Weierstrass inequalities

The inequalities of Weierstrass ( English Weierstrass' inequalities ) belong to the elementary inequalities of the mathematical field of analysis . They go back to the German mathematician Karl Weierstrass .

Weierstraß's inequalities led to a number of advanced investigations which yielded improved and more general inequalities of similar type.

formulation

The inequalities are as follows:

Given a natural number in the open real interval, the real numbers are given . ${\displaystyle N>0}$ ${\displaystyle (0,1)}$ ${\displaystyle N}$ ${\displaystyle a_{1},\ldots ,a_{N}\in (0,1)}$

Then apply:

(W1a) ${\displaystyle \quad \prod _{k=1}^{N}{(1-a_{k})}>1-\sum _{k=1}^{N}{a_{k}}}$

(W1b) ${\displaystyle \quad \prod _{k=1}^{N}{(1-a_{k})}<{\frac {1}{1+\sum _{k=1}^{N}{a_{k}}}}}$

(W2a) ${\displaystyle \quad \prod _{k=1}^{N}{(1+a_{k})}>1+\sum _{k=1}^{N}{a_{k}}}$

(W2b)  , provided ${\displaystyle \quad \prod _{k=1}^{N}{(1+a_{k})}<{\frac {1}{1-\sum _{k=1}^{N}{a_{k}}}}}$ ${\displaystyle \sum _{k=1}^{N}{a_{k}}<1}$

annotation

The above inequalities (W1a) and (W2a) contain a generalization of the Bernoullian inequality .

literature

• 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 ).

References and footnotes

1. ↑ DS Mitrinović: Analytic Inequalities. 1970, p. 210, p. 396
2. ↑ See list (  ( 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 note. ]) In MathSciNet!  
3. ↑ Mitrinović, op.cit., P. 210
4. ↑ See Mitrinović, op.cit., P. 35!