I'm working on a proof for a version of Runge's Approximation Theorem and I'm stuck trying to find an upper estimate of a term. Here are the details:
$U \subset \mathbb{C}$ is an open set, $K \subset U$ a compact subset and $f$ is a function that is holomorphic on $U$. Furthermore, $\gamma \in C^1$ is a curve running in $U\backslash K$. There are complex step functions $\varphi, \psi$ and $\chi: [0,1] \to \mathbb C$, such that
$$ |f(\gamma(t))-\varphi(t)|<\delta, \ |\gamma(t)-\psi(t)|<\delta, \ |\gamma'(t)-\chi(t)|<\delta $$ for $\delta>0$. Now I'm supposed to show that for $z \in K$ and $t \in [0,1]$
$$ |(\psi(t)-z)f(\gamma(t))\gamma'(t)-(\gamma(t)-z)\varphi(t)\chi(t)| \leq const \cdot \delta. $$
My idea is the following: I want to bring the term into a form, in which I can use at least one of the approximations above to get the $\delta$ part. For example, I tried isolating $|\psi(t)-z|-|\gamma(t)-z|$, which is $< \delta$. Then I want to use the fact that step functions only take finitely many values and the continuous functions $f(\gamma(t)),\gamma(t)$ and $\gamma'(t)$ are bounded on the compact set $[0,1]$. However, I have not yet been able to arrange the term in a way, so that I can follow that plan. I would appreciate any help.
Best,
Alex