For $||x|| \leq ||y||$ is it true that $||x||^4 + ||y||^4 - 2\langle x, y \rangle ^2 \leq 3 ||y||^2 ||x-y||^2$?

Question

I have recently come across this seemingly simple Hilbert space inequality

$$||x||^4 + ||y||^4 - 2\langle x, y \rangle ^2 \leq 3 ||y||^2 ||x-y||^2 $$

for $||x|| \leq ||y||$, where $<\cdot, \cdot>$ denotes the inner product and $||\cdot||$ the corresponding norm. I have been trying to prove it but something always seems to be off. If the inequality holds, could you please give me a pointer on how to prove it?

Thank you in advance.

Answer 1

Let's consider our Hilbert space to be $\mathbb{R}$ and let $y=1$, $x=1-h$

The left-hand side is:

$(1-h)^4+1-2(1-h)^2=4h^2-4h^3+h^4$

The right-hand side is $3h^2$. For $h$ smaller than $2-\sqrt3$, the inequality doesn't hold.