This article covers the Chebyshev arithmetic inequality. For the statistical Chebyshev inequality see
Chebyshev inequality .
The Chebyshev inequality (after Pafnuti Lwowitsch Chebyshev ) is an inequality of mathematics .
statement
It says that for monotonically equally ordered n- tuples of real numbers
and
-
,
the relationship
-
.
applies. Are and, however, ordered in opposite directions, for example
and
-
,
so the inequality applies in the opposite direction
-
.
Note that, unlike many other inequalities, no preconditions for the signs of and are necessary.
proofs
Proof from rearrangement inequality
The Chebyshev total inequality can be derived from the rearrangement inequality . If you multiply the right side, the result is
Because of the rearrangement inequality , each of these sums (in the case of identically ordered numbers and ) is now smaller or equal , so overall you get exactly the desired relationship
-
.
In the case of oppositely ordered numbers and the rearrangement inequality only needs to be applied in the opposite direction.
Proof with complete induction
The Chebyshev sum inequality can also be proven with complete induction and application of the rearrangement inequality for the simplest case with two summands. The induction start is easy to manage. Now consider in the induction step
-
.
If we now apply the rearrangement inequality for two summands to the middle term and the induction assumption to the last term, we get (in the case of identically ordered numbers and )
-
In the case of oppositely ordered numbers and the proof is analogous.
Proof from equation formulation
Another proof is straight from the equation
or more generally with weights
-
.
It is true
-
.
Renaming the indices results in
-
,
all in all exactly the assertion:
-
.
generalization
The Chebyshev total inequality can also be expressed in the form
write. In this form, it can also be generalized to more than two equally ordered n- tuples , although the numbers considered must be greater than or equal to zero: For
With
applies
The proof can, for example, be carried out with complete induction according to , since for non-negative numbers ordered with respect to decreasing also their products
are descending and are nonnegative.
variants
Are monotonous to the same direction and is a weight function, i. H. then
-
.
To prove this, one integrates the nonnegative function
multiplied by x and y from 0 to 1. This can be further generalized:
Are monotonous and nonnegative then is
in the same direction
-
.
And are monotonous and nonnegative on the same direction and then is a weight function
-
.
This is the result of substituting x with .
See also
Individual evidence
-
↑ Harro Heuser : Textbook of Analysis Part 1. Wiesbaden, Vieweg + Teubner , Verlag 2003, ISBN 3-322-96828-6 , p. 99.
-
↑ Martin Aigner : Diskrete Mathematik , 6th, corrected edition, Vieweg, Wiesbaden 2006, ISBN 978-3-8348-0084-8 , p. 54.