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.
13755
Is there a harmonic function in the whole plane that is postive everywhere?
48275
$f$ defined $|f(x) - f(y)| \leq |x - y|^{1+ \alpha}$ Prove that $f$ is a constant.
120533
A harmonic function with sublinear growth at infinity is constant
114488
If $|f|+|g|$ is constant on $D,$ prove that holomorphic functions $f,~g$ are constant on $D$.
103125
Show that a non-constant holomorphic function has zeroes in a compact set
141863
If the imaginary part of an entire function is never zero, the function is constant
36444
Bound of a complex-valued function
128547
$f:\mathbb C\rightarrow \mathbb C$ be an arbitrary analytic function satisfying $f(0)=0$ and $f(1)=2.$
97192
product of harmonic functions
84510
If $f(\mathbb{C})\subset \mathbb{C}-[0,1]$ then $f$ is constant
63526
Proving that $(\sup_{x\in R}|f'(x)|)^2\leq 4\sup_{x\in R}|f(x)|\cdot\sup_{x\in R}|f''(x)|$.
26388
proof that function is constant
112517
Proving bounds of a harmonic series
9451
Show bounded and convex function on $\mathbb R$ is constant
48619
Let $f$ be a non-constant analytic function on $\mathbb{D}$ satisfying $|f(z^2)| ≤ |f(z)|$ for $|z| < 1.$
26353
Can any harmonic function on $\{z:0<|z|<1\}$ be extended to $z=0$?
34223
Prove that every function that verifies $|f(x)-f(y)|\leq(x-y)^2$ for all $x,y$ is constant.