# Hilbert inequality

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 [1]. The precise constant was found by I. Schur [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 [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

 [1] H. Weyl, "Singuläre Integralgleichungen mit besonderer Berücksichtigung des Fourierschen Integraltheorems" , Göttingen (1908) (Thesis) [2] I. Schur, "Bemerkungen zur Theorie der beschränkten Bilinearformen mit unendlich vielen Veränderlichen" J. Reine Angew. Math. , 140 (1911) pp. 1–28 [3] G.H. Hardy, J.E. Littlewood, G. Pólya, "Inequalities" , Cambridge Univ. Press (1934) [4] F.F. Bonsall, "Inequalities with non-conjugate parameters" Quart. J. Math. Oxford (2) , 2 (1951) pp. 135–150 [5] V. Levin, "On the two-parameter extension and analogue of Hilbert's inequality" J. London Math. Soc. (1) , 11 (1936) pp. 119–124 [6] N.G. de Bruijn, H.S. Wilf, "On Hilbert's inequality in dimensions" Bull. Amer. Math. Soc. , 68 (1962) pp. 70–73 [7] P.L. Walker, "A note on an inequality with non-conjugate parameters" Proc. Edinburgh Math. Soc. , 18 (1973) pp. 293–294
How to Cite This Entry:
Hilbert inequality. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Hilbert_inequality&oldid=47229
This article was adapted from an original article by E.K. Godunova (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article