If m,n,p are distinct complex numbers of moduli 1, then |m+n|+|n+p|+|p+m| does not exceed 3.
My trial was to put m=cosA+isinA and so on, and I needed to prove that sum of |cos((A-B)/2)|<=3/2.
2026-04-12 07:32:57.1775979177
Is this complex numbers inequality true?
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This inequality is clearly not true. Just take any three complex numbers which are suitably close together on the unit circle. For example $$(m,n,p)=(1,e^{0.1i},e^{-0.1i})$$ then we have $$|m+n|+|n+p|+|p+m|=5.9850094\dots\gt3$$ In fact the value of this expression can be made arbitrarily close to $6$ by choosing suitably close $m,n,p$.