$O \subset \mathbb{C}$ region with ball $\overline{D(a, r)} \subset O$, $f$ analytic on $O$ s.t. $|f|$ constant on $\partial D(a, r)$.
47474
Proving $|f(z)|$ is constant on the boundary of a domain implies $f$ is a constant function
112597
Extreme point of unit balls, the complex case
64748
analytic in open unit disk,corresponding to a bounded sequence and a bounded functional sequence
117908
Is there an analytic function $f$ on $B(0,1)$ the open ball with radius 1 such that $f(1/n)=e^{-n}$ for $n=2,3,4,...$?
146839
Orthogonal Projection onto the $ {L}_{\infty} $ Unit Ball
141834
Schwarz Lemma of Complex Analysis
106081
The Projection to a Projective Line is Holomorphic
142901
"The closure of the unit ball of $C^1[0, 1]$ in $C[0, 1]$" and its compactness
114488
If $|f|+|g|$ is constant on $D,$ prove that holomorphic functions $f,~g$ are constant on $D$.
146654
Real valued analytic function defined on a connected set is constant
154534
consider the following subsets of complex plane
84510
If $f(\mathbb{C})\subset \mathbb{C}-[0,1]$ then $f$ is constant
61810
An application of Borel-Caratheodory theorem
45219
Proving that $C$ is a subset of $f^{-1}[f(C)]$
47748
Analytic function on an open disc.
73547
Prove that is $\bar{f}g$ is analytic then $f$ is constant or...?
49924
Analytic map with two fixed points on a simply connected domain is the identity