I came across this DIY exercise in my lecture notes which requires us to prove the following statement:
Let G be a region in $\mathbb{C}$ and $a,b \in \mathbb{C}$. Suppose G contains the line segment C from $a$ to $b$. Let $f:G \to \mathbb{C}$ be holomorphic. Prove that $\exists \lambda \in \mathbb{C}$ with $|\lambda| \leq1$ and $\theta \in \mathbb{C}$ such that $f(b)-f(a)=\lambda(b-a)f'(\theta)$.
As I have taken a class in analysis, this seem to resemble the Mean Value Theorem. May I clarify if that is the case and if so, what is the proof to this theorem? I am sure this is a standard proof and I have tried to look online for the proof but to no avail.
Hint: The purpose of $G$ containing the line segment from $a$ to $b$ is that this allows you to integrate $f'$ along the straight line linking $a$ and $b$.