Show that $| z + 1 | + ∣ ∣ 1 + z + z^{2} ∣ ∣ + ∣ ∣ 1 + z^{3} ∣ ∣ \geq 1$ $|z+1|+|1+z+z^2|+|1+z^3|>=1$ ?

Question

If $z$ $z$ $\in$ $in$ $CC$ , show that $|z+1|+|1+z+z^2|+|1+z^3|>=1$

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Answer 1 · 2018-05-11T14:43:12

$|z+1|+|z^2+z+1|>=|(z^2+z+1)-(z+1)|=|z^2|=|z|^2>=1$

$|z+1|+|z^2+z+1|>=|z||z+1|+|z^2+z+1|=$

$|z(z+1)|+|z^2+z+1|=|z^2+z|+|z^2+z+1|>=|(z^2+z+1)-(z^2+z)|=1$

Hence, $|z+1|+|1+z+z^2|>=1$ , $z$ $in$ $CC$

and

$|z+1|+|1+z+z^2|+|1+z^3|>=|1+z|+|1+z+z^2|>=1$

,
" $=$ " , $z=-1vvz=e^((2k+1)iπ)$ , $k$ $in$ $ZZ$