Show that $|(f + t_1g)(f + t_0g)| \ge ( |f| - |g| )^2.$

Question

I am trying to follow the steps in (an alternative) proof of Rouché's theorem. I am just missing the following part:

Let $C$ denote a simple closed contour, and suppose that the two functions $f(z)$ and $g(z)$ are analytic inside and on $C$, and $|f(z)| > |g(z)|$ at each point on $C$. Then for any $t_0, t_1 \in [0,1]$ the following inequity holds $$|(f + t_1g)(f + t_0g)| \ge ( |f| - |g| )^2.$$ I tried but no success! How to prove that?

Answer 1

Hint: For $a,b\in \mathbb C,$ $|a+b|\ge |a|-|b|.$