If a holomorphic function on the complex plane has two periods which are linearly independent over $\mathbb R$ then is the function constant?
7878
Proving that a doubly-periodic entire function $f$ is constant.
48222
Composition of a harmonic function with a holomorphic function is still harmonic
39610
Holomorphic functions: is it true that $f(\bar{z})=\overline {f(z)}$?
32723
Open sets and holomorphic functions in Complex analysis
116064
Does every holomorphic function $f$ defined on an open set $U$ have a primitive?
40273
Complex Analysis Question from Stein
137045
The estimate of real and imaginary part of a holomorphic function
35754
Is $f(x)=\sin(x^2)$ periodic?
128477
Entire function constant, where $f(z)=f(z+1)$ and $|f(z)|< e^{|z|}$.
106290
Prove a function is constant. (complex analysis)
42075
If $f$ is analytic then $|f|$ is not constant unless $f$ is constant
96050
$f$ an analytic function on $\mathbb{C}$ which takes values in $\mathbb{C}\backslash(-\infty,0]$ implies constant
152701
A question on periodicity of complex functions
13765
Is a rational-valued continuous function $f\colon[0,1]\to\mathbb{R}$ constant?
19750
If a function $f$ is holomorphic on the closed unit disk centered at the origin and is real valued whenever $|z| = 1$, then $f$ is constant.
47474
Proving $|f(z)|$ is constant on the boundary of a domain implies $f$ is a constant function
75638
Let $u:\mathbb{C}\rightarrow\mathbb{R}$ be a harmonic function such that $0\leq f(z)$ for all $ z \in \mathbb{C}$. Prove that $u$ is constant.