Let $\varphi_t$ be an analytic function on an open domain $\Omega\subseteq\mathbb{C}$.
Let $K \subset \Omega$ be a compact set.
I am trying to prove that for any fixed parameter and fixed values: $\left|\frac{\varphi_t(b) - \varphi_t(a)}{b-a}\right| \leq \sup_{z \in K} \left|\frac{d}{dz}\varphi_t(z)\right|$
For a second, I thought mean value theorem might work here, but then I realized that MVT does not exist for complex functions.
Any ideas for proving the statement?