Any hint for this exercise from Casella´s book:

[![enter image description here](https://i.stack.imgur.com/TkB8N.png)](https://i.stack.imgur.com/TkB8N.png)

I tried with Cauchy Schwarz, Minkovsky Inequality but I am stuck.

I also tried to calculate the Variance but its not clear that they are independent.

Any help?