Hilbert's inequality

The inequality of Hilbert ( English Hilbert's inequality ) is a classic inequality of Analysis , one of the branches of mathematics . It goes back to a work by the German mathematician David Hilbert from 1888 and gives an upper estimate of certain double sums of positive real numbers . Hilbert's inequality has been tightened, generalized and modified by numerous authors. Last but not least, Hermann Weyl - for example in his inaugural dissertation Singular Integral Equations with special consideration of Fourier's integral theorem from 1908 - and Godfrey Harold Hardy in particular subjected them to intensive investigation.

Formulation of the inequality

Hilbert's inequality can be stated as follows:

Is given for a natural number one - tuple of positive real numbers. ${\displaystyle N>0}$ ${\displaystyle (N+1)}$ ${\displaystyle (a_{0},\ldots ,a_{N})}$

Then:

(H)  . ${\displaystyle \sum _{i=0}^{N}{\sum _{j=0}^{N}{\frac {a_{i}a_{j}}{i+j+1}}}\leq \pi \cdot \sum _{i=0}^{N}{{a_{i}}^{2}}}$

Tightening

According to H. Frazer, the last inequality has a tightening, in which the circle number is replaced by a better estimation factor:

(HF)  . ${\displaystyle \sum _{i=0}^{N}{\sum _{j=0}^{N}{\frac {a_{i}a_{j}}{i+j+1}}}\leq (N+1)\cdot \sin \left({\frac {\pi }{N+1}}\right)\cdot \sum _{i=0}^{N}{{a_{i}}^{2}}}$

DV Aries showed the following stronger inequality:

(HW)  . ${\displaystyle \sum _{i=0}^{N}{\sum _{j=0}^{N}{\frac {a_{i}a_{j}}{i+j+1}}}\leq \pi \cdot \sum _{i=0}^{N}{\sum _{j=0}^{N}{{\frac {(i+j)!}{i!j!}}{\frac {a_{i}a_{j}}{2^{i+j+1}}}}}}$

Related inequality

Fu Cheng Hsiang proved the following related inequality:

Given a natural number plus two tuples and of non-negative real numbers. ${\displaystyle N>0}$ ${\displaystyle (N+1)}$ ${\displaystyle (a_{0},\ldots ,a_{N})}$ ${\displaystyle (b_{0},\ldots ,b_{N})}$

Then:

(HHs)  . ${\displaystyle \sum _{i=0}^{N}{\sum _{j=0}^{N}{\frac {a_{i}b_{j}}{2i+2j+1}}}\leq (N+1)\cdot \sin {\frac {\pi }{2(N+1)}}\cdot \left(\sum _{i=0}^{N}{{a_{i}}^{2}}\right)^{\frac {1}{2}}\cdot \left(\sum _{j=0}^{N}{{b_{j}}^{2}}\right)^{\frac {1}{2}}}$

Analogues and extensions

By analogy and extension of the above inequalities one obtains the corresponding for double series and -integrals :

For two sequences and of nonnegative real numbers, which do not have both only as a sequence term , and two positive real numbers with, the following always applies: ${\displaystyle \left(a_{i}\right)_{i\in \mathbb {N} }}$ ${\displaystyle \left(b_{j}\right)_{j\in \mathbb {N} }}$ ${\displaystyle 0}$ ${\displaystyle p,q}$ ${\displaystyle {\tfrac {1}{p}}+{\tfrac {1}{q}}=1}$

(HH_1)  . ${\displaystyle \sum _{i=1}^{\infty }{\sum _{j=1}^{\infty }{\frac {a_{i}b_{j}}{i+j}}}<{\frac {\pi }{\sin \left({\frac {\pi }{p}}\right)}}\cdot \left(\sum _{i=1}^{\infty }{{a_{i}}^{p}}\right)^{\frac {1}{p}}\cdot \left(\sum _{j=1}^{\infty }{{b_{j}}^{q}}\right)^{\frac {1}{q}}}$

For two real functions that are not both the null function and two positive real numbers with : ${\displaystyle f,g\colon [0,\infty )\to [0,\infty )}$ ${\displaystyle p,q}$ ${\displaystyle {\tfrac {1}{p}}+{\tfrac {1}{q}}=1}$

(HH_2)  . ${\displaystyle \int _{0}^{\infty }\int _{0}^{\infty }{\frac {f(x)g(y)}{x+y}}\,\mathrm {d} x\,\mathrm {d} y<{\frac {\pi }{\sin \left({\frac {\pi }{p}}\right)}}\cdot \left(\int _{0}^{\infty }{f^{p}(x)}\,\mathrm {d} x\right)^{\frac {1}{p}}\cdot \left(\int _{0}^{\infty }{g^{q}(x)}\,\mathrm {d} x\right)^{\frac {1}{q}}}$

Addition: For both (HH_1) and (HH_2) the estimation factor is the best possible. ${\displaystyle {\tfrac {\pi }{\sin \left({\frac {\pi }{p}}\right)}}}$

Remarks

• For one speaks in terms of (HH_1) also from Hilbert rule double row set ( English Hilbert's double series theorem ). ${\displaystyle p=q=2}$
• With respect to the general case, it is common today, the above inequalities (HH_1) or (HH_2) as hardy-Hilbert inequality ( English Hardy-Hilbert's inequality ) or as hardy-Hilbert Integralungleichung ( English Hardy-Hilbert's integral inequality hereinafter).

