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Let $\mathbb{D}$ be the open unit disc and f and g are holomorphic on an area of $\overline{\mathbb{D}}$ and do not have any zeros on $\mathbb{D}$.
How can I prove that if $|f|=|g|$ on $\partial \mathbb{D}$ then there is a constant with the propertiees above.
The hint is to first assume that there are no zeros on $\partial \mathbb{D}$ of $g$. I guess no I can look at $\left|\frac{f}{g}\right|=1$. As a holomorphic function I could use the maximum principle that this is the maximum of the function, since $\frac{f}{g}$ is holomorphic. How can I continue from here on? And how can I then leave the restriction to $g$ off?

MilesDefis
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  • For your idea: You've proven that $f/g$ is constant on $\partial \mathbb{D}$. Denote this constant by $c$ and consider $h:=f/g-c$. What happens if you apply the maximum modulus principle to this function? – WoolierThanThou Jun 12 '23 at 12:42
  • You need to clarify your question. – copper.hat Jun 12 '23 at 13:19
  • @copper.hat Thanks for the comment, I corrected the question. – MilesDefis Jun 12 '23 at 13:25
  • What would $| { g \over f } | $ be? – copper.hat Jun 12 '23 at 13:28
  • @WoolierThanThou I know that $h\equiv 0$. By that I know that, $|f/g|=|c|=1$ and therefore $f=cg$, right? – MilesDefis Jun 12 '23 at 13:30
  • @ChessBaker251 Well, the middle conclusion with the absolute value doesn't imply the latter conclusion without it, but you're correct that $h\equiv 0$ implies $f=cg$. – WoolierThanThou Jun 12 '23 at 15:27
  • I gave a solution here: https://math.stackexchange.com/questions/4631621/show-two-analytic-functions-are-related-by-a-constant/ – user8675309 Jun 12 '23 at 16:18
  • @user8675309 in our lecture, we haven't tolkaed about meromorphic functions.. – MilesDefis Jun 13 '23 at 13:21
  • @WoolierThanThou what I yet don't see is, how I can leave the assumption $g(z)\neq 0$ on the boundary of $\mathbb{D}$. Could I use that, $h$ has a removable singularity? I wouldn't see how. – MilesDefis Jun 13 '23 at 13:42
  • @ChessBaker251 -- you don't need to know what a meromorphic function is. You just need to know what a rational function is and what an analytic function is (with particular emphasis on a locally non-zero analytic function), then reason that their product cannot have an essential singularity. – user8675309 Jun 13 '23 at 15:34

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