Minkowski inequality

In mathematical analysis, the Minkowski inequality establishes that the L p {displaystyle L^{p}} spaces satisfy the triangle inequality in the definition of normed vector spaces. The inequality is named after the German mathematician Hermann Minkowski.

Let S {textstyle S} be a measure space, let 1 ≤ p ≤ ∞ {textstyle 1leq pleq infty } and let f {textstyle f} and g {textstyle g} be elements of L p ( S ) . {textstyle L^{p}(S).} Then f + g {textstyle f+g} is in L p ( S ) , {textstyle L^{p}(S),} and we have the triangle inequality

‖ f + g ‖ p ≤ ‖ f ‖ p + ‖ g ‖ p {displaystyle |f+g|_{p}leq |f|_{p}+|g|_{p}}

with equality for 1 < p < ∞ {textstyle 1<p<infty } if and only if f {textstyle f} and g {textstyle g} are positively linearly dependent; that is, f = λ g {textstyle f=lambda g} for some λ ≥ 0 {textstyle lambda geq 0} or g = 0. {textstyle g=0.} Here, the norm is given by:

‖ f ‖ p = ( ∫ | f | p d μ ) 1 p {displaystyle |f|_{p}=left(int |f|^{p}dmu right)^{frac {1}{p}}}

if p < ∞ , {textstyle p<infty ,} or in the case p = ∞ {textstyle p=infty } by the essential supremum

‖ f ‖ ∞ = e s s s u p x ∈ S ⁡ | f ( x ) | . {displaystyle |f|_{infty }=operatorname {ess sup} _{xin S}|f(x)|.}

The Minkowski inequality is the triangle inequality in L p ( S ) . {textstyle L^{p}(S).} In fact, it is a special case of the more general fact

‖ f ‖ p = sup ‖ g ‖ q = 1 ∫ | f g | d μ , 1 p + 1 q = 1 {displaystyle |f|_{p}=sup _{|g|_{q}=1}int |fg|dmu ,qquad {tfrac {1}{p}}+{tfrac {1}{q}}=1}

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:

( ∑ k = 1 n | x k + y k | p ) 1 / p ≤ ( ∑ k = 1 n | x k | p ) 1 / p + ( ∑ k = 1 n | y k | p ) 1 / p {displaystyle left(sum _{k=1}^{n}|x_{k}+y_{k}|^{p}right)^{1/p}leq left(sum _{k=1}^{n}|x_{k}|^{p}right)^{1/p}+left(sum _{k=1}^{n}|y_{k}|^{p}right)^{1/p}}

for all real (or complex) numbers x 1 , … , x n , y 1 , … , y n {textstyle x_{1},dots ,x_{n},y_{1},dots ,y_{n}} and where n {textstyle n} is the cardinality of S {textstyle S} (the number of elements in S {textstyle S} ).

In probabilistic terms, given the probability space ( Ω , F , P ) , {displaystyle (Omega ,{mathcal {F}},mathbb {P} ),} and E {displaystyle mathbb {E} } denote the expectation operator for every real- or complex-valued random variables X {displaystyle X} and Y {displaystyle Y} on Ω , {displaystyle Omega ,} Minkowski's inequality reads

( E [ | X + Y | p ] ) 1 p ⩽ ( E [ | X | p ] ) 1 p + ( E [ | Y | p ] ) 1 p . {displaystyle left(mathbb {E} [|X+Y|^{p}]right)^{frac {1}{p}}leqslant left(mathbb {E} [|X|^{p}]right)^{frac {1}{p}}+left(mathbb {E} [|Y|^{p}]right)^{frac {1}{p}}.}

First, we prove that f + g {textstyle f+g} has finite p {textstyle p} -norm if f {textstyle f} and g {textstyle g} both do, which follows by

| f + g | p ≤ 2 p − 1 ( | f | p + | g | p ) . {displaystyle |f+g|^{p}leq 2^{p-1}(|f|^{p}+|g|^{p}).}

Indeed, here we use the fact that h ( x ) = | x | p {textstyle h(x)=|x|^{p}} is convex over R + {textstyle mathbb {R} ^{+}} (for p > 1 {textstyle p>1} ) and so, by the definition of convexity,

| 1 2 f + 1 2 g | p ≤ | 1 2 | f | + 1 2 | g | | p ≤ 1 2 | f | p + 1 2 | g | p . {displaystyle left|{tfrac {1}{2}}f+{tfrac {1}{2}}gright|^{p}leq left|{tfrac {1}{2}}|f|+{tfrac {1}{2}}|g|right|^{p}leq {tfrac {1}{2}}|f|^{p}+{tfrac {1}{2}}|g|^{p}.}

