MathDB

Find the smallest positive real number

k

such that the following inequality holds

\left|z_{1}+\ldots+z_{n}\right| \geqslant \frac{1}{k}\big(\left|z_{1}\right|+\ldots+\left|z_{n}\right|\big) .

for every positive integer

n \geqslant 2

and every choice

z_{1}, \ldots, z_{n}

of complex numbers with non-negative real and imaginary parts.
[Hint: First find

k

that works for

n=2

. Then show that the same

k

works for any

n \geqslant 2

Problem Statement