Hölder's inequality

In the mathematical analysis is one of the Hölder inequality, named after Otto Hölder, together with the Minkowski inequality and the inequality between jens to the fundamental inequalities for Lp- spaces.

• 2.1 Proof of Holder's inequality
• 2.2 Proof of the generalization
• 2.3 prove the reverse Hölder inequality

• 3.1 Proof of Minkowski 's inequality
• 3.2 interpolation inequality for Lebesgue functions
• 3.3 Proof of Faltungsungleichung Young

Statement

Hölder's inequality

Let be a measure space, with, whether from in and out.

Then

And we have:

This is known as the Hölder exponent conjugate to. The last inequality holds also in the case and are only Lebesgue measurable ( because if the left side is infinite, then it follows that the right side must be infinite).

Special cases

Be the crowd, equipped with the counting measure, we obtain as a special case the inequality

Valid for all real ( or complex ) numbers. Is the set of natural numbers with the counting measure, we obtain the same inequality, only with infinite series instead of finite sums.

For obtained as a special case of the Cauchy- Schwarz inequality.

Generalization

There are as well and for all.

Then follows

And it holds the estimate

Reverse Hölder inequality

It is for almost everyone.

Then for all the reverse Hölder inequality

Evidence

Proof of the Hölder inequality

For (and vice versa) the statement of the Holder's inequality is trivial. We therefore assume that the following holds. Were without limitation. After the young between inequality holds:

For everyone. Set a specifically herein. provides integration

What Hölder inequality implies.

Evidence of generalization

The proof is by mathematical induction on. The case is trivial. So do both now and without restriction. Then two cases can be distinguished:

Case 1: Then is then apply induction hypothesis

Case 2:. Applies after the ( usual ) Holder's inequality for the exponents

Therefore. Is now. From the induction hypothesis, this results in the induction step.

Prove the reverse Hölder inequality

The reverse Hölder inequality arises from the ( usual ) Holder's inequality by choosing as exponents and. Obtained thus:

Forming this inequality yields the reverse Hölder inequality.

Applications

Proof of Minkowski 's inequality

By Holder's inequality can be the Minkowski inequality (which is the triangle inequality in ) easily prove.

Interpolation inequality for Lebesgue functions

Let and, then follows and applies the interpolation inequality

With or for.

Proof: Without loss was. Fix with. Note that and conjugate Hölder exponents. From the Hölder inequality follows

Exponentiation with the inequality and calculating the exponent implies the interpolation inequality.

Proof of Faltungsungleichung Young

Another typical application is the proof of the generalized Young inequality between ( for convolution integrals)

For and.