This means that

| f + g | p ≤ 1 2 | 2 f | p + 1 2 | 2 g | p = 2 p − 1 | f | p + 2 p − 1 | g | p . {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}.}

Now, we can legitimately talk about ‖ f + g ‖ p {textstyle |f+g|_{p}} . If it is zero, then Minkowski's inequality holds. We now assume that ‖ f + g ‖ p {textstyle |f+g|_{p}} is not zero. Using the triangle inequality and then Hölder's inequality, we find that

‖ f + g ‖ p p = ∫ | f + g | p d μ = ∫ | f + g | ⋅ | f + g | p − 1 d μ ≤ ∫ ( | f | + | g | ) | f + g | p − 1 d μ = ∫ | f | | f + g | p − 1 d μ + ∫ | g | | f + g | p − 1 d μ ≤ ( ( ∫ | f | p d μ ) 1 p + ( ∫ | g | p d μ ) 1 p ) ( ∫ | f + g | ( p − 1 ) ( p p − 1 ) d μ ) 1 − 1 p Hölder's inequality = ( ‖ f ‖ p + ‖ g ‖ p ) ‖ f + g ‖ p p ‖ f + g ‖ p {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}}}

We obtain Minkowski's inequality by multiplying both sides by

‖ f + g ‖ p ‖ f + g ‖ p p . {displaystyle {frac {|f+g|_{p}}{|f+g|_{p}^{p}}}.}

Given t ∈ ( 0 , 1 ) {displaystyle tin (0,1)} , one has, by convexity (Jensen's inequality), for every x ∈ S {displaystyle xin S}

| f ( x ) + g ( x ) | p = | ( 1 − t ) f ( x ) 1 − t + t g ( x ) t | p ≤ ( 1 − t ) | f ( x ) 1 − t | p + t | g ( x ) t | p = | f ( x ) | p ( 1 − t ) p − 1 + | g ( x ) | p t p − 1 . {displaystyle |f(x)+g(x)|^{p}={Bigl |}(1-t){frac {f(x)}{1-t}}+t{frac {g(x)}{t}}{Bigr |}^{p}leq (1-t){Bigl |}{frac {f(x)}{1-t}}{Bigr |}^{p}+t{Bigl |}{frac {g(x)}{t}}{Bigr |}^{p}={frac {|f(x)|^{p}}{(1-t)^{p-1}}}+{frac {|g(x)|^{p}}{t^{p-1}}}.}

By integration this leads to

∫ S | f + g | p d μ ≤ 1 ( 1 − t ) p − 1 ∫ S | f | p d μ + 1 t p − 1 ∫ S | g | p d μ . {displaystyle int _{S}|f+g|^{p},mathrm {d} mu leq {frac {1}{(1-t)^{p-1}}}int _{S}|f|^{p},mathrm {d} mu +{frac {1}{t^{p-1}}}int _{S}|g|^{p},mathrm {d} mu .}

One takes then

t = ‖ g ‖ p ‖ f ‖ p + ‖ g ‖ p {displaystyle t={frac {Vert gVert _{p}}{Vert fVert _{p}+Vert gVert _{p}}}}

to reach the conclusion.

Suppose that ( S 1 , μ 1 ) {textstyle (S_{1},mu _{1})} and ( S 2 , μ 2 ) {textstyle (S_{2},mu _{2})} are two ?-finite measure spaces and F : S 1 × S 2 → R {textstyle F:S_{1}times S_{2}to mathbb {R} } is measurable. Then Minkowski's integral inequality is:[1][2]

[ ∫ S 2 | ∫ S 1 F ( x , y ) μ 1 ( d x ) | p μ 2 ( d y ) ] 1 p ≤ ∫ S 1 ( ∫ S 2 | F ( x , y ) | p μ 2 ( d y ) ) 1 p μ 1 ( d x ) , p ∈ [ 1 , ∞ ) {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),quad pin [1,infty )}

with obvious modifications in the case p = ∞ . {textstyle p=infty .} If p > 1 , {textstyle p>1,} and both sides are finite, then equality holds only if | F ( x , y ) | = φ ( x ) ψ ( y ) {textstyle |F(x,y)|=varphi (x),psi (y)} a.e. for some non-negative measurable functions φ {textstyle varphi } and ψ {textstyle psi } .

