I have been given a problem as practice that I do not understand how to solve. Here is the question:
Given $x$,$p$ and $y$ are real numbers, and $0\leq p\leq 1$, prove that $|x|^p +|y|^p \geq |x+y|^p$
I have tried writing down everything I know that could help, which lead me to these inequalities:
$|a|^p + |b|^p \leq |a| + |b|$
$|a+b|^p \leq |a+b| $
$|a+b| \leq |a| + |b|$
However, these don't seem to be able to lead to the desired result. If anyone could tell me what I'm missing, that would be greatly appreciated!
As suggested by one of the comments, use Minkowski inequality: That is, if $q\geq 1$ then we have \begin{align} (|x|^q+|y|^q)^{1/q} \leq |x|+|y|. \end{align} To show the inequality for $0<p<1$, it suffices to show \begin{align} |x|+|y| \leq (|x|^p+|y|^p)^{1/p}. \end{align}
Set $s = |x|^{p}$ and $t=|y|^{p}$. Then observe that \begin{align} |x|+|y| = t^q+s^q \end{align} where $q=1/p>1$. Then by Minkowski inequality, we have \begin{align} t^q+s^q \leq (t+s)^{q} \ \ \Longleftrightarrow \ \ \ |x|+|y| \leq (|x|^p+|y|^p)^{1/p}. \end{align}