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I'm preparing myself to take a qualifying exam for my math PHD. I was trying to solve the previous exams and I found a complex analysis problem I couldn't solve:

Let $0

So far, I tried to define a function that scaled the range and domain to fit into the unitary disc and then apply Schwarz's Lemma, but when I tried to prove the differentiability at 0 I noticed it depended on the derivative of $f$ in $re^{i\theta}$, and I can't define $f'$ in the boundary of the annulus, so I gave up on that idea. Also, the problem has a hint that states: "consider $\log|f(z)|-\gamma\log|z|$ for a convenient $\gamma$". I tried to approach defining a logarithm function, but the domain isn't simply connected so I don't know how to proceed. Any help is appreciated. Thanks

P.S.:This is my first time asking a question on mathstack. If I'm breaking any rule please point it out so I can stop doing it in the future.

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    Can you show that $f$ is entire? If so, can you apply the Maximum modulus principle? – 2017-02-15
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    @Bacon Aren't entire functions defined in the whole complex plane? this one has a bounded domain. – 2017-02-15
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    My ansatz would be to try it with a Laurent series. Your function is holomorphic on a annulus. Hence there is a Laurent series $Q(z)=\sum_{n=-\infty}^{\infty}{c_n z^n}$ such that $Q=f$ on $A$. Maybe you can proceed now with the Cauchy integral formula (for Laurent series). – 2017-02-15

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I would do this using some theory about harmonic functions. Solve the corresponding Dirichlet problem: $$ \begin{cases} \Delta u = 0 \\ u = \log \alpha, & |z|=r \\ u = \log \beta, & |z|=R. \end{cases} $$ Almost by inspection, the solution is $$ u = a + b\log|z| $$ where $a$ and $b$ are chosen to match the boundary conditions. On the other hand, since $f$ is zero-free, $\log|f|$ is harmonic and solves the same Dirichlet problem. By uniqueness (or the maximum principle for harmonic functions), $u = \log |f|$, so $$ |f| = c|z|^b. $$ If $f$ is to be holomorphic, argue that $b$ must be an integer. Then finish off by using that if $f$ and $g$ are holomorphic and $|f|=|g|$, then $f = g$ up to a unimodular constant.