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Inequality involving complex

Source: Science ON 2021 grade X/1

March 8, 2021
algebracomplex numbersInequality

Problem Statement

Consider the complex numbers x,y,zx,y,z such that ∣x∣=∣y∣=∣z∣=1|x|=|y|=|z|=1. Define the number a=(1+xy)(1+yz)(1+zx).a=\left (1+\frac xy\right )\left (1+\frac yz\right )\left (1+\frac zx\right ). <spanclass=′latex−bold′>(a)</span><span class='latex-bold'>(a)</span> Prove that aa is a real number. <spanclass=′latex−bold′>(b)</span><span class='latex-bold'>(b)</span> Find the minimal and maximal value aa can achieve, when x,y,zx,y,z vary subject to ∣x∣=∣y∣=∣z∣=1|x|=|y|=|z|=1.
(Stefan Bălăucă & Vlad Robu)
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