Question

If $\left|z\right|<\frac{1}{2},$ show that $|(1+i)z^3+iz|<\frac{3}{4}$.

Answer 1

$|z|<1/2\Rightarrow |z{|}^3<\frac{1}{8}$

To show

$|(1+i)z^3+iz|<3/4$

we know,

$|a+b|\le |a|+|b|$

Therefore we can say that

$|(1+i)z^3+iz|\le |(1+i)z^3|+|iz|$

$|(1+i)z^3+iz|\le |z^3|+|i{z}^3|+|iz|$

$|(1+i)z^3+iz|\le |z{|}^3+|z{|}^3+|z|$

$(|a^3|=|a{|}^3)$

$|(1+i)z^3+iz|\le \frac{1}{8}+\frac{1}{8}+\frac{1}{2}$

$|(1+i)z^3+iz|\le \frac{3}{4}$