Chebyshev's sum inequality

The Chebyshev - Summenungleichung (after Pafnuti Lvovitch Chebyshev ) is an inequality of mathematics. In older transcriptions is found occasionally even the spelling of Chebyshev.

Definition

It says that for monotonous sibling n- tuple of real numbers

And

The relationship

Applies. Are contrast and opposite sorts of things, for example,

And

So the inequality holds in the reverse direction:

Note that in contrast to many other inequalities are no prerequisites for the signs of and necessary.

Evidence

Evidence of rearrangement inequality

The Chebyshev Summenungleichung can be derived from the rearrangement inequality. Multiplying from the right side, we obtain

Because of the rearrangement inequality is now each of these sums ( in the case of sibling numbers and ) less than or equal to total so you get exactly the desired relationship

In the case of opposite minor numbers and needs the rearrangement inequality only be applied in the reverse direction.

Proof by mathematical induction

The Chebyshev Summenungleichung can prove by mathematical induction and application of the rearrangement inequality for the simplest case of two summands. The base case is easy to carry. In the induction step we consider now

Applying now to the middle summand the rearrangement inequality for two terms and the last term the induction hypothesis, so there is ( in the case of sibling numbers and )

In the case of opposite minor numbers and the proof is analogous.

Proof of equation formulation

Another proof follows directly from the equation

Or more generally with weights

It is namely

With the name change of the indices obtained

A total of exactly the assertion:

Generalization

The Chebyshev Summenungleichung can also be in the form

. Write In this form they can be applied to more than two siblings generalize n-tuple, where the observed numbers must, however, be greater than or equal to zero: for

With

Applies

The proof can be done by, for example, by induction, since for regarding falling ordered non-negative numbers whose products also

Decreasing order and are non-negative.

Variants

Are in the same direction monotonous and is a weight function, ie then

To prove you integrate the non-negative function multiplied out to x and y are each from 0 to 1, this can be generalized further:

Are in the same direction monotonic and non-negative then

And are monotonic and non-negative on the same direction and is a weight function then.

This is evident if one substituted by x.