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 $B$
31522
Proving $f$ has at least one zero inside unit disk
63255
Proving an entire, complex function with below bounded modulus is constant.
75390
How does one prove that if function's partial derivative respect to every variable is zero, function is constant?
100765
If $f'(x) = 0$ for all $x \in \mathbb{Q}$, is $f$ constant?
69804
Prove that if $f$ is entire and $\vert f(z^2) \vert \leq 2 \vert f(z) \vert$, then $f$ is constant
5365
Bound on $|f'(0)|$ of a holomorphic funtion in the unit disk
39583
ii) Show that $u$ is not a real part of the function which analytic on $\mathbb{C} \backslash \lbrace 0 \rbrace$
17343
Function on $[a,b]$ that satisfies a Hölder condition of order $\alpha > 1 $ is constant
48619
Let $f$ be a non-constant analytic function on $\mathbb{D}$ satisfying $|f(z^2)| ≤ |f(z)|$ for $|z| < 1.$
73547
Prove that is $\bar{f}g$ is analytic then $f$ is constant or...?
111760
If $f^2$ is an analytic function then so is $f$
94680
Holomorphic function $f$ such that $f'(z_0) \neq 0$
23663
Show that holomorphic $f_1, . . . , f_n $ are constant if $\sum_{k=1}^n \left| f_k(z) \right|$ is constant.
62640
Show a given analytic function is constant
62901
Bound for Analytic Function on Unit Disk
87743
Is a function whose derivative vanishes at rationals constant?
19178
$f$ continuous, $f^N$ analytic on a domain D implies $f$ analytic on D