Let $f$ be an entire function and suppose , $|f(z)+(1-2i)|\geq \epsilon,$ for all $z\in \mathbb{C}$. Prove that $f$ is constant.

Question

Let $f$ be an entire function and suppose that for some $\epsilon>0$, $$|f(z)+(1-2i)|\geq \epsilon,$$ for all $z\in \mathbb{C}$. Prove that $f$ is constant.

What I know so far is that, Liouvilles Theorem will be used in proving the $f$ is constant. Since $f$ is already entire, I have to show that it is bounded. I know that it is bounded below by $\epsilon$, however, I can't bound it above. My Professor said that this has to do something with annular domain. But I can't figure it out.

Do you have any ideas?

Answer 1

Hint: Consider the function $$\frac{1}{f(z) + (1-2i)}.$$ On what domain is this function analytic?

Answer 2

Hint: Consider $1/g$ where $g$ is your function $f$ plus the constant., prove this is bounded, and apply Liouville’s theorem. If $1/f$ is constant, then so is $g$, hence $f$.