How does one prove: $$ E(|X+Y|^r)\leq c_r\bigg[E(|X|^r)+E(|Y|^r)\bigg] $$ where $c_r=1$ if $0<r\leq 1$ and $2^{r-1}$ if $r>1$?
This is a classic result whose proof I once knew but have since forgotten. Can someone please explain so I could learn it again? Thank you.
For $a,b\geqslant 0$, show and use the inequalities $$(a+b)^r\leqslant a^r+b^r\mbox{ if }0\lt r\leqslant 1 $$ and $$(a+b)^r\leqslant 2^{r-1}(a^r+b^r)\mbox{ if }r\gt1.$$
This uses concavity (for $r\leqslant 1$) and convexity (for $r\gt 1$) of the function $t\mapsto t^r$.
