# If $D\subset\mathbb C$ is a bounded domain, and $f,g:D\to\mathbb C$ are holomorphic such that $|f|\le|g|$ on $\partial D$, does $|f|\le|g|$ on $D$?

Suppose we have a bounded domain $$D\subset\mathbb C$$ with smooth boundary, and holomorphic functions $$f,g:D\to\mathbb C$$ which are continuous up to the boundary, and such that $$|f|\le|g|$$ on $$\partial D$$. Does this imply that $$|f|\le|g|$$ on $$D$$?

I would like to just take logs and use the maximum principle for harmonic functions, but it seems difficult to make this work when $$f$$ may have zeros in $$\overline D$$.

Take $$f(z) = \frac12, g(z) = z$$ and look at the unit disc. Multiply both by $$z$$ (or basically any other holomorphic function) for a less trivial example.
• Ah I see. So it’s essential then that $g$ be nonvanishing. – Monstrous Moonshine Apr 16 at 14:39