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.