Zeros of $f_{\epsilon}(z) = f(z) + \epsilon g(z)$ with $f$ and $g$ holomorphic

I'm stuck with this problem from Stein-Shakarchi:

Suppose $f$ and $g$ are holomorphic in a region containing the disc $|z| \leq 1 $. Suppose that $f$ has a simple zero at $z = 0$ and vanishes nowhere else in $|z| \leq 1 $. Let $$f_{\epsilon}(z) = f(z) + \epsilon g(z)$$

Show that if $\epsilon$ is sufficiently small then:

a) $f_{\epsilon}(z)$ has a unique zero in $|z| \leq 1 $

b) if $z_{\epsilon}$ is this zero, the mapping $\epsilon \rightarrow z_{\epsilon}$ is continous

Any idea ?

Solution 1:

a) $|f_{\epsilon}(z) - f(z)| = |\epsilon g(z)| < |f(z)|$ on $\partial D(0,1)$ for $\epsilon$ sufficiently small. So we apply Rouché's theorem and we conclude.

b) We use the generalized argument principle: with the usual hypotheses and $g$ holomorphic $$\frac{1}{2\pi i}\int_{\gamma} \frac{f'(z)}{f(z)}g(z) dz = \sum_{k}g(z_{k}) - g(p_{k})$$ where $z_{k}$ are the zeros of $f$ inside $\gamma$ and $p_{k}$ are the poles.

So in our case we have $$\frac{1}{2\pi i}\int_{\gamma} \frac{f'(z)+\epsilon g'(z)}{f(z) + \epsilon g(z)}z \ dz = z_{\epsilon}$$ where $\gamma = \partial D(0, 1)$