Mathieu inequalities

The mathieuschen inequalities ( English Mathieu’s inequalities ) are two classic inequalities that belong to the mathematical branch of analysis . They are named after the French mathematician Émile Léonard Mathieu .

Mathieu's inequalities supply a lower and an upper estimate for certain series of positive numbers , of which the upper one was assumed by Mathieu in 1890, but not proven. This upper estimate is used in mathematical physics , where with its help series developments for solving boundary value problems in elasticity studies can be derived.

The first complete proof of the upper estimate suggested by Mathieu was provided in 1952 by the German mathematician Lothar Berg . As a result, numerous papers were written, of which the one by the Hungarian mathematician Endre Makai (1915–1987) from 1957 deserves special mention, as this is where the author presented the first completely elementary proof of the mathematician conjecture.

formulation

The math inequalities say:

The estimates apply to every real number ${\displaystyle x\neq 0}$

${\displaystyle {\frac {1}{x^{2}+{\frac {1}{2}}}}<\sum _{n=1}^{\infty }{\frac {2n}{(n^{2}+x^{2})^{2}}}<{\frac {1}{x^{2}}}}$   .

Evidence sketch

According to Makai, the proof can be sketched as follows:

For every real one , two infinite sequences and are defined, where for a natural number ${\displaystyle x\neq 0}$ ${\displaystyle \left(a_{n}\right)_{n=1,2,3,\ldots }}$ ${\displaystyle \left(b_{n}\right)_{n=1,2,3,\ldots }}$ ${\displaystyle n>0}$

${\displaystyle a_{n}={\frac {1}{(n-{\frac {1}{2}})^{2}+x^{2}-{\frac {1}{4}}}}}$

and

${\displaystyle b_{n}={\frac {1}{(n-{\frac {1}{2}})^{2}+x^{2}+{\frac {1}{4}}}}}$

are set.

By means of algebraic transformations arising

${\displaystyle a_{n}-a_{n+1}={\frac {2n}{(n^{2}+x^{2})^{2}-n^{2}}}>{\frac {2n}{(n^{2}+x^{2})^{2}}}}$

and accordingly

${\displaystyle b_{n}-b_{n+1}={\frac {2n}{(n^{2}+x^{2}+{\frac {1}{2}})^{2}-n^{2}}}={\frac {2n}{(n^{2}+x^{2})^{2}+x^{2}+{\frac {1}{4}}}}<{\frac {2n}{(n^{2}+x^{2})^{2}}}}$   .

Now one forms the two associated telescope sums and thus obtains the chain of inequalities

${\displaystyle {\begin{aligned}{\frac {1}{x^{2}+{\frac {1}{2}}}}&=b_{1}\\&=\sum _{n=1}^{\infty }{(b_{n}-b_{n+1})}\\&<\sum _{n=1}^{\infty }{\frac {2n}{(n^{2}+x^{2})^{2}}}\\&<\sum _{n=1}^{\infty }{(a_{n}-a_{n+1})}\\&=a_{1}\\&={\frac {1}{x^{2}}}\\\end{aligned}}}$

and from this the asserted estimates.

annotation

In the treatise of 1949, the mathematician Kurt Schröder pointed out that he could not see the correctness of the above mathematical inequality.

Instead, he proved the weaker (but sufficient for his objective) inequality

${\displaystyle \sum _{n=1}^{\infty }{\frac {n}{(n^{2}+x^{2})^{2}}}<{\frac {2}{3x^{2}}}}$   .

literature

• L. Berg: About an assessment by Mathieu . In: Mathematical News . tape 7 , 1952, pp. 257-259 , doi : 10.1002 / mana.19520070502 .
• E. Makai: On the inequality of Mathieu . In: Publicationes Mathematicae Debrecen . tape 5 , 1957, pp. 204-205 ( MR0091361 ).
• E. Mathieu: Traité de physique mathématique . VI – VII, part 2. Paris 1890, chap. X.
• 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, Heidelberg, New York 1970, ISBN 3-540-62903-3 ( MR0274686 ).
• Kurt Schröder: The problem of the clamped rectangular elastic plate . In: Mathematical Annals . tape 121 , 1959, pp. 247-326 .

References and footnotes

1. ↑ Kurt Schröder: The problem of the clamped rectangular elastic plate. Math. Ann. 121, p. 247 ff, p. 258 ff
2. ↑ See entry in the Hungarian Wikipedia !
3. ↑ DS Mitrinović: Analytic Inequalities. 1970, pp. 360-361, p. 392
4. ↑ See list ( list in MathSciNet )!
5. ↑ ^{a b} E. Makai: On the inequality of Mathieu. Publ. Math. Debrecen 5, pp. 204-205
6. ↑ Mitrinović, op.cit., P. 360
7. ↑ Mitrinović, op. Cit., Pp. 360–361
8. ↑ Schröder, op. Cit. P. 260
9. ↑ Schröder, op. Cit. P. 258