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Subject: Analysis
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Recomm. for undergrads: yes
Posted by: feanor
on: April 7th, 2010
Solved by: fedorpetrov here in comments
Conjecture   There exists a real positive $ c $, such that for any $ n\in\mathbb{N} $ and any $ z_i\in\mathbb{C} $ where $ |z_i|\le 1 $ for $ 1\le i\le n $ and $ \~z:=\frac{1}{n}\sum^n_{k=1}z_k $, the following holds: $$\left|\prod^n_{k=1}z_k - \~z^n\right| \le c\cdot\sum^n_{k=1}|z_k-\~z|^2$$

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