Inequality that established Lp spaces are normed vector spaces
In mathematical analysis, the Minkowski inequality establishes that the Lp spaces are normed vector spaces. Let
be a measure space, let
and let
and
be elements of
Then
is in
and we have the triangle inequality
![{\displaystyle \|f+g\|_{p}\leq \|f\|_{p}+\|g\|_{p}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a1d3de762058656808721fc899c4b223914c6c3f)
with equality for
![{\displaystyle 1<p<\infty }](https://wikimedia.org/api/rest_v1/media/math/render/svg/5540fd86346b4798d7447b8de70fe98cf6243d28)
if and only if
![{\displaystyle f}](https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61)
and
![{\displaystyle g}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d3556280e66fe2c0d0140df20935a6f057381d77)
are positively linearly dependent; that is,
![{\displaystyle f=\lambda g}](https://wikimedia.org/api/rest_v1/media/math/render/svg/bc49d15992f169714e4f461c6eb889c9a6ce5c27)
for some
![{\displaystyle \lambda \geq 0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0c26004859ae51dde7800b3f3a960c73f81cd583)
or
![{\displaystyle g=0.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2499af95528f92ec5e6bb5883769878367fa6b9d)
Here, the norm is given by:
![{\displaystyle \|f\|_{p}=\left(\int |f|^{p}d\mu \right)^{\frac {1}{p}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d98b372b16f4185ffc2c0c58aafcd0ec93589593)
if
![{\displaystyle p<\infty ,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/68e45e8071b07eba7a7043b920a47201060a68ce)
or in the case
![{\displaystyle p=\infty }](https://wikimedia.org/api/rest_v1/media/math/render/svg/42846cbf8fc7db1ff19e51d0d43ca84c249b062b)
by the essential supremum
![{\displaystyle \|f\|_{\infty }=\operatorname {ess\ sup} _{x\in S}|f(x)|.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1f514691a17ada8a419574d4cfc813f4b765d4e9)
The Minkowski inequality is the triangle inequality in
In fact, it is a special case of the more general fact
![{\displaystyle \|f\|_{p}=\sup _{\|g\|_{q}=1}\int |fg|d\mu ,\qquad {\tfrac {1}{p}}+{\tfrac {1}{q}}=1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3d56ccda83307e558835e221c57955bf9909a227)
where it is easy to see that the right-hand side satisfies the triangular inequality.
Like Hölder's inequality, the Minkowski inequality can be specialized to sequences and vectors by using the counting measure:
![{\displaystyle {\biggl (}\sum _{k=1}^{n}|x_{k}+y_{k}|^{p}{\biggr )}^{1/p}\leq {\biggl (}\sum _{k=1}^{n}|x_{k}|^{p}{\biggr )}^{1/p}+{\biggl (}\sum _{k=1}^{n}|y_{k}|^{p}{\biggr )}^{1/p}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fa53e54b15011c268df0c9885c5a9dd450e2b9ba)
for all real (or complex) numbers
![{\displaystyle x_{1},\dots ,x_{n},y_{1},\dots ,y_{n}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c4a9669d1490134a6b16a8eacf7433d467482069)
and where
![{\displaystyle n}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b)
is the cardinality of
![{\displaystyle S}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4611d85173cd3b508e67077d4a1252c9c05abca2)
(the number of elements in
![{\displaystyle S}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4611d85173cd3b508e67077d4a1252c9c05abca2)
).
The inequality is named after the German mathematician Hermann Minkowski.
