Minkowski's Inequality
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Theorem
Theorem for Sums
Let $a_1, a_2, \ldots, a_n, b_1, b_2, \ldots, b_n \ge 0$ be real numbers.
Let $p \in \R$ be a real number.
If $p > 1$, then:
- $\displaystyle \left[{\sum_{k=1}^n \left({a_k + b_k}\right)^p}\right]^{1/p} \le \left({\sum_{k=1}^n a_k^p}\right)^{1/p} + \left({\sum_{k=1}^n b_k^p}\right)^{1/p}$
If $p < 1$ and $p \ne 0$, then the reverse inequality holds.
Theorem for Integrals
Let $f, g$ be integrable functions in $X \subseteq \R^n$ with respect to the volume element $dV$.
Riemann integrable or Lebesgue integrable?
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- $(1):\quad$ Let $p > 1$. Then:
- $\displaystyle \left({\int_X \left\vert{f + g}\right\vert^p \mathrm d V}\right)^{1/p} \le \left({\int_X \left\vert{f}\right\vert^p \mathrm d V}\right)^{1/p} + \left({\int_X \left\vert{g}\right\vert^p \mathrm d V}\right)^{1/p}$
- $(2):\quad$ Let $p < 1, p \ne 0$. Then:
- $\displaystyle \left({\int_X \left\vert{f + g}\right\vert^p \mathrm d V}\right)^{1/p} \ge \left({\int_X \left\vert{f}\right\vert^p \mathrm d V}\right)^{1/p} + \left({\int_X \left\vert{g}\right\vert^p \mathrm d V}\right)^{1/p}$
Source of Name
This entry was named for Hermann Minkowski.
