Suppose $X \geq 0$ and $Y \geq 0$ are random variables and that $p\geq 0$

Question

Suppose $X \geq 0$ and $Y \geq 0$ are random variables and that $p\geq 0$.

1. Prove $$E((X+Y)^p)\leq 2^p (E(X^p)+E(Y^p))$$

   Proof

Since $(X+Y)^p \leq (2 \> \max\{X,Y\})^p=2^p \> \max \{X^p,Y^p\}\leq 2^p(X^p+Y^p)$ $ \implies E((X+Y)^p)\leq 2^p (E(X^p)+E(Y^p))$

2. If $p>1$ the factor $2^p$ may be replaced by $2^{p-1}$

   Proof

$E\left(\left(\frac{X+Y}{2}\right)^p\right) \leq (E(X^p)+E((Y^p))$ I think I want to use Jensen's inequality but lost.

3. If $0 \leq p \leq 1$ the factor $2^p$ can be replaced by $1$

Need help with part 1, 2 and 3 any suggestions.

Answer 1

For 2., since $p>1$, the map $t\mapsto t^p$ is convex on $[0,+\infty)$ hence for each non-negative $a$ and $b$, $$\left(\frac{a+b}2\right)^p\leqslant\frac{a^p}2+\frac{b^p}2 .$$ Apply this to $a:=X(\omega)$ and $b:=Y(\omega)$ and integrate.

For 3., you have to use the fact that $(a+b)^p\leqslant a^p+b^p$ for all non-negative numbers $a$ and $b$.