Proof
First, we prove that
has finite
-norm if
and
both do, which follows by
![{\displaystyle |f+g|^{p}\leq 2^{p-1}(|f|^{p}+|g|^{p}).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/09381447aedbbd09c375faf0b939561316b6abb0)
Indeed, here we use the fact that
![{\displaystyle h(x)=|x|^{p}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a8b453ba0b5a9473a7f040c6613670bf8e06499f)
is convex over
![{\displaystyle \mathbb {R} ^{+}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/97dc5e850d079061c24290bac160c8d3b62ee139)
(for
![{\displaystyle p>1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0f127e7a5f2449ddf3edb8164c2d2439120641f9)
) and so, by the definition of convexity,
![{\displaystyle \left|{\tfrac {1}{2}}f+{\tfrac {1}{2}}g\right|^{p}\leq \left|{\tfrac {1}{2}}|f|+{\tfrac {1}{2}}|g|\right|^{p}\leq {\tfrac {1}{2}}|f|^{p}+{\tfrac {1}{2}}|g|^{p}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a3bcc20e05c6db9c56fe7377bb3b34cdfa6ebd31)
This means that
![{\displaystyle |f+g|^{p}\leq {\tfrac {1}{2}}|2f|^{p}+{\tfrac {1}{2}}|2g|^{p}=2^{p-1}|f|^{p}+2^{p-1}|g|^{p}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/05a16c801672e5b5e6546958e5949853ddf7e62d)
Now, we can legitimately talk about
If it is zero, then Minkowski's inequality holds. We now assume that
is not zero. Using the triangle inequality and then Hölder's inequality, we find that
![{\displaystyle {\begin{aligned}\|f+g\|_{p}^{p}&=\int |f+g|^{p}\,\mathrm {d} \mu \\&=\int |f+g|\cdot |f+g|^{p-1}\,\mathrm {d} \mu \\&\leq \int (|f|+|g|)|f+g|^{p-1}\,\mathrm {d} \mu \\&=\int |f||f+g|^{p-1}\,\mathrm {d} \mu +\int |g||f+g|^{p-1}\,\mathrm {d} \mu \\&\leq \left(\left(\int |f|^{p}\,\mathrm {d} \mu \right)^{\frac {1}{p}}+\left(\int |g|^{p}\,\mathrm {d} \mu \right)^{\frac {1}{p}}\right)\left(\int |f+g|^{(p-1)\left({\frac {p}{p-1}}\right)}\,\mathrm {d} \mu \right)^{1-{\frac {1}{p}}}&&{\text{ Hölder's inequality}}\\&=\left(\|f\|_{p}+\|g\|_{p}\right){\frac {\|f+g\|_{p}^{p}}{\|f+g\|_{p}}}\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a318b7b6967ffd2aa0f4ffa4335cf08c6cc88181)
We obtain Minkowski's inequality by multiplying both sides by
![{\displaystyle {\frac {\|f+g\|_{p}}{\|f+g\|_{p}^{p}}}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/188653a338e08015f0d6a85bcb099a129608bc05)
Minkowski's integral inequality
Suppose that
and
are two 𝜎-finite measure spaces and
is measurable. Then Minkowski's integral inequality is:
![{\displaystyle \left[\int _{S_{2}}\left|\int _{S_{1}}F(x,y)\,\mu _{1}(\mathrm {d} x)\right|^{p}\mu _{2}(\mathrm {d} y)\right]^{\frac {1}{p}}~\leq ~\int _{S_{1}}\left(\int _{S_{2}}|F(x,y)|^{p}\,\mu _{2}(\mathrm {d} y)\right)^{\frac {1}{p}}\mu _{1}(\mathrm {d} x),}](https://wikimedia.org/api/rest_v1/media/math/render/svg/74694688ad2e3f6e8169d4b150b30e9daa7ba18a)
with obvious modifications in the case
![{\displaystyle p=\infty .}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3e84b68700f76da4a8befa9b0b17662b9daca968)
If
![{\displaystyle p>1,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4b18ad57c656bab68c7bd45e2bf4596d09de7c0a)
and both sides are finite, then equality holds only if
![{\displaystyle |F(x,y)|=\varphi (x)\,\psi (y)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6c315eecc13990a4de3b245369145e7ebde8ae60)
a.e. for some non-negative measurable functions
![{\displaystyle \varphi }](https://wikimedia.org/api/rest_v1/media/math/render/svg/33ee699558d09cf9d653f6351f9fda0b2f4aaa3e)
and
If
is the counting measure on a two-point set
then Minkowski's integral inequality gives the usual Minkowski inequality as a special case: for putting
for
the integral inequality gives
![{\displaystyle \|f_{1}+f_{2}\|_{p}=\left(\int _{S_{2}}\left|\int _{S_{1}}F(x,y)\,\mu _{1}(\mathrm {d} x)\right|^{p}\mu _{2}(\mathrm {d} y)\right)^{\frac {1}{p}}\leq \int _{S_{1}}\left(\int _{S_{2}}|F(x,y)|^{p}\,\mu _{2}(\mathrm {d} y)\right)^{\frac {1}{p}}\mu _{1}(\mathrm {d} x)=\|f_{1}\|_{p}+\|f_{2}\|_{p}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/36fd6b33d9a5711e2ee4f70322405c2848ffda44)
If the measurable function
is non-negative then for all
