Proving that a locally constant function is constant
40208
Locally Constant Functions on Connected Spaces are Constant
34223
Prove that every function that verifies $|f(x)-f(y)|\leq(x-y)^2$ for all $x,y$ is constant.
158407
derivative bounded by a constant multiple of the function
89050
If $|f(z)|\leq |f(z^2)|$ on the unit disk, show that $f$ is a constant function.
69804
Prove that if $f$ is entire and $\vert f(z^2) \vert \leq 2 \vert f(z) \vert$, then $f$ is constant
153855
Complex analysis, showing a function is constant
115636
Show that $f$ is either injective or a constant function.
114488
If $|f|+|g|$ is constant on $D,$ prove that holomorphic functions $f,~g$ are constant on $D$.
141863
If the imaginary part of an entire function is never zero, the function is constant
90199
Prove that there exists a constant $c>0$ such that the following happens
62640
Show a given analytic function is constant
7878
Proving that a doubly-periodic entire function $f$ is constant.
105658
Show that a function that is locally increasing is increasing?
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.
110112
Proving a function is Lipschitz continuous
51934
Let $f$ be entire and suppose that $\text{Im} f\ge0$. Show that $\text{Im} f$ is constant.
32789
Show bounded harmonic function on $\mathbb{C}$ is constant.