Proving an analytic function is constant
10194
If $f,g$ are analytic in the unit disk, and $|f|^2+|g|^2=1$, then $f,g$ constant.
111760
If $f^2$ is an analytic function then so is $f$
50342
Proving that a Function is Analytic Given that it is Equal to the Complex Conjugate of an analytic Function
64650
If $a\operatorname{Re}(f(z))+b\operatorname{Im}(f(z))=c$ then f is constant
43177
How do I rigorously show $f(z)$ is analytic if and only if $\overline{f(\bar{z})}$ is?
64172
Show that if $f$ is analytic in a domain $D$ and $|f(z)|$ is constant in $D$, then the function $f(z)$ is constant in $D$
47474
Proving $|f(z)|$ is constant on the boundary of a domain implies $f$ is a constant function
58664
Proving a function is continuous for a fixed variable
153411
If $|f|$ is constant, so is $f$ for $f$ analytic on a domain $D$.
148247
Prove the existence of an analytic function
73547
Prove that is $\bar{f}g$ is analytic then $f$ is constant or...?
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.
132663
Show $f$ is analytic if $f^8$ is analytic
132478
Prove that continuous functions mapping irrationals to rationals 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 $x$
23663
Show that holomorphic $f_1, . . . , f_n $ are constant if $\sum_{k=1}^n \left| f_k(z) \right|$ is constant.
3818
Proving two entire functions are constant.