In mathematical analysis, the Minkowski inequality establishes that the L^p spaces are normed vector spaces. Let S be a measure space, let 1 ≤ p ≤ ∞ and let f and g be elements of L^p(S). Then f + g is in L^p(S), and we have

$\backslash |f+g\backslash |\_p\; \backslash le\; \backslash |f\backslash |\_p\; +\; \backslash |g\backslash |\_p$

If 1 < p < ∞, equality holds only if f = k g or g = k f with k ≥ 0.

The Minkowski inequality is the triangle inequality in L^p(S). Its proof uses Hölder's inequality.

Like Hölder's inequality, the Minkowski inequality can be specialized to sequences and vectors by using the counting measure:

$\backslash left(\; \backslash sum\_\{k=1\}^n\; |x\_k\; +\; y\_k|^p\; \backslash right)^\{1/p\}\; \backslash le\; \backslash left(\; \backslash sum\_\{k=1\}^n\; |x\_k|^p\; \backslash right)^\{1/p\}\; +\; \backslash left(\; \backslash sum\_\{k=1\}^n\; |y\_k|^p\; \backslash right)^\{1/p\}$

for all real (or complex) numbers x₁, ..., x_n, y₁, ..., y_n and where n is the dimension of S.

Proof of Integral form

We consider the $p$-th power of $\backslash |f+g\backslash |\_p$ and find

$\backslash left(\backslash int\_\{a\}^\{b\}|f(x)+g(x)|^\{p\}dx\backslash right)^\{\backslash frac\{1\}\{p\}*p\}=\backslash int\_\{a\}^\{b\}|f(x)+g(x)||f(x)+g(x)|^\{p-1\}dx$

(expanding $|f(x)+g(x)|$ using the triangle equality)

$\backslash leq\backslash int\_\{a\}^\{b\}|f(x)||f(x)+g(x)|^\{p-1\}dx+\backslash int\_\{a\}^\{b\}|g(x)||f(x)+g(x)|^\{p-1\}dx$

(using Hölder's inequality) $\backslash leq\backslash left(\backslash int\_\{a\}^\{b\}|f(x)|^\{p\}dx\backslash right)^\{\backslash frac\{1\}\{p\}\}\backslash left(\backslash int\_\{a\}^\{b\}|f(x)+g(x)|^\{q\backslash left(p-1\backslash right)\}dx\backslash right)^\{\backslash frac\{1\}\{q\}\}+\; \backslash left(\backslash int\_\{a\}^\{b\}|g(x)|^\{p\}dx\backslash right)^\{\backslash frac\{1\}\{p\}\}\backslash left(\backslash int\_\{a\}^\{b\}|f(x)+g(x)|^\{q\backslash left(p-1\backslash right)\}dx\backslash right)^\{\backslash frac\{1\}\{q\}\}$

$=\backslash left*\backslash left(\backslash int\_\{a\}^\{b\}|f(x)+g(x)|^\{qp-q\}dx\backslash right)^\{\backslash frac\{1\}\{q\}\}$

(using that $p=qp-q$, since $\backslash frac\{1\}\{p\}+\backslash frac\{1\}\{q\}=1$)

$\backslash leq\backslash left*\backslash left(\backslash int\_\{a\}^\{b\}|f(x)+g(x)|^\{p\}dx\backslash right)^\{\backslash frac\{1\}\{p\}*\backslash frac\{p\}\{q\}\}$

Now we divide the first and last expression in this sequence of inequalities by the second factor of latter. This gives us

$\backslash left(\backslash int\_\{a\}^\{b\}|f(x)+g(x)|^\{p\}dx\backslash right)^\{\backslash frac\{1\}\{p\}\backslash left(p-\backslash frac\{p\}\{q\}\backslash right)\}\backslash leq\backslash left*$

and as $p-\backslash frac\{p\}\{q\}=1$, we finally arrive at

$\backslash left(\backslash int\_\{a\}^\{b\}|f(x)+g(x)|^\{p\}dx\backslash right)^\{\backslash frac\{1\}\{p\}\}\backslash leq\backslash left*$

which completes the proof.

Inequalities

Minkowski-Ungleichung | Неравенство Минковского | Minkowski Eşitsizliği