Are you sure of your value?

Geometry Level pending

There is a constant C, independent of $$n$$, such that if $$\{z_j\}$$ are complex numbers and if

$$\sum_{j=1}^n |z_j| \geq 1,$$

then there is a sub-collection $$\{z_{j=1},...,z_{j_h}\}\subseteq\{z_1,...,z_n\}$$ such that

$$\Bigg | \sum_{m=1}^{k} z_{j_m} \Bigg | \geq C.$$

What is the best (largest) constant $$C$$?

For example, we can surely allow $$C=0$$ and have the necessary equations always satisfied, but would that be the largest value?

taken from 'Function Theory of One Complex Variable' by Robert Greene

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