if $\frac{\theta_1-\theta_2}{2\pi}$ is irrational then $f$ is a constant. Let $f: B(0,1)\to\mathbb{C}$ be analytic with the property that there exists $\theta_1, \theta_2\in\mathbb{R}$ such that $$|f(re^{i\theta_1})|=|f(0)|=|f(re^{i\theta_2})|$$ for all $r\in(0,1)$. Show that if $\frac{\theta_1-\theta_2}{2\pi}$ is irrational then $f$ is a constant.
Could anyone kindly help? I have been thinking for a long time and still have no clue. How to use $\frac{\theta_1-\theta_2}{2\pi}$ is irrational to prove $f$ is constant? Thanks so much!
 A: Suppose $f(0)=0$. Then the result is immediate, as $f$ has zeros at every point on a ray and thus must be identically zero.
Suppose $f(0)\ne 0$. Then there exists a neighborhood around $0$ for which $f\ne 0$, and hence an analytic logarithm of $f$ can be defined. Let $g = \log f$ in this neighborhood. Since the real part of $\log z$ is $\log{|z|}$, it follows that the real part of $g$ is constant along the rays $\{re^{i\theta_1}\}$ and $\{re^{i\theta_2}\}$. It suffices to prove the following:
Claim: If $g$ is analytic around $0$ and $\text{Re}(g)$ is constant along the rays $\{re^{i\theta_1}\}$ and $\{re^{i\theta_2}\}$ with $\frac{\theta_2-\theta_1}{2\pi}\not\in\mathbb{Q}$, then $g$ is constant.
Proof: WLOG let $g(0)=0$ so that the real part of $g$ along both rays is zero, and assume $g$ is not identically zero. Then the zeros of $g$ are discrete, so by taking a small enough neighborhood we can assume that $g$ has no other zeros. Since $g(0)=0$ and $g$ is not identically zero, there exists a positive integer $m$ and a complex number $C\ne 0$ such that
$$ g(z) = Cz^m + O(|z|^{m+1}) $$
near $z=0$. Now, for small enough $r>0$, we know that $g(re^{i\theta_1})$ and $g(re^{i\theta_2})$ are both not zero, and furthermore
\begin{align} g(re^{i\theta_1}) &= Cr^me^{im\theta_1} + O(r^{m+1}) \\
 g(re^{i\theta_2}) &= Cr^me^{im\theta_2} + O(r^{m+1}). \end{align}
It follows that
$$\frac{g(re^{i\theta_2})}{g(re^{i\theta_1})} = e^{im(\theta_2-\theta_1)} + O(r). $$
Since the real parts of both $g(re^{i\theta_1})$ and $g(re^{i\theta_2})$ are zero, both numbers are purely imaginary, so their quotient must be real for all $r>0$. On the other hand, as $r\rightarrow 0$, the imaginary part of the right-hand side approaches $\sin(m(\theta_2-\theta_1))$. It follows that $\sin(m(\theta_2-\theta_1)) = 0$, contradicting the assumption that $\frac{\theta_2-\theta_1}{2\pi}$ is irrational.
Thus, it follows that $g$ must be identically zero under the assumption that $g(0)=0$, so in the more general case it follows that $g$ must be constant. Since $g$ was defined as a logarithm of $f$, it follows that $f = e^g$ must be constant as well.
