Q1: Let $f: D \rightarrow \mathbb{C}$ be a holomorphic function where $\overline{B_r(z_0)} \subseteq D$. What is the correct Cauchy’s inequality?

A1: $$| f^\prime(z_0) | = \frac{1}{r} \sup_{ z \in \partial B_r(z_0)} | f(z) |$$

A2: $$| f^\prime(z_0) | = \frac{1}{r^2} \sup_{ z \in \partial B_r(z_0)} | f(z) |$$

A3: $$| f^\prime(z_0) | = \frac{1}{r^3} \sup_{ z \in \partial B_r(z_0)} | f(z) |$$

A4: $$| f^\prime(z_0) | = \frac{4}{r^2} \sup_{ z \in \partial B_r(z_0)} | f(z) |$$

Q2: Let $f: \mathbb{C} \rightarrow \mathbb{C}$ be a holomorphic function given by $f(z) = \cos(z)$. Is there a $z_1$ with $|\cos(z_1)| \geq 10$?

A1: Yes, there is.

A2: Yes, but $z \in \mathbb{R}$.

A3: No, $\cos$ is bounded.

Q3: Let $f: D \rightarrow \mathbb{C}$ be a holomorphic function that is bounded and not constant. Is it possible that $D = \mathbb{C}$?

A1: No, that is not possible.

A2: Yes, there are a lot of examples.