Suppose $f$ and $g$ are holomorphic on a bounded domain $D$ and continuous on $\bar D$. Suppose also $|f(z)|=|g(z)|\neq0$ on $\partial D$ and $\frac{|g(z)|}{3}\leq|f(z)|\leq 3|g(z)|$ for all $z\in D$. What can we say about $f$ and $g$?
I wanted to consider $\frac{f(z)}{g(z)}$ and try to apply Maximum modulus principle. On the boundary $\frac{f(z)}{g(z)}=1$, but in $D$ there is no guarantee that $g(z)\neq0$ to divide the inequality throughout by $|g(z)|$. How do I use the inequality or are there other things to consider? Any hints will be appreciated.
