# Determine a Constant

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.

• Are $\lambda,\mu,\tau$ all complex numbers? – TheSimpliFire May 17 at 7:40
• @TheSimpliFire Yes, except for $\tau \in R$ – A Slow Learner May 17 at 7:42
• @TheSimpliFire Question edited. The radius of the ball is less than $\mu$ – A Slow Learner May 17 at 7:44

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.