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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 $\overline D$.

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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.

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  • $\begingroup$ Ah I see. So it’s essential then that $g$ be nonvanishing. $\endgroup$ – Monstrous Moonshine Apr 16 at 14:39

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