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
![{\ displaystyle f, g \ colon X \ to {\ overline {\ mathbb {R}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ad38ba1df2a27e01e946b11357b4e4f5b850bf3e)
For and with the convention one defines
![p \ in [1, \ infty)](https://wikimedia.org/api/rest_v1/media/math/render/svg/9c107de2d3bb99b7eb3f4ca1b7f68e0c57a0be3b)
![{\ displaystyle \ infty ^ {p} = \ infty ^ {\ frac {1} {p}} = \ infty}](https://wikimedia.org/api/rest_v1/media/math/render/svg/403bdf4b7c38e28ce6afab24b0a160f4f9835c28)
![{\ displaystyle H_ {p} (f) = \ left (\ int _ {X} | f | ^ {p} \ mathrm {d} \ mu \ right) ^ {\ tfrac {1} {p}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7cf9b093cdba995a203d164de062aa5d6df94793)
and
![{\ displaystyle H _ {\ infty} (f) = \ mathrm {ess} \ sup _ {x \ in X} | f (x) |}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d1a99c470ebc5d539b3701ae2bb42021f71a4496)
the essential supremum . The Hölder inequality then reads: for with , where is agreed, applies
![{\ displaystyle 1 \ leq p, q \ leq \ infty}](https://wikimedia.org/api/rest_v1/media/math/render/svg/619dde51a6203901e6bfaafef50b553b33a0eba6)
![{\ tfrac {1} {p}} + {\ tfrac {1} {q}} = 1](https://wikimedia.org/api/rest_v1/media/math/render/svg/b0258202dfc8069365da867d2ef863a2a53f04ef)
![{\ displaystyle {\ tfrac {1} {\ infty}} = 0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fe28645754c7b979ddb36fda2fdacb433a70824e)
![{\ displaystyle H_ {1} (fg) \ leq H_ {p} (f) \ cdot H_ {q} (g)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/80df07be0686bd5c0c7622380ab5c9b9b94e0bf2)
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
![q](https://wikimedia.org/api/rest_v1/media/math/render/svg/06809d64fa7c817ffc7e323f85997f783dbdf71d)
![{\ displaystyle {\ mathcal {L}} ^ {p} (X, {\ mathcal {A}}, \ mu)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b4db4d17fa6a72dc200b58f84b60822da9e10c91)
![p](https://wikimedia.org/api/rest_v1/media/math/render/svg/81eac1e205430d1f40810df36a0edffdc367af36)
![{\ displaystyle \ | \ cdot \ | _ {p}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a44e951f6e15fdb1bdd7d223bbbe628955251a20)
![{\ displaystyle f \ in {\ mathcal {L}} ^ {p} (X, {\ mathcal {A}}, \ mu), g \ in {\ mathcal {L}} ^ {q} (X, { \ mathcal {A}}, \ mu)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/303641b797cbee37949e8c4dab1d333f48d01d56)
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.
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![{\ displaystyle ([a, b], {\ mathcal {B}} ([a, b]), \ lambda)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9c2b59b485eb0b1398fb95f3227903c8d73ac2c0)
![{\ displaystyle f, g \ in {\ mathcal {L}} ^ {2} ([a, b], {\ mathcal {B}} ([a, b]), \ lambda)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/cac32c5185a1c64eb10cfcc3ffb7ce0b5e6e9cb5)
![{\ displaystyle \ int _ {a} ^ {b} | fg | \ mathrm {d} \ lambda \ leq \ left (\ int _ {a} ^ {b} | f | ^ {2} \ mathrm {d} \ lambda \ right) ^ {\ tfrac {1} {2}} \ cdot \ left (\ int _ {a} ^ {b} | g | ^ {2} \ mathrm {d} \ lambda \ right) ^ { \ tfrac {1} {2}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8ebc98ea16ba9f504e579e07bd7b205d6a6b3e6d)
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
![\ {1, \ ldots, n \}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0401c38cf1a2e51b30b38f4b93b5285aa77f8fad)
![\ sum _ {{k = 1}} ^ {n} | x_ {k} y_ {k} | \ leq \ left (\ sum _ {{k = 1}} ^ {n} | x_ {k} | ^ {p} \ right) ^ {{1 / p}} \ left (\ sum _ {{k = 1}} ^ {n} | y_ {k} | ^ {q} \ right) ^ {{1 / q }},](https://wikimedia.org/api/rest_v1/media/math/render/svg/2a36d0b2bfc041fa67d09950f69f0e5d2df96da2)
valid for all real (or complex) numbers . For one obtains the Cauchy inequality (or the discrete formulation of the Cauchy-Schwarz inequality)
![x_ {1}, \ ldots, x_ {n}, y_ {1}, \ ldots, y_ {n}](https://wikimedia.org/api/rest_v1/media/math/render/svg/20365cafe1dd65c593f13a3119769d8c11ebe48f)
![p = q = 2](https://wikimedia.org/api/rest_v1/media/math/render/svg/78ddeff7adfcc6b70743b38ab1504661f0cabec0)
![{\ displaystyle | \ langle x, y \ rangle | \ leq \ | x \ | _ {2} \ cdot \ | y \ | _ {2}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c9b1c5394fd587118dad82958b3063660a279d7f)
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
![\ mathbb {N}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fdf9a96b565ea202d0f4322e9195613fb26a9bed)
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for real or complex sequences . In the borderline case this corresponds
![{\ displaystyle (a_ {k}) _ {k \ in \ mathbb {N}}, (b_ {k}) _ {k \ in \ mathbb {N}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e68d19ca77682560ba3562908d4b2d2536be7df4)
![{\ displaystyle q = \ infty}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ac371e92e110d21e647ef278207dce5d2191790a)
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generalization
There are as well and for all .
