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Suppose we have holomorphic functions $h_1$, $h_2$ in some connected, open subset $\Omega$ of $\mathbb{C}$ containing $0$. Assume further that $h_1(0)=h_2(0)=0$ and most importantly that $$|h_1(z)|=|h_2(z)|^\alpha$$ for any $z \in \Omega$ and $\alpha\ge0$.

My question is how strongly related are $h_1$ and $h_2$?

In case when $\alpha \in \mathbb{Z}$ the problem is easy since either both functions are identical zero or in a neighborhood of $0$ their ratio is of a constant modulus so $h_1=\xi_k \cdot h_2^k$ for $\xi_k$ being a square root of unity.

Unfortunately the power function is not well defined for rational and irrational $\alpha$ so one can not repeat that reasoning. I would be grateful for Your attempts/ideas.

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If $\Omega$ didn't contain $0$ we could have interesting examples. Take a small open region $U$ away from any zeros of either function. If it's small enough, there will be branches of $\ln h_1$ and $\ln h_2$ there. Then $\Re(\ln h_1)-\alpha\Re(\ln h_2)=0$ on $U$. This means that $\Im(\ln h_1)-\alpha\Im(\ln h_2)$ is constant on $U$. So locally $h_1$ is a constant of absolute value $1$ times an $\alpha$-th power of $h_2$.

But $0\in\Omega$, and $|h_1(z)|\sim C_1|z|^a$ and $ |h_2(z)|\sim C_2|z|^b$ as $z\to0$, with $a$, $b\in\Bbb N$. This means that $\alpha=a/b\in\Bbb Q$.

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  • $\begingroup$ The second part of Your answer is very helpful. Any ideas for rational case then ? $\endgroup$ – J.E.M.S Jun 4 '17 at 10:34
  • $\begingroup$ In the rational case, there will be some integers $r$ and $s$ with $ra+sb=g=\gcd(m,n)$. Consider $h_3=h_1^rh_2^s$. Up to constants, $h_1$ and $h_2$ will be powers of $h_3$. $\endgroup$ – Lord Shark the Unknown Jun 4 '17 at 10:54

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