Question

If $|z|<\frac{1}{2}$ and $4|(1+i)z^3+iz|<k$, find the least integral value of $k$.

Answer 1

Given that $|z|<\frac{1}{2}$
& $4|(1+i)z^3+iz|$
Now, $4|(1+i)z^3+iz|\le 4|(1+i)z^3|+4\left|iz\right|=4\sqrt{2}{\left|z\right|}^3+4\left|z\right|$
$\Rightarrow 4|(1+i)z^3+iz|<\frac{4\sqrt{2}}{8}+\frac{4}{2}=2+\frac{1}{\sqrt{2}}<3$
Therefore, $k=3$
Ans: 3