Jtdylktuy

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.)

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 ${ w mid exp left(frac{-2 pi^2}{log theta}right) < |w| < 1 }$.

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} = rdfrac{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.