Inequality from complex variable

Question

I have been struggling with the following inequality:

Let $\lbrace z_1,z_2,\dots,z_n \rbrace$ and $\lbrace w_1,w_2,\dots,w_n \rbrace$ be complex numbers. Let

$$ |\sum_j z_jw_j| \leq 1, $$

and $\sum_j |w_j|^2 \leq 1$. Prove that $\sum_j |z_j|^2 \leq 1$.

I tried several things, including Cauchy-Schwartz inequality but I couldn't prove it yet.

score 3 · Answer 1 · answered Apr 01 '17 at 03:41

3

Your claim is False. Consider \begin{align} \{w_1, w_2, \ldots, w_n\} = \{1, 0, \ldots, 0\} \end{align} and \begin{align} \{z_1, z_2, \ldots, z_n\} = \{1, 2, \ldots, n\}. \end{align}

answered Apr 01 '17 at 03:41

Jacky Chong

25,739

Yes, it's equal to 1. That doesn't break the hypothesis. – Jacky Chong Apr 01 '17 at 03:45
Sorry, I meant http://math.stackexchange.com/questions/1930236/if-sum-limits-j-1n-w-j2-leq-1-implies-left-sum-limits-j-1n-z – user2820579 Apr 01 '17 at 03:46
Then what's your question? The post you cite already has answers. – Jacky Chong Apr 01 '17 at 03:49
I just found it, and there is a proof for the exercise, without saying that it cannot be fulfilled. One of you two must be wrong... – user2820579 Apr 01 '17 at 03:50
Your original question is different from the question of that post. We are correct. – Jacky Chong Apr 01 '17 at 03:52
Copied wrong... – user2820579 Apr 01 '17 at 03:55
Let us continue this discussion in chat. – user2820579 Apr 01 '17 at 04:03

Inequality from complex variable

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