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Inequality on Complex Numbers

Source: China Mathematical Olympiad 2015 Q1

December 20, 2014
inequalitiestrigonometrycomplex numbersinequalities proposed

Problem Statement

Let z1,z2,...,znz_1,z_2,...,z_n be complex numbers satisfying ∣zi−1∣≤r|z_i - 1| \leq r for some rr in (0,1)(0,1). Show that
∣∑i=1nzi∣⋅∣∑i=1n1zi∣≥n2(1−r2). \left | \sum_{i=1}^n z_i \right | \cdot \left | \sum_{i=1}^n \frac{1}{z_i} \right | \geq n^2(1-r^2).
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