I'm studying Complex Analysis and I found that interesting question
https://math.stackexchange.com/questions/256830/hadamards-three-circle-theorem#=
I understand that, it the equality holds, we get $M(r)=cr^\lambda$. However, I couldn't follow the discussion when he concluded that the unique functions that satisfies this condition are $f(z)=cz^\lambda$.
Could anybody help me?
If we know somehow that $\lambda$ is an integer this is very simple: Let $$g(z)=f(z)/z^\lambda.$$ For every $r$ the maximum of $|g|$ on the circle $|z|=r$ is $c$, so in particular $|g|$ has a local maximum, hence $g$ is constant.
Of course we're not given that $\lambda$ is an integer. We can avoid multi-valued functions and analytic continuation by using a covering map: Define $$g(z)=f(e^{iz}),$$so $g$ is holomorphic in a certain horizontal strip. More or less the same argument as above shows that $g(z)/e^{i\lambda z}$ is constant; now $g(z+2\pi)=g(z)$ shows that $\lambda\in\Bbb Z$.
