

$$ 
\leq  \ 
K  ^  \lambda  \left (
\int\limits _ { 0 } ^  \infty   f ^ { p } ( x)  dx \right )  ^ {1/p}  \left
( \int\limits _ { 0 } ^  \infty   g  ^ {r} ( y)  dy \right )  ^ {1/r} ,
$$

where  $  K( x, y) $
is a non-negative kernel, homogeneous of degree  $  - 1 $, 
$  p > 1 $, 
$  r > 1 $, 
$  \lambda = p  ^ {-} 1 + r  ^ {-} 1 \leq  1 $, 
$  f, g \geq  0 $, 
and

$$ 
K  =  \int\limits _ { 0 } ^  \infty  
u ^ {- 1/ \lambda q } K ( 1, u)  du;
$$

and the previously obtained special case of this inequality [[#References|[4]]] with kernel  $  K( x, y) = 1/( x + y) $(
the so-called double-parametric Hilbert inequality) and constant  $  K  ^  \lambda  = ( \pi / \sin  {\lambda q } )  ^  \lambda  $. 
The preciseness of this constant has been proved for  $  r/( r- 1) = p $. 
It is also asymptotically precise as  $  p \rightarrow 1 $
for an arbitrary admissible fixed  $  r $. 
The problem of the asymptotic behaviour of the constant in (*) for finite sums ( $  1 \leq  n, m \leq  N $) 
has not been solved (1988); it is only known that if  $  p = q = 2 $, 
the constant is

$$ 
\pi - 
\frac{\pi  ^ {5} }{2}

(  \mathop{\rm ln}  N)  ^ {2} + O (  \mathop{\rm ln}   \mathop{\rm ln}  \{ N \ 
(  \mathop{\rm ln}  N)  ^ {-} 3 \} ).
$$

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  H. Weyl,   "Singuläre Integralgleichungen mit besonderer Berücksichtigung des Fourierschen Integraltheorems" , Göttingen  (1908)  (Thesis)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  I. Schur,   "Bemerkungen zur Theorie der beschränkten Bilinearformen mit unendlich vielen Veränderlichen"  ''J. Reine Angew. Math.'' , '''140'''  (1911)  pp. 1–28</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  G.H. Hardy,   J.E. Littlewood,   G. Pólya,   "Inequalities" , Cambridge Univ. Press  (1934)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  F.F. Bonsall,   "Inequalities with non-conjugate parameters"  ''Quart. J. Math. Oxford (2)'' , '''2'''  (1951)  pp. 135–150</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  V. Levin,   "On the two-parameter extension and analogue of Hilbert's inequality"  ''J. London Math. Soc. (1)'' , '''11'''  (1936)  pp. 119–124</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top">  N.G. de Bruijn,   H.S. Wilf,   "On Hilbert's inequality in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047270/h04727023.png" /> dimensions"  ''Bull. Amer. Math. Soc.'' , '''68'''  (1962)  pp. 70–73</TD></TR><TR><TD valign="top">[7]</TD> <TD valign="top">  P.L. Walker,   "A note on an inequality with non-conjugate parameters"  ''Proc. Edinburgh Math. Soc.'' , '''18'''  (1973)  pp. 293–294</TD></TR></table>