1. ## analytic function

Find all analytic functions in |z|<3 satisfying f(i)=-3i and |f(z)|<=3.
Can I have some ideas on how to solve this question? Thank you

2. For $z=i$ the funcion $|f|$ has an absolute maximum. Now, apply the Maximum Modulus Principle.

Fernando Revilla

3. If I using the maximum modulus principle, then i'll get |f(z)|<=|f(i)| for all z. But how could I find all the analytic funtions in |z|<3? Thank you

4. The function $f$ must be constant in $|z|<3$ . As $f(i)=-3i$ there is only one function satisfyng the given hypothesis: $f(z)=-3i$ .