Wallis inequalities

In the analysis , one of the part areas of mathematics is known as wallissche inequalities ( English Wallis's inequalities ) are those inequalities which the one after the mathematician John Wallis named product formula related. These inequalities provide estimates that illuminate the relationship between the double faculty function and the number of circles . The Valais inequalities were subjected to further investigations in a large number of papers . ${\displaystyle \pi }$

Representation of the inequalities

Two of the most common Wallis inequalities are:

The estimates apply to every natural number ${\displaystyle n>0}$

${\displaystyle {\frac {1}{\sqrt {\pi \cdot (n+{\frac {1}{2}})}}}<{\frac {(2n-1)!!}{(2n)!!}}={\frac {(2n-1)\cdot (2n-3)\cdot \mathrm {} \cdots \mathrm {} \cdot 3\cdot 1}{2n\cdot (2n-2)\cdot \mathrm {} \cdots \mathrm {} \cdot 4\cdot 2}}<{\frac {1}{\sqrt {\pi \cdot n}}}}$   .

Inferences

From the above inequalities the following inequalities can be derived which, apart from a few small indices , are weaker than the previous two:

For every natural number one has ${\displaystyle n>0}$

${\displaystyle {\sqrt {\frac {\frac {5}{4}}{4n+1}}}<{\frac {(2n-1)!!}{(2n)!!}}<{\sqrt {\frac {\frac {3}{4}}{2n+1}}}}$   .

As Robert Alexander Rankin shows in his monograph An Introduction to Mathematical Analysis , the last-mentioned inequalities can also be obtained directly with an induction proof .

Tightening

A mathematician named Donat K. Kazarinoff showed in 1956 a tightening of the upper estimate, namely:

The following applies to every natural number ${\displaystyle n>0}$

${\displaystyle {\frac {(2n-1)!!}{(2n)!!}}<{\frac {1}{\sqrt {\pi \cdot (n+{\frac {1}{4}})}}}}$   .

In 2005, the two mathematicians Chen Chao-Ping and Qi Feng proved that the lower estimate was tightened, namely:

The following applies to every natural number ${\displaystyle n>0}$

${\displaystyle {\frac {1}{\sqrt {\pi \cdot (n+{\frac {4}{\pi }}-1)}}}\leq {\frac {(2n-1)!!}{(2n)!!}}}$   .

Connection with the Valais product

The above-mentioned relationship between the double faculty function and the circle number is obtained when the following result is taken into account, which can be found in the differential and integral calculus II by GM Fichtenholz (and also in the aforementioned monograph by Rankin):

For every natural number is ${\displaystyle n>0}$

${\displaystyle {\Biggl [}{\frac {(2n)!!}{(2n-1)!!}}{\Biggr ]}^{2}\cdot {\frac {1}{2n+1}}<{\frac {\pi }{2}}<{\Biggl [}{\frac {(2n)!!}{(2n-1)!!}}{\Biggr ]}^{2}\cdot {\frac {1}{2n}}}$

and consequently

${\displaystyle {\begin{aligned}{\frac {\pi }{2}}&=\lim _{n\rightarrow \infty }\left({\Biggl [}{\frac {(2n)!!}{(2n-1)!!}}{\Biggr ]}^{2}\cdot {\frac {1}{2n}}\right)\\&=\lim _{n\rightarrow \infty }\left({\Biggl [}{\frac {(2n)!!}{(2n-1)!!}}{\Biggr ]}^{2}\cdot {\frac {1}{2n+1}}\right)\\&=\lim _{n\rightarrow \infty }\left({\frac {2^{2}\cdot 4^{2}\cdot \mathrm {} \cdots \mathrm {} \cdot (2n-2)^{2}\cdot (2n)^{2}}{1\cdot [1\cdot 3]\cdot [3\cdot 5]\cdot \mathrm {} \cdots \mathrm {} \cdot [(2n-3)\cdot (2n-1)]\cdot [(2n-1)\cdot (2n+1)]}}\right)\\\end{aligned}}}$   .

literature

• GM Fichtenholz: Differential and Integral Calculus II . Translation from Russian and scientific editing: Dipl.-Math. Brigitte Mai, Dipl.-Math. Walter Mai (=  university books for mathematics . Volume 62 ). 6th edition. VEB Deutscher Verlag der Wissenschaften , Berlin 1974.
• Chen Chao-Ping, Qi Feng: Best upper and lower bounds in Wallis' inequality . In: Journal of the Indonesian Mathematical Society (MIHMI) . tape 11 , 2005, p. 137-141 ( MR2168684 ).
• DK Kazarinoff: On Wallis' formula . In: Edinburgh Mathematical Notes . 1956, no. 40, 1956, p. 19-21 ( MR0082501 ).
• 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 ).
• Robert A. Rankin: An Introduction to Mathematical Analysis (=  International Series of Monographs on Pure and Applied Mathematics . Volume 165 ). Pergamon Press , Oxford, London, New York, Paris 1963.

Individual evidence

1. ↑ DS Mitrinović: Analytic Inequalities. 1970, pp. 192-193, p. 287
2. ↑ See list in MathSciNet  ( page no longer available , search in web archives )  Info: The link was automatically marked as defective. Please check the link according to the instructions and then remove this notice. !  
3. ↑ ^{a b c} Mitrinović, op.cit., P. 192
4. ^ Robert A. Rankin: An Introduction to Mathematical Analysis. 1963, p. 13
5. ↑ GM Fichtenholz: Differential- und Integralrechner II. 1974, pp. 149–150
6. ↑ Rankin, op.cit., P. 380
7. ↑ As Fichtenholz explains, is namely the difference between the two outer expressions   . ${\displaystyle <{\frac {\pi }{4n}}}$