![p_ {j} \ in [1, \ infty], j = 1, \ ldots, m](https://wikimedia.org/api/rest_v1/media/math/render/svg/a11d9f053eeada9bedf8241ee2a0ffb488b5f8ef)
![\ textstyle {\ frac {1} {r}}: = \ sum _ {{j = 1}} ^ {m} {\ frac {1} {p_ {j}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e5c2db6d430015e2df37c07d29a183a4ec957ed9)
![f_ {j} \ in L ^ {{p_ {j}}} (S)](https://wikimedia.org/api/rest_v1/media/math/render/svg/b76f95f05d6928f1bdfc24a1f6565ebb4a246814)
![j = 1, \ ldots, m](https://wikimedia.org/api/rest_v1/media/math/render/svg/745ead100813850e9d68e2a6cb6be14124998279)
Then follows
![\ prod _ {{j = 1}} ^ {m} f_ {j} \ in L ^ {r} (S)](https://wikimedia.org/api/rest_v1/media/math/render/svg/a71eb2ccd9772f0119056a4a1f7b8ff328e9faa2)
and the estimate applies
![{\ displaystyle \ left \ | \ prod _ {j = 1} ^ {m} f_ {j} \ right \ | _ {r} \ leq \ prod _ {j = 1} ^ {m} \ left \ | f_ {j} \ right \ | _ {p_ {j}}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f62cfcef58894f8d4c209cbda6c8111849eb9e6f)
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
![{\ displaystyle (a_ {i, j}) _ {i \ in \ {1,2, \ dots, n \}, j \ in \ {1,2, \ dots, m \}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/42e28e6efc54c069d7adffe46cb680f2840e79c6)
![m](https://wikimedia.org/api/rest_v1/media/math/render/svg/0a07d98bb302f3856cbabc47b2b9016692e3f7bc)
![{\ displaystyle (\ lambda _ {j}) _ {j \ in \ {1, \ dots, m \}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/efd85b253156e7d88995c2183d943103a3286a69)
![{\ displaystyle \ sum _ {j = 1} ^ {m} \ lambda _ {j} = 1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/792534d9b5c1a22f3553e00521c9f4fb2e3c6666)
![{\ displaystyle \ sum _ {i = 1} ^ {n} \ prod _ {j = 1} ^ {m} a_ {i, j} ^ {\ lambda _ {j}} \ leq \ prod _ {j = 1} ^ {m} \ left (\ sum _ {i = 1} ^ {n} a_ {i, j} \ right) ^ {\ lambda _ {j}}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/cc60dd034368026e7d1dddd5ffa660f740f026ee)
Inverted Hölder's inequality
It is for almost everyone .
