TITLE: Name of $|x|^p+|y|^p\le (|x|+|y|)^p$ ($p\ge 1$)?
QUESTION [6 upvotes]: I checked these 
What is the difference between square of sum and sum of square?
Prove $(|x| + |y|)^p \le |x|^p + |y|^p$ for $x,y \in \mathbb R$ and $p \in (0,1]$.
It is easy to see $p$-th power ($p\ge 1$) version, i.e., $|x|^p+|y|^p\le (|x|+|y|)^p$ ($p\ge 1$), holds as well using the argument by Quang Hoang in Prove $(|x| + |y|)^p \le |x|^p + |y|^p$ for $x,y \in \mathbb R$ and $p \in (0,1]$. Is there a name for this inequality so that I can just quote? (It might be an elementary result but people around me bother to put "from Cauchy--Schwarz,..." when it is clearly Cauchy--Schwarz, so.)

REPLY [4 votes]: I think this should be called an special case of Minkowski's Inequality. 
1 - For finite sequences, the Minkowski Inequality states that 
$$ \left(\sum_{k=1}^n |x_k + y_k |^p \right) ^{1/p} \le \left(\sum_{k=1}^n |x_k|^p \right)^{1/p} + \left(\sum_{k=1}^n |y_k|^p\right)^{1/p} $$ 
Your inequality then follows if $x_1 = x, x_i = 0, \; i \ge 2$, and $y_2 = y, y_i = 0, i \ne 2$.
2 - More Generally, the case above can be easily extended to infinite sequences. The most general result is that, if $f,g : \Omega \to \Bbb{R}$ are measurable functions in a measure space $(\Omega, \Sigma, \mu)$, then the inequality 
$$ \| f+g \|_{L^p (d\mu)} \le \|f\|_{L^p(d\mu)} + \|g\|_{L^p(d\mu)} $$
Where $ \|f\|_{L^p(d\mu)}^p = \int_ {\Omega} |f(x)|^p d \mu (x) $
3 - There was a little lie at topic 2, because there is yet another generalization of this result, called Minkowski's Inequality for Integrals. In the link above, you might check a proof of this result, based on these previously presented ones.