Holomorphic function on a upper half plane that is scale invariant with respect to a positive real number. Let $\theta \in \mathbb{R}^{+}$ be a fixed number and $\theta \neq 1$. Let $f:\mathbb{H} \to \mathbb{C}$ be a holomorphic function on the upper half plane such that $\forall z \in \mathbb{H}$, $f(z)=f(\theta z)$.
Then, must $f$ be a constant? (I highly doubt it since $f$ is not defined at $0$, but I don't know how to rigorously construct a counterexample.)
 A: No, $f$ does not need to be constant.
Define $g$ in the strip $S = \{ w \mid 0 < \operatorname{Im} w < i \pi \} $ as
$g(w) = f(e^w)$. Then for a given positive real number $\theta \ne 1$
$$
 f(z) = f(\theta z) \quad \text{for all } z \in \Bbb H
$$
if and only if
$$
 g(w) = g(w + \log \theta) \quad \text{for all } w \in S
$$
so that the problem reduces to find all $\log \theta$ - periodic functions in the strip $S$. The most simple example would be
$$
  g(w) = \exp \left(\frac{2 \pi i w}{\log \theta} \right)
$$
corresponding to
$$
  f(z) = \exp \left(\frac{2 \pi i \log z}{\log \theta} \right)
$$
where $\log z = \log |z| + i \operatorname{Arg} z$ is the main branch of the logarithm with $0 <  \operatorname{Arg} z < \pi$ for $z \in \Bbb H$.
All $\log \theta$ - periodic functions in $S$ are obtained by composing $g$ with a holomorphic function, this leads to the general solution 
$$
f(z) = h \left(\exp \left(\frac{2 \pi i \log z}{\log \theta} \right) \right)
$$
where $h$ is holomorphic in the annulus $\{  w \mid \exp \left(\frac{-2 \pi^2}{\log \theta}\right) < |w| < 1 \}$.
A: If we write
$f(z) = f(x + iy) = u(x, y) + iv(x, y) = u(r, \phi) + iv(r, \phi), \tag 1$
where $(r, \phi)$ are standard polar coordinates on $\Bbb H$, then we have
$\dfrac{\partial u}{\partial r} + i \dfrac{\partial v}{\partial r} = \dfrac{\partial f(z)}{\partial r} = \displaystyle \lim_{\Delta \theta \to 0} \dfrac{f((1 + \Delta \theta)z) - f(z)}{\Delta \theta} = \lim_{\Delta \theta \to 0} \dfrac{f(z) - f(z)}{\Delta \theta} = 0, \tag 2$
since varying $\theta$ moves $\theta z$ in the radial direction; thus,
$\dfrac{\partial u}{\partial r} = \dfrac{\partial v}{\partial r} = 0; \tag 3$
now according to the Cauchy-Riemann equations in the polar coordinates $(r, \phi)$,
$ \dfrac{\partial v}{\partial \phi} = r\dfrac{\partial u}{\partial r} = 0, \tag 4$
$ \dfrac{\partial u}{\partial \phi} = -r \dfrac{\partial v}{\partial r} = 0; \tag 5$
these two equations together with (3) show that
$\nabla u(z) = \nabla v(z) = 0, \forall z \in \Bbb H;  \tag 6$
it follows that $u(z)$ and $v(z)$ are constant in $\Bbb H$, since it is a connected open set; thus so also $f(z)$ is constant on $\Bbb H$.
The above derivation may be written in a slightly more compact fashion if we recall the polar form of the Cauchy-Riemann equations may be expressed in terms of the single equation in $f$,
$\dfrac{\partial f}{\partial r} = \dfrac{1}{ir} \dfrac{\partial f}{\partial \phi}; \tag 7$
then (2) shows that
$\dfrac{\partial f}{\partial r} = 0, \tag 8$
from which we find that
$\dfrac{\partial f}{\partial \phi} = 0, \tag 9$
and the constancy of $f(z)$ immediately follows.
