Let $f$ be an entire function and $Re[f(z)] \leq M$, then $f$ is constant.
17408
An entire function whose real part is bounded must be constant.
50115
Show that the holomorphic function $u + iv$ must be constant if $u = h \circ v$. Problem from Remmert.
82001
How to prove the function is a constant
51934
Let $f$ be entire and suppose that $\text{Im} f\ge0$. Show that $\text{Im} f$ is constant.
82371
$f$ is an entire function with Im $f\geq 0$
84206
How do I show that if $f$ is entire and $\{\lvert f(z)\rvert < M\}$ is connected for all $M$, then $f$ is a power function?
148981
Entire $f$ and $g$ constant if $e^{f(z)}+e^{g(z)}=1$
34223
Prove that every function that verifies $|f(x)-f(y)|\leq(x-y)^2$ for all $x,y$ is constant.
89050
If $|f(z)|\leq |f(z^2)|$ on the unit disk, show that $f$ is a constant function.
155531
Entire functions such that $f(z+\lambda_1)=af(z) \; , \; f(z+\lambda_2)=bf(z) $
146506
If a complex function $f$ is Entire, and there exist $k,R > 0$, $n \in N$ such that $|f(z)| > k|z|^n$ for all $|z| > R$, then f is a polynomial.
107518
Entire function $f$ such that $|f(z)| ≤ K |z|^3$ for $|z|\ge1$ and $f(z) = f(iz)$ for every $z∈C$
132113
Does $f(x) = f(2x)$ for all real $x$, imply that $f(x)$ 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.
124346
Where is $f(z)=\Re (z)$ differentiable?
17343
Function on $[a,b]$ that satisfies a Hölder condition of order $\alpha > 1 $ is constant
137043
To show that function is constant