$f$ holomorphic in $\mathbb{C}\setminus\{0\}$ and $|f(z)|\le \sqrt{|z|} + \frac{1}{\sqrt{|z|}}$ then $f$ is constant
106290
Prove a function is constant. (complex analysis)
46514
Proving that if $f: \mathbb{C} \to \mathbb{C} $ is a continuous function with $f^2, f^3$ analytic, then $f$ is also analytic
94680
Holomorphic function $f$ such that $f'(z_0) \neq 0$
47474
Proving $|f(z)|$ is constant on the boundary of a domain implies $f$ is a constant function
32723
Open sets and holomorphic functions in Complex analysis
27772
Let $f$ be an analytic function such that $\operatorname{Im}(f(z))=\operatorname{Re}(f(z))^2$. Prove that $f$ is constant.
56978
How to Prove that if $f(z)$ is entire, and $f(z+i) = f(z), f(z+1) = f(z)$, then $f(z)$ is constant?
89050
If $|f(z)|\leq |f(z^2)|$ on the unit disk, show that $f$ is a constant function.
74622
Composition of holomorphic functions is holomorphic
137045
The estimate of real and imaginary part of a holomorphic function
64172
Show that if $f$ is analytic in a domain $D$ and $|f(z)|$ is constant in $D$, then the function $f(z)$ is constant in $D$
111076
If $f$ is non-constant and entire, prove that there exists a $z_0 \in C$ such that $f(z_0)$ is a positive real number
62640
Show a given analytic function is constant
142079
If $f$ is a non-constant analytic function on $B$ such that $|f|$ is a constant on $\partial B$, then $f$ must have a zero in $B$
50115
Show that the holomorphic function $u + iv$ must be constant if $u = h \circ v$. Problem from Remmert.
73547
Prove that is $\bar{f}g$ is analytic then $f$ is constant or...?
93400
Prove that an analytic function with a constant imaginary part is constant itself