Given that $f(z)$ and $|f(z)|$ are each analytic in a domain D, prove that $f(z)$ is constant in D.
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 $D$
56127
Prove a function is constant under certain conditions
148474
on the boundary of analytic functions
102553
How to show that $f(z) = f(i)$
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
108846
Estimate of analytic function in a unit disk
55138
An analytic function $f$ on $\{z \in\mathbb{C} : |z|>1 \}$
74202
Show that $|f(z_1)|=|f(z_2)|≤|z_1|^2=|z_2|^2$.
34719
Show this holomorphic function is constant
34223
Prove that every function that verifies $|f(x)-f(y)|\leq(x-y)^2$ for all $x,y$ 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 $B$
96050
$f$ an analytic function on $\mathbb{C}$ which takes values in $\mathbb{C}\backslash(-\infty,0]$ implies constant
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 $D$.
597
f is differentiable on convex domain D and Re(f')>0 implies f is injective on D
129343
What is the maximum value of $|f(z)|$ on the unit disc $D=\{z \in \mathbb C:|z|\leq 1\}$?
8179
If $f$ is entire and $|f|\geq 1$, then show $f$ is constant.
84510
If $f(\mathbb{C})\subset \mathbb{C}-[0,1]$ then $f$ is constant