Minkowski inequality (also known as Brunn Minkowski inequality) states that if two functions ‘f’ and ‘g’ and their sum (f + g) is measurable, then for 1 ≤ p < ∞,
||f + g||p ≤ ||f||p + ||g||p which implies ${\left( \int \left| f+g\right| ^{p}\right) ^{\dfrac{1}{p}}\leq \left( \int \left| f\right| ^{p}\right) ^{\dfrac{1}{p}}+\left( \int \left| g\right| ^{p}\right) ^{\dfrac{1}{p}}}$
Mathematically,
If f, g Є Lp and (f + g) Є Lp, then for 1 ≤ p < ∞,
||f + g||p ≤ ||f||p + ||g||p
⇒ ${\left( \int \left| f+g\right| ^{p}\right) ^{\dfrac{1}{p}}\leq \left( \int \left| f\right| ^{p}\right) ^{\dfrac{1}{p}}+\left( \int \left| g\right| ^{p}\right) ^{\dfrac{1}{p}}}$
This is the generalized Minkowski inequality for integrals.
In the field of mathematical analysis, the Minkowski inequality confirms that Lp spaces qualify as normed vector spaces.
Proof
Firstly, we will verify that the inequality holds for p = 1, and then for 1 < p < ∞
When p = 1,
||f + g|| = ${\int \left| f+g\right|}$ ≤ ${\int \left| f\right| +\int \left| g\right|}$ = ||f||1 + ||g||1
When p > 1,
|f + g|p = (|f + g|) ⋅ (|f + g|)p – 1 ≤ (|f| + |g|) ⋅ (|f + g|)p – 1 = |f| ⋅ (|f + g|)p – 1 + |g| ⋅ (|f + g|)p – 1
⇒ |f + g|p ≤ |f| ⋅ (|f + g|)p – 1 + |g| ⋅ (|f + g|)p – 1 …..(i)
From Hölder’s inequality, we know
${\dfrac{1}{p}+\dfrac{1}{q}=1}$
⇒ ${q=\dfrac{p}{p-1}}$
⇒ ${p=q\left( p-1\right)}$
Now, |f + g|q(p – 1) = |f + g|p
⇒ ${\int \left| f+g\right| ^{q\left( p-1\right) }=\int \left| f+g\right| ^{p}}$ …..(ii)
By definition of Lp space, R.H.S. of (ii) is finite, and thus, L.H.S. of (ii) is also finite.
⇒ ${\int \left| f+g\right| \left( p-1\right)}$ Є Lq
Now, by integrating (i), we get
${\int \left| f+g\right| ^{p}\leq \int \left| f\right| \cdot \left| f+g\right| ^{p-1}+\int \left| g\right| \cdot \left| f+g\right| ^{p-1}}$
Using Hölder’s inequality in R.H.S, we get
${\int \left| f+g\right| ^{p}\leq \left\| f\right\| _{p}\cdot \left\| f+g\right\| _{q}^{p-1}+\left\| g\right\| _{p}\cdot \left\| f+g\right\| _{q}^{p-1}}$
⇒ ${\left\| f+g\right\| ^{p}\leq \left( \left\| f\right\| _{p}+\left\| g\right\| _{p}\right) \cdot \left\| f+g\right\| _{q}^{p-1}}$ …..(iii)
If ${\left\| f+g\right\| _{q}^{p-1}}$ = 0, then from (iii), ${\int \left| f+g\right| ^{p}=0}$
⇒ ${\left\| f+g\right\| ^{p}=0=\left\| f+g\right\| _{q}^{p-1}}$
If possible let ${\left\| f+g\right\| _{q}^{p-1}}$ ≠ 0
Its value of integration in (iii) is ${\int \left( \left| f+g\right| ^{p-1}\right) ^{q}}$
Now, dividing the inequality (iii) by ${\left( \int \left| f+g\right| ^{p}\right) ^{\dfrac{1}{q}}}$, we get
${\dfrac{\int \left| f+g\right| ^{p}}{\left( \int \left| f+g\right| ^{p}\right) ^{\dfrac{1}{q}}}\leq \dfrac{\left( \left\| f\right\| _{p}+\left\| g\right\| _{p}\right) \left( \int \left| f+g\right| ^{\left( p-1\right) q}\right) ^{\dfrac{1}{q}}}{\left( \int \left| f+g\right| ^{p}\right) ^{\dfrac{1}{q}}}}$
⇒ ${\dfrac{\int \left| f+g\right| ^{p}}{\left( \int \left| f+g\right| ^{p}\right) ^{\dfrac{1}{q}}}\leq \left( \left\| f\right\| _{p}+\left\| g\right\| _{p}\right)}$ (Since q(p – 1) = p)
⇒ ${\left( \int \left| f+g\right| ^{p}\right) ^{1-\dfrac{1}{q}}\leq \left\| f\right\| _{p}+\left\| g\right\| _{p}}$ …..(iv)
Since ${\dfrac{1}{p}+\dfrac{1}{q}=1}$, which means ${1-\dfrac{1}{q}=\dfrac{1}{p}}$
Thus, from (iv), we get ${\left( \int | f+g\right) ^{\dfrac{1}{p}}\leq \left\| f\right\| _{p}+\left\| g\right\| _{p}}$
⇒ ${\left\| f+g\right\| _{p}\leq \left\| f\right\| _{p}+\left\| g\right\| _{p}}$
Thus, Minkowski’s inequality is proved.
