Interesting inequality of the modules of the complex numbers: $\bigl|\sum_{j\in J} z_j\bigr|\ge \frac{1}{4\sqrt 2} \sum_{j=1}^n |z_j|.$

Asked May 05 '23 at 00:54

Active May 25 '23 at 16:27

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The problem

Let $z_1,z_2,\dots,z_n$ be complex numbers. Show that there exists $J\subseteq\{1,2,\dots,n\}$ such that $$\left|\sum_{j\in J} z_j\right|\ge \frac{1}{4\sqrt 2} \sum_{j=1}^n |z_j|.$$

My steps:
I have been trying to solve this problem, my first idea is to do it by induction, which by assuming n = 1 is true, and by taking n+1, playing with the products and analyzing cases, I can get that it is true in some cases, in others I am not sure how to justify it, but my question is if there is another way to solve this problem?

edited May 25 '23 at 16:27

Xander Henderson

32,453

asked May 05 '23 at 00:54

ruka

At least a quarter of the $z_k$ are in one of the four quadrants defined by the lines $x=\pm y$; show that if $w_k$ are in one such quadrant the inequality with only $1/\sqrt 2$ holds considering either the real or imaginary parts and then conclude – Conrad May 05 '23 at 02:04
Searching with Approach0 returns this post: Inequality with complex numbers. The linked questions might be of interest, too. – Martin Sleziak May 05 '23 at 05:12

Interesting inequality of the modules of the complex numbers: $\bigl|\sum_{j\in J} z_j\bigr|\ge \frac{1}{4\sqrt 2} \sum_{j=1}^n |z_j|.$

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