Let $A = \{1/2 < |z| < 2\}.$ Is there an analytic function $f$ on $\mathbb{C} \setminus \{0\}$ so that $Im(f) < −1$ on $∂A$ and $f(1) = 0$? 
Let $A = \{1/2 < |z| < 2\}.$ Is there an analytic function $f$ on $\mathbb{C} \setminus \{0\}$ so that the imaginary part $Im(f) < −1$ on $∂A$ and $f(1) = 0$? Explain your answer.

I am not sure how to begin this problem.  I was thinking the answer is no and that I could use Maximum modulus maybe?
Any suggestions?

Since $Im (f)$ is harmonic on $A$, it satisfies the maximum principle there. Hence $Im(f) < -1$ on $\partial A$ implies $Im(f) < -1$ on all $A$, and $f(1) = 0$ is impossible. 
