Let $f \in O(D)$ for some domain $D$. Prove that if $|f(z)|$ is a constant, then $f(z) =$ const on $D$.
140602
If the absolute value of an analytic function $f$ is a constant, must $f$ be a constant?
39921
show that f is constant on I
96050
$f$ an analytic function on $\mathbb{C}$ which takes values in $\mathbb{C}\backslash(-\infty,0]$ implies constant
91980
Prove that f is one-to-one on D
142138
Show that $f(x)$ is a constant function if $\lim\limits_{h\to 0}\frac{1}{h^3}\int_{-h}^{h}f(x+t)\cdot t\,dt=0$ for all $x$
114488
If $|f|+|g|$ is constant on $D,$ prove that holomorphic functions $f,~g$ are constant on $D$.
59850
If $f$,$g$ and $\overline{f}g$ are holomorphic on $\Omega$, then $g=0$ or $f$ is constant
34719
Show this holomorphic function is 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 $D$.
86517
If $|f(z)|\le |g(z)|$, can we say $f$ is constant?
84313
$|f(z)|\le 1-|z|\forall z\in D$, we need to show $f\equiv 0$
104369
If $f$ continuous and $f(x^2) = f(x)$, then $f$ is a const
93400
Prove that an analytic function with a constant imaginary part is constant itself
114826
Zero Partials imply Constant Function Theorem or Proof
11158
Proof that a continuous function with $f(x) = f(x^2)$ is constant.
158561
Let $f (z) $ be an entire function such that $|f (z)|≤K|z|$, $∀z∈\mathbb{C}$, for some $K>0$. If $f (1) =i$, then$f (i) $ is
19178
$f$ continuous, $f^N$ analytic on a domain D implies $f$ analytic on D