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complex inequalities, |z|+|w|≤|z+w|+|z-w|

Source: France 1986 P3

May 19, 2021
inequalitiesComplex numbercomplex inequalityalgebra

Problem Statement

(a) Prove or find a counter-example: For every two complex numbers z,wz,w the following inequality holds: ∣z∣+∣w∣≤∣z+w∣+∣z−w∣.|z|+|w|\le|z+w|+|z-w|.(b) Prove that for all z1,z2,z3,z4∈Cz_1,z_2,z_3,z_4\in\mathbb C: ∑k=14∣zk∣≤∑1≤i<j≤4∣zi+zj∣.\sum_{k=1}^4|z_k|\le\sum_{1\le i<j\le4}|z_i+z_j|.
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