# A Hölder-looking inequality of random variables

I am trying to show that, if $X,Y$ are positive random variables, $$E((X+Y)^p)\leq 2^p\left( E(X^p)+E(Y^p)\right)$$where, if $0\leq p <1$, the $2^p$ can be replaced by $1$. I've been given the hint to think about the proof to Hölder's inequality, but I am not finding this tractable at all. Can anyone push me in the right direction?

• $p \geq 1$: By Hölder's inequality, it holds for any real numbers $x,y \in \mathbb{R}$ that $$(x \cdot 1+y \cdot 1)^p \leq 2^p (x^p+y^p).$$ Set $x=X(\omega)$, $y=Y(\omega)$ and integrate both sides.
• $p \in (0,1)$: Since the mapping $x \mapsto f(x) := x^p$ is concave, it is in particular sub-additive, i.e. $$f(x+y) \leq f(x)+f(y).$$ Again, integrating both sides yields the desired inequality.
• $p>1$ in Hölder's inequality, right? – Cm7F7Bb Oct 3 '14 at 8:05
• @V.C. Ah, right, I totally missed that the OP also wants to consider $p \in (0,1)$. Should be fine now. – saz Oct 3 '14 at 8:12
The case when $0<p\le1$ follows from the inequality $$(a+b)^p\le a^p+b^p,$$ where $a$ and $b$ are non-negative real numbers and $0<p\le1$.
Suppose that $a\ne0$. Define a function $$f(a)=(a+b)^p-a^p-b^p,$$ where $a>0$ and $b$ is some fixed non-negative real number. Then $$f'(a)=p[(a+b)^{p-1}-a^{p-1}]<0$$ for all $a>0$. We have that $f(0)=0$ and $f'(a)<0$ for all $a>0$. Hence, $f(a)\le0$.