![{\displaystyle \left\|\left\|F(\,\cdot ,s_{2})\right\|_{L^{p}(S_{1},\mu _{1})}\right\|_{L^{q}(S_{2},\mu _{2})}~\leq ~\left\|\left\|F(s_{1},\cdot )\right\|_{L^{q}(S_{2},\mu _{2})}\right\|_{L^{p}(S_{1},\mu _{1})}\ .}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a987a6a478046a49212bebb65df7d8aba33a782f)
This notation has been generalized to
![{\displaystyle \|f\|_{p,q}=\left(\int _{\mathbb {R} ^{m}}\left[\int _{\mathbb {R} ^{n}}|f(x,y)|^{q}\mathrm {d} y\right]^{\frac {p}{q}}\mathrm {d} x\right)^{\frac {1}{p}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/378fb353fca1c3274d623037524f0b25d84d29f4)
for
![{\displaystyle f:\mathbb {R} ^{m+n}\to E,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c6c51c2e4288ddf1a86c5a7d11ecec29317750db)
with
![{\displaystyle {\mathcal {L}}_{p,q}(\mathbb {R} ^{m+n},E)=\{f\in E^{\mathbb {R} ^{m+n}}:\|f\|_{p,q}<\infty \}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8a7b83ff3dff49cafba0fbb18f300315d8fae5dd)
Using this notation, manipulation of the exponents reveals that, if
![{\displaystyle p<q,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/af0d8ec521bb4b8bfcf844182a9ae2b99c081ccc)
then
Reverse inequality
When
the reverse inequality holds:
![{\displaystyle \|f+g\|_{p}\geq \|f\|_{p}+\|g\|_{p}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e8bfee0623cb28ecb7edc7bb7d6d8ace2c9c207e)
We further need the restriction that both
and
are non-negative, as we can see from the example
and
The reverse inequality follows from the same argument as the standard Minkowski, but uses that Holder's inequality is also reversed in this range.
Using the Reverse Minkowski, we may prove that power means with
such as the harmonic mean and the geometric mean are concave.
Generalizations to other functions
The Minkowski inequality can be generalized to other functions
beyond the power function
The generalized inequality has the form
![{\displaystyle \phi ^{-1}\left(\textstyle \sum \limits _{i=1}^{n}\phi (x_{i}+y_{i})\right)\leq \phi ^{-1}\left(\textstyle \sum \limits _{i=1}^{n}\phi (x_{i})\right)+\phi ^{-1}\left(\textstyle \sum \limits _{i=1}^{n}\phi (y_{i})\right).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d1b6505522783a0db1b6f4e74b857e3b744e00bc)
Various sufficient conditions on
have been found by Mulholland[4] and others. For example, for
one set of sufficient conditions from Mulholland is
is continuous and strictly increasing with ![{\displaystyle \phi (0)=0.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ee1c5eb6dea6ccd082da53c67672b9bfe1725010)
is a convex function of ![{\displaystyle x.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d07e9f568a88785ae48006ac3c4b951020f1699a)
is a convex function of ![{\displaystyle \log(x).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fe2a2c39494d7e3c980f4577e0e3497454bf502c)
See also
References
- ^ Mulholland, H.P. (1949). "On Generalizations of Minkowski's Inequality in the Form of a Triangle Inequality". Proceedings of the London Mathematical Society. s2-51 (1): 294–307. doi:10.1112/plms/s2-51.4.294.
- Bahouri, Hajer; Chemin, Jean-Yves; Danchin, Raphaël (2011). Fourier Analysis and Nonlinear Partial Differential Equations. Grundlehren der mathematischen Wissenschaften. Vol. 343. Berlin, Heidelberg: Springer. ISBN 978-3-642-16830-7. OCLC 704397128.
- Hardy, G. H.; Littlewood, J. E.; Pólya, G. (1988). Inequalities. Cambridge Mathematical Library (second ed.). Cambridge: Cambridge University Press. ISBN 0-521-35880-9.
- Minkowski, H. (1953). Geometrie der Zahlen. Chelsea..
- Stein, Elias (1970). Singular integrals and differentiability properties of functions. Princeton University Press..
- M.I. Voitsekhovskii (2001) [1994], "Minkowski inequality", Encyclopedia of Mathematics, EMS Press
- Lohwater, Arthur J. (1982). "Introduction to Inequalities".
Further reading
- Bullen, P. S. (2003), "The Power Means", Handbook of Means and Their Inequalities, Dordrecht: Springer Netherlands, pp. 175–265, doi:10.1007/978-94-017-0399-4_3, ISBN 978-90-481-6383-0, retrieved 2022-06-23
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