Let $\theta$ and $\phi$ two real $C^h$ functions on $[0,2\pi]$, $h\geq1$, satisfying for all not critical values $b$ of $\phi$ $$ \sum_{x \in \phi^{-1}(b)} \frac{\text{e}^{i \theta(x)}}{|\phi'(x)|}=0, $$ then, up to sign, values of $\text{e}^{i\theta}$ and $\phi$ and all their derivatives must match at the extreme points of the interval.

I can prove this statement if $a:=\phi(0)$ is not a critical value. In that case we find $\delta>0$ such that $\phi$ is invertible on the finitely many components of $\phi^{-1}([a-\delta,a+\delta])$. We index from left to right as $\{p_j\}_{j \in [1,...,N_+]}$ the restrictions of $\phi$ to $\phi^{-1}([a,a+\delta])$ and analogously as $\{m_j\}_{j \in [1,...,N_-]}$ those to $\phi^{-1}([a-\delta,a])$. By our hipotesis on the level sets, we have $$ \lim_{\varepsilon \to 0^+} \sum_j \frac{\mathrm{e}^{i\theta\big(p_j^{-1}(a+\varepsilon)\big )}}{|\phi'(p_j^{-1}(a+\varepsilon))|} = \lim_{\varepsilon \to 0^-} \sum_j \frac{\mathrm{e}^{i\theta\big(m_j^{-1}(a+\varepsilon)\big )}}{|\phi'(m_j^{-1}(a+\varepsilon))|}=0. $$ Without loss of generality we can assume $\phi'(0)>0$. Observing that in the limit contributions from the restrictions to intervals in the interior of $[0,2\pi]$ are equal in the two sums, we have no other choice than $\phi(0)=\phi(2\pi)$. Now, if also $\phi'(2\pi)>0$, for $0< \varepsilon <\delta$, by our condition, we have \begin{align*} \frac{\mathrm{e}^{i\theta\big(p_1^{-1}(a+\varepsilon)\big )}}{\phi'(p_1^{-1}(a+\varepsilon))}&=- \; \sum_{1<j} \frac{\mathrm{e}^{i\theta\big(p_j^{-1}(a+\varepsilon)\big )}}{|\phi'(p_j^{-1}(a+\varepsilon))|}, \\ \frac{\mathrm{e}^{i\theta\big(m_{N_-}^{-1}(a-\varepsilon)\big )}}{\phi'(m_{N_-}^{-1}(a-\varepsilon))}&=-\sum_{j<N_-} \frac{\mathrm{e}^{i\theta\big(m_j^{-1}(a-\varepsilon)\big )}}{|\phi'(m_j^{-1}(a-\varepsilon))|}. \end{align*} The sums on the right side of the equations are $C^h$ and equal in the limit of all their derivatives, which proves that also the derivatives at extreme points must be equal. If $\phi'(0)\phi'(2\pi)<0$ a similar argument shows that the derivatives are equal up to alternating sign.

If $\phi(0)$ is a critical value I have been trying hard without success and I kind of feel I am missing something obvious. Does anyone have a proof at least for $\theta$ and $\phi$ analytic functions? Assuming the periodicity of $\text{e}^{i\theta}$ by hipotesis would also be fine for my purposes (but less beautiful).

An answer in the positive would have nice geometrical implications as you can read at Given $\theta$, find $f$ such that $\int_{\mathbb{T}} \text{e}^{i\theta} \cos(h \cdot f) = 0,$ for all $h \in \mathbb{N}$. Thanks.


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