![g (x) \ neq 0](https://wikimedia.org/api/rest_v1/media/math/render/svg/304c689556e38a33dde280823338d0ef90735d89)
![x \ in S](https://wikimedia.org/api/rest_v1/media/math/render/svg/51186ba8afb2067573a9082d55dd383df1ea9214)
Then the reverse of Hölder's inequality applies to all of them
![\ int _ {S} | f (x) g (x) | dx \ geq \ left (\ int _ {S} | f (x) | ^ {{{\ frac {1} {r}}}} dx \ right) ^ {r} \ left (\ int _ {S} | g (x) | ^ {{- {\ frac {1} {r-1}}}} dx \ right) ^ {{- (r -1)}}.](https://wikimedia.org/api/rest_v1/media/math/render/svg/18237282a7cd6354d8ebd350155ea6a6f5890395)
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 :
![p = 1, q = \ infty](https://wikimedia.org/api/rest_v1/media/math/render/svg/7b06a0ee234842520e040e3c17ccc6077bd86ef4)
![1 <p, q <\ infty](https://wikimedia.org/api/rest_v1/media/math/render/svg/ecd79fbece22707c6d5ead11afbd6dffeccde731)
![\ | f \ | _ {p}> 0](https://wikimedia.org/api/rest_v1/media/math/render/svg/adeed488df666c22e47abbcebb42cec570974020)
![\ | g \ | _ {q}> 0](https://wikimedia.org/api/rest_v1/media/math/render/svg/e5af6045968616ef807a2ba1ee8d842e76b7b152)
![AB \ leq {\ frac {A ^ {p}} {p}} + {\ frac {B ^ {q}} {q}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8eb17503d5c9b8f20bb1af8b664e22794ce35b32)
for everyone . Use this specifically . Integration delivers
![A, B \ geq 0](https://wikimedia.org/api/rest_v1/media/math/render/svg/eaa663f3ae395e714dc6df465c3204174f85abfd)
![A: = {\ tfrac {| f (x) |} {\ | f \ | _ {p}}}, \, B: = {\ tfrac {| g (x) |} {\ | g \ | _ {q}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/04db22844053ef87f623f20b14ea240fcfe56dd5)
![{\ displaystyle {\ frac {1} {\ | f \ | _ {p} \ | g \ | _ {q}}} \ int _ {S} | fg | \ mathrm {d} \ mu \ leq {\ frac {1} {p}} + {\ frac {1} {q}} = 1,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ce0d5718e4232ab470c5af727a4e30632899404c)
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:
![m](https://wikimedia.org/api/rest_v1/media/math/render/svg/0a07d98bb302f3856cbabc47b2b9016692e3f7bc)
![m = 1](https://wikimedia.org/api/rest_v1/media/math/render/svg/b6100c5ebd48c6fd848709f2be624465203eb173)
![m \ geq 2](https://wikimedia.org/api/rest_v1/media/math/render/svg/eca2e437e89ef4565a87f1a6d90ed37eef1d8ce3)
![p_ {1} \ leq \ cdots \ leq p_ {m}](https://wikimedia.org/api/rest_v1/media/math/render/svg/624eaf324afe98438d30f1aa8ddfa8b7d50f33f7)
Case 1: Then, by the induction hypothesis, then holds
![p_ {m} = \ infty.](https://wikimedia.org/api/rest_v1/media/math/render/svg/b2b254813a05d4fc630a5d104c3b966b5e473be2)
![\ textstyle {\ frac {1} {r}} = \ sum _ {{j = 1}} ^ {{m-1}} {\ frac {1} {p_ {j}}}.](https://wikimedia.org/api/rest_v1/media/math/render/svg/2b95c0f278b1b35530f67b67d24745bacd78dc30)
![\ | f_ {1} \ cdots f_ {m} \ | _ {r} \ leq \ | f_ {m} \ | _ {\ infty} \ | f_ {1} \ cdots f _ {{m-1}} \ | _ {r} \ leq \ | f_ {m} \ | _ {\ infty} \ | f_ {1} \ | _ {{p_ {1}}} \ cdots \ | f _ {{m-1}} \ | _ {{p _ {{m-1}}}}.](https://wikimedia.org/api/rest_v1/media/math/render/svg/4a8b48bd760ca488395669013e2321becdb5b705)
Case 2: . According to the (usual) Hölder inequality for the exponents ,
![p_ {m} <\ infty](https://wikimedia.org/api/rest_v1/media/math/render/svg/2d82c01e1754a22026f2792f1d2f4ed5ce1c552a)
![{\ tfrac {p_ {m}} {p_ {m} -r}}, {\ tfrac {p_ {m}} {r}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3a6a59dcfb5994b2db779d035b818b454f673578)
![{\ displaystyle \ int _ {S} | f_ {1} \ cdots f_ {m-1} | ^ {r} | f_ {m} | ^ {r} \ mathrm {d} \ mu \ leq \ left (\ int _ {S} | f_ {1} \ cdots f_ {m-1} | ^ {\ frac {rp_ {m}} {p_ {m} -r}} \ mathrm {d} \ mu \ right) ^ { \ frac {p_ {m} -r} {p_ {m}}} \ left (\ int _ {S} | f_ {m} | ^ {p_ {m}} \ mathrm {d} \ mu \ right) ^ {\ frac {r} {p_ {m}}},}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7f84f355a0dde3b8649bde0f8688ad4b7e90f934)
so . Well is . The induction step thus results from the induction assumption.