Two more related inequalities

To deliver as part of efforts simplest possible proof of the Hilbert rule double row set, were - starting in the 1920 and 1925 work by GH Hardy and Edmund Landau - found two related inequalities for series and integrals and derived, both under the heading of Hardy's Inequality ( English Hardy's inequality ) became known. These are the following:

For a sequence of nonnegative real numbers, which are not all equal , and a real number we always have: ${\displaystyle \left(a_{i}\right)_{i\in \mathbb {N} }}$ ${\displaystyle 0}$ ${\displaystyle p>1}$

(H_1)  . ${\displaystyle \sum _{i=1}^{\infty }\left({\frac {a_{1}+\ldots +a_{i}}{i}}\right)^{p}<\left({\frac {p}{p-1}}\right)^{p}\cdot \sum _{i=1}^{\infty }{{a_{i}}^{p}}}$

For a real function that is not the null function and a real number, the following always applies: ${\displaystyle f\colon [0,\infty )\to [0,\infty )}$ ${\displaystyle p>1}$

(H_2)  . ${\displaystyle \int _{0}^{\infty }\left({\frac {\int _{0}^{x}f(t)\,\mathrm {d} t}{x}}\right)^{p}\,\mathrm {d} x<\left({\frac {p}{p-1}}\right)^{p}\cdot \int _{0}^{\infty }f^{p}(x)\,\mathrm {d} x}$

Addition: Both for (H_1) and (H_2) the estimation factor is the best possible. ${\displaystyle \left({\frac {p}{p-1}}\right)^{p}}$

literature

• H. Frazer: Note on Hilbert's inequality . In: The Journal of the London Mathematical Society . tape 21 , 1946, pp. 7-9 ( MR0018226 ).
• GH Hardy : Note on a theorem of Hilbert . In: Mathematical Journal . tape 6 , 1920 ( MR1544414 ).
• GH Hardy: Note on a theorem of Hilbert concerning series of positive terms . In: Proceedings of the London Mathematical Society (2) . tape 23 , 1925.
• GH Hardy, JE Littlewood , G. Pólya : Inequalities . Reprint (of the 2nd edition 1952). Cambridge University Press , Cambridge 1973.
• David Hilbert: About the representation of definite forms as the sum of form squares . In: Mathematical Annals . tape 32 , 1888, pp. 342-350 ( MR1510517 ).
• Fu Cheng Hsiang: An inequality for finite sequences . In: Mathematica Scandinavica . tape 5 , 1957, pp. 12-14 ( [1] ).
• E. Landau : A note on a theorem concerning series of positive terms . In: The Journal of the London Mathematical Society . tape 1 , 1926, pp. 38-39 .
• DS Mitrinović : Analytic Inequalities . In cooperation with PM Vasić (=  The basic teachings of the mathematical sciences in individual representations with special consideration of the areas of application . Volume 165 ). Springer Verlag , Berlin ( inter alia ) 1970, ISBN 3-540-62903-3 ( MR0274686 ).
• Waadallah Tawfeeq Sulaiman: Hardy-Hilbert's integral inequalities via homogeneous functions and some other generalizations . In: Acta et Commentationes Universitatis Tartuensis de Mathematica . tape 11 , 2007, p. 23-32 ( MR2391968 ).
• DV Widder: An Inequality Related to One of Hilbert’s . In: The Journal of the London Mathematical Society . tape 4 , 1929, pp. 194-198 ( MR1575045 ).
• Bicheng Yang, Qiang Chen: A new extension of Hardy-Hilbert's inequality in the whole plane . In: Journal of Function Spaces . 2016 ( MR3548430 - Art. ID 9197476, 8 pages).

References and footnotes

1. ↑ DS Mitrinović: Analytic Inequalities. 1970, pp. 357-358
2. ^ GH Hardy, JE Littlewood, G. Pólya: Inequalities. 1973, p. 226 ff
3. ↑ ^{a b c d} Mitrinović, op.cit., P. 357
4. ↑ ^{a b} Hardy et al., Op.cit., P. 226
5. ↑ As to the transition of double sums in double rows should be noted that the pair of volumes and another in bijection stand and that for ever is. ${\displaystyle \mathbb {N} _{0}\times \mathbb {N} _{0}}$ ${\displaystyle \mathbb {N} \times \mathbb {N} }$ ${\displaystyle i,j\in \mathbb {N} }$ ${\displaystyle {\tfrac {1}{i+1+j+1}}<{\tfrac {1}{i+j+1}}}$
6. ↑ Hardy et al., Op. Cit., Pp. 239 ff