If $Y_n$ converges in mean square to $Y$, then prove/disprove $Y_n$ converges in $r$th mean to $Y$

Question

Let $ Y_1,\ldots,Y_n$ be a series of random variables such that $Y_n$ converges in mean square to $Y$. Prove or disprove that for $ 2 \ge r \ge 1$, $Y_n$ converges in $r$th mean to $Y$.

I have refered to this already, but the thing is we have not spoken about the mentioned Lyapunov’s inequality in my course and this was an exercise.

I have tried playing around with Jensen's inequality but no sucess. any ideas ?

Answer 1

It's just Holder inequality. Let $t>0$ s.t. $\frac{1}{r}=\frac{1}{2}+\frac{1}{t}.$ By Holder inequality

$$\mathbb E[|Y-Y_n|^r]\leq \mathbb E[|Y-Y_n|^2]^{r/2}\underset{n\to \infty }{\longrightarrow }0.$$