If $f: \mathbb{C} \rightarrow \mathbb{C}$ is holomorphic and $|f(z)| \geq 1 \forall z \in \mathbb{C}$, show $f$ is constant
8179
If $f$ is entire and $|f|\geq 1$, then show $f$ is constant.
95632
Where is this function holomorphic?
72856
Show $f$ is constant if $|f(x)-f(y)|\leq (x-y)^2$.
55138
An analytic function $f$ on $\{z \in\mathbb{C} : |z|>1 \}$
50115
Show that the holomorphic function $u + iv$ must be constant if $u = h \circ v$. Problem from Remmert.
148981
Entire $f$ and $g$ constant if $e^{f(z)}+e^{g(z)}=1$
87723
Show a holomorphic function on unit disc with with $|f'(z) - f'(0)| < |f'(0)|$ is injective
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.
84103
Let $f(z)$ be entire function. Show that if $f(z)$ is real when $|z| = 1$, then $f(z)$ must be a constant function using Maximum Modulus theorem
111635
Show that if $f(z)$ is a continuous function on a domain $D$ such that $f(z)^N$ is analytic for some integer $N$, then $f(z)$ is analytic on $D$.
32723
Open sets and holomorphic functions in Complex analysis
20955
Let $f(z)$ be analytic on $\mathbb{D}$ = {${z\in\mathbb{C}:|z-1|<1}$} such that $f(1) = 1$, if $f(z) = f(z^2)$ for all $z\in\mathbb{D}$,
77267
Given $f$ is continuous and $f(x)=f(e^{t}x)$ for all $x\in\mathbb{R}$ and $t\ge0$, show that $f$ is constant function
117418
Showing that the image of a function is $\mathbb{C}$ if it satisfies a nice functional equation
158519
Hints for a complex limit: Prove if $\lim_{z \to \infty} f(z)/z = 0$ then $f(z)$ is constant.
14117
Holomorphic function has at most countably zeros
125251
Show that a function from a Riemann Surface $g:Y\to\mathbb{C}$ is holomorphic iff its composition with a proper holomorphic map is holomorphic.