Let \(f\) be analytic in a domain \(D\) and fix \(z_0 \in D\) with \(w_0 \definedas f(z_0)\). Suppose \(z_0\) is a zero of \(f(z) - w_0\) with finite multiplicity \(m\). Show that there exists \(\delta >0\) and \(\eps > 0\) such that for each \(w\) such that \(0 < \abs{w-w_0} < \eps\), the equation \(f(z) - w = 0\) has exactly \(m\) distinct solutions inside the disc \(\abs{z-z_0} < \delta\).
Solution
Write \(g(z) \da f(z) - w_0\), then \(g\) is holomorphic on \(D\) and thus \(w_0\) is an isolated zero. Choose \(\delta\) small enough so that \(g\) is nonvanishing on \(\DD_\delta(z_0)\smts{z_0}\). Let \[ \gamma \da \ts{\abs{\xi - z_0} = \delta }= \bd\DD_{\delta}(z_0) .\] Choose \(\eps < \inf\ts{w\in f(\delta)}\) so that \(\abs{f(z) - w_0} > \eps\) in \(\DD_\eps(w_0)\smts{w_0}\) for every \(z\in \gamma\). Let \[ \gamma' \da \bd \DD_{\eps}(w_0) = \ts{\abs{z-w_0} = \eps} ,\] and define the solution counting function: \[ F(w) \da {1\over 2\pi i} \oint_{\gamma'} \logd(g(z)) \dz = {1\over 2\pi i } \oint_{\gamma'} {g'(z)\over g(z) }\dz = {1\over 2\pi i} \oint_{\gamma'} {f'(z)\over f(z) - w} \dz ,\] which counts the zeros of \(g\) (since it has no poles) and consequently the number of solutions to \(f(z) = w\) in \(\DD_\eps(w_0)\). This is now a continuous integer valued function on \(\DD_\eps(w_0)\), and is thus constant. Since \(f(z_0) = w_0\) with \(z_0\) enclosed by \(\gamma\) and \(w_0\) enclosed by \(\gamma'\), the constant is exactly the multiplicity of the zero of \(f(z) - w_0\) at \(z_0\), which is \(m\).