# Is the following function a constant function

Suppose that $f: \mathbb{C} \rightarrow \mathbb{C}$ is entire and bounded on the set $\{z \in \mathbb{C}; Re(z) \leq 0\}$. Is $f$ a constant function.

I know by Picards theorem that a non-constant entire function assumes all but one value in the complex plane. Can this result be tweaked and applied here? Any hints?

• here's the full question, asked much earlier. – Jesse P Francis Jun 22 '15 at 11:28

Take $f(z) = e^z$. Then, for any $z$ such that $Re(z) \leq 0$, we have that $f(z)$ is contained in the (closed) unit circle, yet $f$ is not constant.
• ..with radius $1$.. – Berci Jun 22 '15 at 11:27