# Should I use Liouville thm or continuity?

Make an examples $$f:\Bbb{C}\to\Bbb{C}$$ holomorph (If it exists) with the properties:(Justify your answer)

$$|f(z)|\leq \frac{1}{100}$$ if $$|z|\leq 1$$ and

$$|f(z)-1|\leq \frac{1}{100}$$ if $$|z-3|\leq 1$$.

My attemp: I confuesd to use Liouville thm or continuity? Please guide me how should I think about it!

• If $g$ is an entire function, it is continuous in particular, and hence, bounded by some constant $a = a(D) > 0$ on the disc $D.$ – Will M. Jan 9 at 19:57

Let us define $$g(z)=0$$ on $$|z|\le 1$$ and $$g(z)=1$$ on $$|z-3|\le 1$$. Then $$g$$ is analytic on a neigoborhood of $$K=\{|z|\le 1\}\cup \{|z-3|\le 1\}$$. Note that $$\Bbb C\setminus K$$ is connected. By Runge's theorem, for every $$\epsilon>0$$, there exists a polynomial $$p(z)$$ such that $$|g(z)-p(z)|\le\epsilon,\quad\forall z\in K.$$ If we let $$\epsilon =\frac{1}{100}$$, then $$p$$ is a function we are looking for.
What about the constant function $$f(z)=1/200$$ ?