In mathematical analysis , Hölder's inequality, together with Minkowski's inequality and Jensen's inequality, are among the fundamental inequalities for L p spaces . It was first proven by Leonard James Rogers in 1888; it is named after Otto Hölder , who published it a year later.
statement
Hölder's inequality
Let a dimension space and measurable functions be given
For and with the convention one defines
and
the essential supremum . The Hölder inequality then reads: for with , where is agreed, applies
It is called the Hölder exponent to be conjugated . The inequality is formulated more specifically as follows: If the space of -fold Lebesgue integrable functions (see Lp space ) and is the Lp norm , then applies forever
-
.
Special cases
Schwarz's inequality
If you choose a measurement space , i.e. a real interval with the Lebesgue measure and two functions , then the Hölder inequality reads with
This is exactly the Schwarz inequality or the integral formulation of the Cauchy-Schwarz inequality.
Cauchy's inequality
If one chooses the finite set as the measure space , provided with the power set and equipped with the counting measure , the inequality is obtained as a special case
valid for all real (or complex) numbers . For one obtains the Cauchy inequality (or the discrete formulation of the Cauchy-Schwarz inequality)
Hölder's inequality for series
If one chooses the natural numbers as the basic set of the measure space , again provided with the power set and the counting measure, then one obtains Hölder's inequality for series
-
.
for real or complex sequences . In the borderline case this corresponds
-
.
generalization
There are as well and for all .
Then follows
and the estimate applies
The corollary of this generalization is the following theorem.
If is a family of sequences of non-negative real numbers, and there are non-negative real numbers with , then
Inverted Hölder's inequality
It is for almost everyone .
Then the reverse of Hölder's inequality applies to all of them
proofs
Proof of Hölder's inequality
For (and vice versa) the statement of Hölder's inequality is trivial. We therefore assume that holds. Be and without restriction . According to Young's inequality :
for everyone . Use this specifically . Integration delivers
what the Hölder's inequality implies.
Proof of generalization
The proof is by mathematical induction on out. The case is trivial. So be now and be without restriction . Then two cases have to be distinguished:
Case 1: Then, by the induction hypothesis, then holds
Case 2: . According to the (usual) Hölder inequality for the exponents ,
so . Well is . The induction step thus results from the induction assumption.
Proof of the reverse Hölder's inequality
The reverse Hölder inequality results from the (usual) Holder's inequality by as exponents and selected. This gives:
Rearranging and exponentiating this inequality with yields the reversed Hölder inequality.
Applications
Proof of the Minkowski inequality
With Hölder's inequality one can easily prove the Minkowski inequality (that is the triangle inequality im ).
Interpolation inequality for Lebesgue functions
Let be and , then it follows and the interpolation inequality applies
with or for .
Proof: be without restriction . Fix with . Note that and are the conjugate Hölder exponents. It follows from Hölder's inequality
-
.
Raising the inequality by and calculating the exponents implies the interpolation inequality .
Proof of Young's convolution inequality
Another typical application is the proof of the generalized Young's inequality (for convolution integrals )
for and .
literature
- Herbert Amann, Joachim Escher : Analysis III . 1st edition. Birkhäuser-Verlag, Basel / Boston / Berlin 2001, ISBN 3-7643-6613-3 .
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Jürgen Elstrodt : Measure and integration theory . 6th, corrected edition. Springer-Verlag, Berlin Heidelberg 2009, ISBN 978-3-540-89727-9 , doi : 10.1007 / 978-3-540-89728-6 .
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Achim Klenke : Probability Theory . 3. Edition. Springer-Verlag, Berlin Heidelberg 2013, ISBN 978-3-642-36017-6 , doi : 10.1007 / 978-3-642-36018-3 .
Individual evidence
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↑ Elstrodt: Measure and Integration Theory. 2009, p. 277.