We want to prove

.

Multiply both sides by 4:

.

Since , we can make (2) homogenous of degree three by writing it as

.

Multiply out the brackets in (3) and rearrange it a bit, and you will get

.

So far, the argument is reversible, so if we can prove (4) then (1) will also hold.

We are told that , and . Therefore

.

Multiply out the brackets in (5), rearrange it a bit, and you get exactly the inequality (4), as wanted.