A real harmonic function on $\Bbb C-\{0\}$ which is bounded above must be a constant.
89380
A positive harmonic function on the punctured plane is constant
146654
Real valued analytic function defined on a connected set is constant
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 $B$
5365
Bound on $|f'(0)|$ of a holomorphic funtion in the unit disk
48222
Composition of a harmonic function with a holomorphic function is still harmonic
12556
A non-trivial, non-negative, function bounded below by its derivative with $f(0)=0$?
62901
Bound for Analytic Function on Unit Disk
39386
Harmonic non-surjective functions are constant
17408
An entire function whose real part is bounded must be constant.
15797
Holomorphic function having finitely many zeros in the open unit disc
59850
If $f$,$g$ and $\overline{f}g$ are holomorphic on $\Omega$, then $g=0$ or $f$ is constant
105172
Prove that the entire function is constant over certain conditions.
62568
Does there exists an entire function $f: \mathbb C \to \mathbb C$ which is bounded on real line and imaginary line?
92056
An inequality on holomorphic functions
128911
Continuous extension of a real function
93270
is that function must be constant under the following conditions
74501
$\delta : C([0,1])\to\mathbb{R}$, $f\mapsto f(0)$, with sup-norm is bounded but unbounded with one-norm.