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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