Clearly |z|≠1
Now
|z|z|−1|=|z|||z|−1|
Let
z=r.eiθ
Hence |z|||z|−1| =r||r|−1|=r1−r For 0<r<1
and for r>1 we get rr−1
Now as r increases this value decreases, and maximum is when |r|=2
Now considering arg(z)
|arg(z)|ϵ[0,2π]
=[0,6.28]
Which is generally greater than ||z||z|−1|.
Hence the following inequality is true under certain condtions.