Let $f$ be analytic in a domain $D$ with $|f(z)| = k$, $k$ is real constant with $z \in D$. Prove that $f$ is constant.
42075
If $f$ is analytic then $|f|$ is not constant unless $f$ is constant
73547
Prove that is $\bar{f}g$ is analytic then $f$ is constant or...?
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 $z\in\mathbb{D}$,
56127
Prove a function is constant under certain conditions
28120
Are intervally constant analytic functions always constant?
158647
If $|f(z)|\le |f(z^2)|$ then prove that $f$ is constant
154034
Suppose that $f(x)$ is continuous on $(0, \infty)$ such that for all $x > 0$,$f(x^2) = f(x)$. Prove that $f$ is a constant function.
141115
$f$ be an analytic function defined on $\mathbb{D}=\{z\in\mathbb{C}:|z|<1\}$
153855
Complex analysis, showing a function is constant
154742
If $f(z)$ analytic in the domain $D$ and $f'(z)=0$ so $f(z)=$constant?
9844
Proof composition of analytic functions is analytic
34223
Prove that every function that verifies $|f(x)-f(y)|\leq(x-y)^2$ for all $x,y$ is constant.
35378
Prove that $f$ analytic, $f(x) \in \mathbb{R}$ for all $x \in \mathbb{R}$ implies $f(\overline{z})=\overline{f(z)}$
80650
Complex differentiability of $f(z)=|z|$
82001
How to prove the function is a constant
8179
If $f$ is entire and $|f|\geq 1$, then show $f$ is constant.
63255
Proving an entire, complex function with below bounded modulus is constant.