Summation Form
When expressed in the summation form, it is written as
If x, y Є Lp and (x + y) Є Lp, then for 1 ≤ p < ∞,
${\left( \sum ^{\infty }_{i=1}\left| x_{i}+y_{i}\right| ^{p}\right) ^{\dfrac{1}{p}}\leq \left( \sum ^{\infty }_{i=1}\left| x_{i}\right| ^{p}\right) ^{\dfrac{1}{p}}+\left( \sum ^{\infty }_{i=1}\left| y_{i}\right| ^{p}\right) ^{\dfrac{1}{p}}}$
Proof
When p = 1,
The inequality follows from the triangular inequality of real numbers.
When p > 1,
${\sum ^{\infty }_{i=1}\left| x_{i}+y_{i}\right| ^{p}=\sum ^{\infty }_{i=1}\left| x_{i}+y_{i}\right| \cdot \left| x_{i}+y_{i}\right| ^{p-1}\leq \sum ^{\infty }_{i=1}\left( \left| x_{i}\right| +\left| y_{i}\right| \right) \cdot \left| x_{i}+y_{i}\right| ^{p-1}}$
⇒ ${\sum ^{\infty }_{i=1}\left| x_{i}+y_{i}\right| ^{p}\leq \sum ^{\infty }_{i=1}\left| x_{i}\right| \cdot \left| x_{i}+y_{i}\right| ^{p-1}+\sum ^{\infty }_{i=1}\left| y_{i}\right| \cdot \left| x_{i}+y_{i}\right| ^{p-1}}$ …..(i)
From Hölder’s inequality, we know
${\dfrac{1}{p}+\dfrac{1}{q}=1}$
⇒ ${q=\dfrac{p}{p-1}}$
⇒ ${p=q\left( p-1\right)}$
Using Hölder’s inequality in R.H.S. of (i), we get
${\sum ^{\infty }_{i=1}\left| x_{i}+y_{i}\right| ^{p}\leq \left( \sum ^{\infty }_{i=1}\left| x_{i}\right| ^{p}\right) ^{\dfrac{1}{p}}\left( \sum ^{\infty }_{i=1}\left| x_{i}+y_{i}\right| ^{q\left( p-1\right) }\right) ^{\dfrac{1}{q}}+\left( \sum ^{\infty }_{i=1}\left| y_{i}\right| ^{p}\right) ^{\dfrac{1}{p}}\left( \sum ^{\infty }_{i=1}\left| x_{i}+y_{i}\right| ^{q\left( p-1\right) }\right) ^{\dfrac{1}{q}}}$ …..(ii)
Now, dividing the inequality (ii) by ${\left( \sum ^{\infty }_{i=1}\left| x_{i}+y_{i}\right| ^{p}\right) ^{\dfrac{1}{q}}}$, we get
${\dfrac{\sum ^{\infty }_{i=1}\left| x_{i}+y_{i}\right| ^{p}}{\left( \sum ^{\infty }_{i=1}\left| x_{i}+y_{i}\right| ^{p}\right) ^{\dfrac{1}{q}}}\leq \dfrac{\left( \sum ^{\infty }_{i=1}\left| x_{i}\right| ^{p}\right) ^{\dfrac{1}{p}}\left( \sum ^{\infty }_{i=1}\left| x_{i}+y_{i}\right| ^{q\left( p-1\right) }\right) ^{\dfrac{1}{q}}}{\left( \sum _{i=1}\left| x_{i}+y_{i}\right| \right) ^{p}\dfrac{1}{q}}+\dfrac{\left( \sum ^{\infty }_{i=1}\left| y_{i}\right| ^{p}\right) \dfrac{1}{p}\left( \sum ^{\infty }_{i=1}\left| x_{i}+y_{i}\right| ^{q\left( p-1\right) }\right) ^{\dfrac{1}{q}}}{\left( \sum _{i=1}\left| x_{i}+y_{i}\right| \right) ^{p}\dfrac{1}{q}}}$
⇒ ${\dfrac{\sum ^{\infty }_{i=1}\left| x_{i}+y_{i}\right| ^{p}}{\left( \sum ^{\infty }_{i=1}\left| x_{i}+y_{i}\right| ^{p}\right) ^{\dfrac{1}{q}}}\leq \left( \sum ^{\infty }_{i=1}\left| x_{i}\right| ^{p}\right) ^{\dfrac{1}{p}}+\left( \sum ^{\infty }_{i=1}\left| y_{i}\right| ^{p}\right) ^{\dfrac{1}{p}}}$ (Since q(p – 1) = p)
⇒ ${\left( \sum ^{\infty }_{i=1}\left| x_{i}+y_{i}\right| ^{p}\right) ^{1-\dfrac{1}{q}}\leq \left( \sum ^{\infty }_{i=1}\left| x_{i}\right| ^{p}\right) ^{\dfrac{1}{p}}+\left( \sum ^{\infty }_{i=1}\left| y_{i}\right| ^{p}\right) ^{\dfrac{1}{p}}}$ …..(iii)
Since ${\dfrac{1}{p}+\dfrac{1}{q}=1}$, which means ${1-\dfrac{1}{q}=\dfrac{1}{p}}$
Thus, from (iii), we get
${\left( \sum ^{\infty }_{i=1}\left| x_{i}+y_{i}\right| ^{p}\right) ^{\dfrac{1}{p}}\leq \left( \sum ^{\infty }_{i=1}\left| x_{i}\right| ^{p}\right) ^{\dfrac{1}{p}}+\left( \sum ^{\infty }_{i=1}\left| y_{i}\right| ^{p}\right) ^{\dfrac{1}{p}}}$
Thus, Minkowski’s inequality is proved.