![\ textstyle \ | f_ {1} \ cdots f_ {m} \ | _ {r} \ leq \ | f_ {1} \ cdots f _ {{m-1}} \ | _ {{{{\ tfrac {rp_ {m }} {p_ {m} -r}}}} \ | f_ {m} \ | _ {{p_ {m}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/cddd7e1d37cb4a84522cbb19e1e15b255b84080c)
![\ textstyle \ sum _ {{j = 1}} ^ {{m-1}} {\ frac {1} {p_ {j}}} = {\ frac {1} {r}} - {\ frac {1 } {p_ {m}}} = {\ frac {p_ {m} -r} {rp_ {m}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5dbbecaa93c433475ed2d30bbd96aca536c7c286)
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:
![p: = r](https://wikimedia.org/api/rest_v1/media/math/render/svg/3f466e4575c873c359015d85a2deab6e0456ccdb)
![q: = r '= {\ tfrac {p} {p-1}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2a6315683153b87048fe4652041e10aaadc86a67)
![{\ displaystyle \ int _ {S} | f | ^ {\ frac {1} {r}} \ mathrm {d} \ mu = \ int _ {S} \ left (| fg | ^ {\ frac {1} {r}} \ cdot | g | ^ {- {\ frac {1} {r}}} \ mathrm {d} \ mu \ right) \ leq \ left (\ int _ {S} | fg | \ mathrm { d} \ mu \ right) ^ {\ frac {1} {r}} \ left (\ int _ {S} | g | ^ {- {\ frac {r '} {r}}} \ mathrm {d} \ mu \ right) ^ {\ frac {1} {r '}}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/91a49bc284c2238be8cbaed333d05c6e430a0b03)
Rearranging and exponentiating this inequality with yields the reversed Hölder inequality.
![r](https://wikimedia.org/api/rest_v1/media/math/render/svg/0d1ecb613aa2984f0576f70f86650b7c2a132538)
Applications
Proof of the Minkowski inequality
With Hölder's inequality one can easily prove the Minkowski inequality (that is the triangle inequality im ).
![L ^ {p}](https://wikimedia.org/api/rest_v1/media/math/render/svg/cf2317aaca1ecee4b8ccf667bc1001059eae5850)
Interpolation inequality for Lebesgue functions
Let be and , then it follows and the interpolation inequality applies![f \ in L ^ {p} (S) \ cap L ^ {q} (S)](https://wikimedia.org/api/rest_v1/media/math/render/svg/6570e801f279ac2c0e9cbd898ad160ca446cbdc5)
![1 \ leq q \ leq r \ leq p](https://wikimedia.org/api/rest_v1/media/math/render/svg/e1ea3827cc7be5838a8e163ee30f05030300bfd7)
![\ | f \ | _ {r} \ leq \ | f \ | _ {p} ^ {{1- \ theta}} \ | f \ | _ {q} ^ {\ theta}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6ed775c510ca698ee06a3b8b3b1d4f019526d2a6)
with or for .
![{\ tfrac {1} {r}} =: {\ tfrac {1- \ theta} {p}} + {\ tfrac {\ theta} {q}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/cb787949482a617037e32ad548f0e25a3c0c44d8)
![\ theta: = {\ tfrac {q} {r}} {\ tfrac {pr} {pq}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6bfeef39f82da2db011f8500d761fb049f46ee0d)
![q \ neq p](https://wikimedia.org/api/rest_v1/media/math/render/svg/b15493bd4b96c9d3ccf0a3e9ca7e8b12b5ae99a0)
Proof: be without restriction . Fix with . Note that and are the conjugate Hölder exponents. It follows from Hölder's inequality
![q <r <p](https://wikimedia.org/api/rest_v1/media/math/render/svg/b012ae3d174b9a6b50441e592f60febd8943c23d)
![t \ in (0.1)](https://wikimedia.org/api/rest_v1/media/math/render/svg/47e93bf1b33e520ce1c54ecd00d977a91cc29075)
![r = tp + (1-t) q](https://wikimedia.org/api/rest_v1/media/math/render/svg/5f1078f892e012203565f36a2c22df0640d93bba)
![\ tfrac {1} {t}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b4d2851f22ca31e9ddea8662f3c79833cb29093e)
![{\ tfrac {1} {1-t}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/391cbe13601abd028f83cbc455961c5e30edd894)
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.
Raising the inequality by and calculating the exponents implies the interpolation inequality .
![{\ tfrac {1} {r}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/dc177c75043de5867200268f273b23f2796162fa)
Proof of Young's convolution inequality
Another typical application is the proof of the generalized Young's inequality (for convolution integrals )
![\ | f \ star g \ | _ {r} \ leq \ | f \ | _ {p} \ | g \ | _ {q}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c92c75fa38ed6e463812b6ab3aa8044f7b20611a)
for and .
![{\ tfrac {1} {p}} + {\ tfrac {1} {q}} = 1 + {\ tfrac {1} {r}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a64e44d0a382e20df6f73453f3045ab0ea5ebb61)
![p, q, r \ geq 1](https://wikimedia.org/api/rest_v1/media/math/render/svg/09bf8f25da38b3adfe3b0e0798afac22011893b3)
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.