# What does "attain values" mean for an analytic function here?

Here's a question in complex analysis that I don't get, neither do I understand the first part of its answer given in the book. I would really appreciate some help.

Question:

Show that if $$f(z)$$ is an entire function, and there is a nonempty disk such that $$f(z)$$ does not attain any values in the disk, then $$f(z)$$ is constant.

Answer:

If $$f$$ does not attain values in the disk $$|w - c| < \epsilon$$, then $$1/(f - c)$$ is bounded, hence constant by Liouville's theorem, and $$f$$ is constant.

Entire function: a function that is analytic on the entire complex plane.
Liouville's theorem: every bounded entire function is a constant.

Source of the question: Gamelin, Complex Analysis.

Happy New Year!

• "$f$ attains the value $v$" means $v$ is in the image of $f$, that is, $v=f(z)$ for some $z$. Dec 31, 2020 at 19:35

## 1 Answer

It means that for some disc $$B(a,r)$$ we have that

$$f(\mathbb{C}) \cap B(a,r) = \emptyset$$

Then, for all $$z \in \mathbb{C}$$,

$$| f(z) - a | \geq r$$

and you can consider its multiplicative inverse and apply Liouville.