Finding another proof of Rouche's Theorem.

Question

Let $C$ denote a simple closed contour (positive orientation), and suppose that

$(a)$ two functions $f (z)$ and $g(z)$ are analytic inside and on $C$;

$(b) |f (z)| > |g(z)|$ at each point on $C$.

Then define the function, $$\Phi(t) = \frac{1}{2\pi i} \int_C \frac{f'(z) +tg'(z)}{f(z) + tg(z)} dz$$

Given $0 \le t \le 1$.

How can we ensure the denominator is not zero anywhere so that the integrand is defined?

This is a part of a full problem. For reference one can look at Brown Churchill complex variables Pg 298 No10.

I am also stuck at problem b where to show that there exists a constant $A$.

Answer 1

If $f(z)+tg(z)=0$ for some $z\in C$ then $$|f(z)|=t|g(z)|\leq |g(z)|$$ which contradicts condition (b).