If μ 1 {textstyle mu _{1}} is the counting measure on a two-point set S 1 = { 1 , 2 } , {textstyle S_{1}={1,2},} then Minkowski's integral inequality gives the usual Minkowski inequality as a special case: for putting f i ( y ) = F ( i , y ) {textstyle f_{i}(y)=F(i,y)} for i = 1 , 2 , {textstyle i=1,2,} the integral inequality gives

‖ f 1 + f 2 ‖ p = ( ∫ S 2 | ∫ S 1 F ( x , y ) μ 1 ( d x ) | p μ 2 ( d y ) ) 1 p ≤ ∫ S 1 ( ∫ S 2 | F ( x , y ) | p μ 2 ( d y ) ) 1 p μ 1 ( d x ) = ‖ f 1 ‖ p + ‖ f 2 ‖ p . {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}.}

If the measurable function F : S 1 × S 2 → R {textstyle F:S_{1}times S_{2}to mathbb {R} } is non-negative then for all 1 ≤ p ≤ q ≤ ∞ , {textstyle 1leq pleq qleq infty ,} [3]

‖ ‖ F ( ⋅ , s 2 ) ‖ L p ( S 1 , μ 1 ) ‖ L q ( S 2 , μ 2 ) ≤ ‖ ‖ F ( s 1 , ⋅ ) ‖ L q ( S 2 , μ 2 ) ‖ L p ( S 1 , μ 1 ) . {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})} .}

This notation has been generalized to

‖ f ‖ p , q = ( ∫ R m [ ∫ R n | f ( x , y ) | q d y ] p q d x ) 1 p {displaystyle |f|_{p,q}=left(int _{mathbb {R} ^{m}}left[int _{mathbb {R} ^{n}}|f(x,y)|^{q}mathrm {d} yright]^{frac {p}{q}}mathrm {d} xright)^{frac {1}{p}}}

for f : R m + n → E , {textstyle f:mathbb {R} ^{m+n}to E,} with L p , q ( R m + n , E ) = { f ∈ E R m + n : ‖ f ‖ p , q < ∞ } . {textstyle {mathcal {L}}_{p,q}(mathbb {R} ^{m+n},E)={fin E^{mathbb {R} ^{m+n}}:|f|_{p,q}<infty }.} Using this notation, manipulation of the exponents reveals that, if p < q , {textstyle p<q,} then ‖ f ‖ q , p ≤ ‖ f ‖ p , q {textstyle |f|_{q,p}leq |f|_{p,q}} .

When p < 1 {textstyle p<1} the reverse inequality holds:

‖ f + g ‖ p ≥ ‖ f ‖ p + ‖ g ‖ p . {displaystyle |f+g|_{p}geq |f|_{p}+|g|_{p}.}

We further need the restriction that both f {textstyle f} and g {textstyle g} are non-negative, as we can see from the example f = − 1 , g = 1 {textstyle f=-1,g=1} and p = 1 {textstyle p=1}

‖ f + g ‖ 1 = 0 < 2 = ‖ f ‖ 1 + ‖ g ‖ 1 . {textstyle |f+g|_{1}=0<2=|f|_{1}+|g|_{1}.}

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 p ≤ 1 , {textstyle pleq 1,} such as the harmonic mean and the geometric mean are concave.

The Minkowski inequality can be generalized to other functions ϕ ( x ) {textstyle phi (x)} beyond the power function x p {textstyle x^{p}} . The generalized inequality has the form

ϕ − 1 ( ∑ i = 1 n ϕ ( x i + y i ) ) ≤ ϕ − 1 ( ∑ i = 1 n ϕ ( x i ) ) + ϕ − 1 ( ∑ i = 1 n ϕ ( y i ) ) . {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).}

Various sufficient conditions on ϕ {textstyle phi } have been found by Mulholland[4] and others. For example, for x ≥ 0 {textstyle xgeq 0} one set of sufficient conditions from Mulholland is

1. ϕ ( x ) {textstyle phi (x)} is continuous and strictly increasing with ϕ ( 0 ) = 0. {textstyle phi (0)=0.}
2. ϕ ( x ) {textstyle phi (x)} is a convex function of x . {textstyle x.}
3. log ⁡ ϕ ( x ) {textstyle log phi (x)} is a convex function of log ⁡ ( x ) . {textstyle log(x).}

• Cauchy-Schwarz inequality - Mathematical inequality relating inner products and norms
• Hölder's inequality - Inequality between integrals in Lp spaces
• Mahler's inequality
• Young's convolution inequality - Mathematical inequality about the convolution of two functions
• Young's inequality for products - Mathematical concept

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• 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.