Is the inequality $| \sqrt[3]{x^2} - \sqrt[3]{y^2} | \le \sqrt[3]{|x -y|^2}$ true? I'm having some trouble deciding whether this inequality is true or not...
$| \sqrt[3]{x^2} - \sqrt[3]{y^2} | \le \sqrt[3]{|x -y|^2}$ for $x, y \in \mathbb{R}.$

it boils down to showing that $|1-u^{2/3}|\leq(1-u)^{2/3}$, $1<u<1$, $u=x/y$, $0<x<y<1$.This is the same as showing that $1<u^{2/3}+(1-u)^{2/3}$. Since $0<u$, $1-u<1$ we know that $u<u^{2/3}$, $1-u<(1-u)^{2/3}$. Adding up we get the affirmative result.
