Analytic image of conjugate of a point

Question

Let $f:C \rightarrow \{|z|\leq2021 \}$ be an analytic function. If $f(11+9i)=\frac{1}{2i}$, then find the value of $f(11-9i)$.

How should I proceed with the problem ?

Answer 1

Theorem. (Liouville) Every bounded entire function must be constant.

An entire function is a function which is holomorphic on the entire complex plane. A consequence of Cauchy's integral formula is that the concept of analyticity is equivalent to holomorphism. And since $f:\mathbb C \rightarrow \{|z|\leq2021 \}$ is obviously bounded, by Liouville's theorem $f$ must be constant.

We know that $f(11+9i)=\frac{1}{2i}$. The function $f$ is constant, which means that $$f(z)=\frac{1}{2i},\quad\forall z\in\mathbb C.$$