Let $ f(x)=x^r, 1<r<\infty, x\in \mathbb{R^+}=[0,\infty) ~and ~n\in \mathbb{N}.$ Define $$\pi(x)= 1 ~~if~~ x\leq n ~and =0 ~if~ x< (n+1).$$ Then for any $x,y\in\mathbb{R},~~ $ I want to prove that $$\|\pi(\|x\|_{L^r})x^r-\pi(\|y\|_{L^r})y^r\|_{L^1}\leq C\|x-y\|_{L^r}, $$ where $C$ is constant and strictly less than $1.$
Can Any one help me to prove it? Any help will be appreciated.