Determine a Constant

Question

Let $\lambda \in C$ and that $\lambda \in B(\mu,r)$, a closed ball centered at $\mu \in C$, with radius $r < \mu$. I am trying to determine the value of a constant $\tau$ to guarantee that: $$ |1-\tau\lambda|<1 $$ Attempt: All I can see is that: $$ |1-\lambda| \leq |1-\mu|+r $$ I understand that we have to scale $\lambda$ somehow by multiplying $\tau$ to it. However, I am stuck here.

Answer 1

Since $\lambda=\Re\lambda+i\Im\lambda$, we solve $$|1-\tau\lambda|=|(1-\tau\Re\lambda)-i\tau\Im\lambda|=\sqrt{(1-\tau\Re\lambda)^2+(\tau\Im\lambda)^2}<1.$$ Squaring both sides and tidying yields $$1-2\tau\Re\lambda+\tau^2|\lambda|^2<1\implies\tau(|\lambda|^2\tau-2\Re\lambda)<0.$$ Thus $$0<\tau<\frac{2\Re\lambda}{|\lambda|^2},$$ and note that $-r+\Re\mu<\Re\lambda<r+\Re\mu$.