If |X|r < (, then show that |B X|r ( B |X|r a.s., r ( 1. One

If ε|X|r < (, then show that |εB X|r ( εB |X|r a.s., r ( 1.
One way to establishing this inequality is to use the Jensen inequality (Theorem 8). For this purpose, take g(x) = |X|r, r ( 1, and observe that it is convex. (It is convex for x ( 0 and symmetric about the y-axis, hence convex in (.).