Bounding p-norm expression using p-norm inequality

[ENgez]

problem statement:

need to show:
$$||w||_p^2+||u||_p^2-2(||u||_p^p)^{\frac{2}{p}-1}\Sigma_i(u(i)^{p-1}w(i))$$

can be bounded as a function of

$$||w-u||_p^2$$

where $$p\in[2,\infty)$$

work done:

the expressions are equal for p=2, and i suspect that

$$||w||_p^2+||u||_p^2-2(||u||_p^p)^{\frac{2}{p}-1}\Sigma_i(u(i)^{p-1}w(i)) \leq||w-u||_p^2$$

but i get stuck here. Is there some kind of p-norm inequality I can apply here?Thank you!

 

[StoneTemplePython]

Forum rules require you to show your work. From what I see, you just took the problem statement and re-wrote it with "I suspect that" it holds as an upper bound with a scalar multiple of one.

What kind of work can you show to at least justify your suspicion?