Let $U: \mathbb C \to \mathbb R$ be a harmonic function such that $U(z)\ge 0 \ \forall z\in \mathbb C$ ; then $U$ is constant.
13755
Is there a harmonic function in the whole plane that is postive everywhere?
47977
If $u$ is harmonic and bounded in $0 < |z| < \rho$, show that the origin is a removable singularity
140602
If the absolute value of an analytic function $f$ is a constant, must $f$ be a constant?
97192
product of harmonic functions
142079
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 $B$
15797
Holomorphic function having finitely many zeros in the open unit disc
129324
Let $f:[0;1]\to\mathbb R$ be continiously differentiable
105700
Proving $\,f$ is constant.
10194
If $f,g$ are analytic in the unit disk, and $|f|^2+|g|^2=1$, then $f,g$ constant.
11158
Proof that a continuous function with $f(x) = f(x^2)$ is constant.
25520
Why: A holomorphic function with constant magnitude must be constant.
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 $x$
43547
If a function is convergent and periodic, then it is the constant function.
153411
If $|f|$ is constant, so is $f$ for $f$ analytic on a domain $D$.
67485
The ratio $\frac{u(z_2)}{u(z_1)}$ for positive harmonic functions is uniformly bounded on compact sets
82367
Prove that a holomorphic function with postive real part is constant
153855
Complex analysis, showing a function is constant