If |u|<1,|v|<1, and z=u−v1+¯uv, then least value of |z| is
|u|−|v|1+|u||v|
Let u=r1eiϕ,v=r2eiθ
where r1<1,r2<11+¯uv=1+r1r2ei(θ−ϕ)
|1+¯uv|2=(1+r1r2cos(θ−ϕ))2+r21r22sin2(θ−ϕ)=1+2r1r2cos(θ−ϕ)+r21r22≤(1+r1r2)2=(1+|u||v|)2⇒|1+¯uv|≤1+|u||v|⇒11+|u||v|≤1|1+¯uv|
Also, ||u|−|v||<|u−v|
Thus, ||u|−|v||1+|u||v|≤∣∣u−v1+¯uv∣∣