Complex Analytic functions that are lines are constant
134199
Holomorphic function mapping a set onto a straight line
154742
If $f(z)$ analytic in the domain $D$ and $f'(z)=0$ so $f(z)=$constant?
34719
Show this holomorphic function is constant
115142
Holomorphic functions on a complex compact manifold are only constants
48619
Let $f$ be a non-constant analytic function on $\mathbb{D}$ satisfying $|f(z^2)| ≤ |f(z)|$ for $|z| < 1.$
86517
If $|f(z)|\le |g(z)|$, can we say $f$ is constant?
12525
complex conjugates of holomorphic functions
3349
For $f$ an analytic function, what is
148247
Prove the existence of an analytic function
50342
Proving that a Function is Analytic Given that it is Equal to the Complex Conjugate of an analytic Function
47474
Proving $|f(z)|$ is constant on the boundary of a domain implies $f$ is a constant function
19750
If a function $f$ is holomorphic on the closed unit disk centered at the origin and is real valued whenever $|z| = 1$, then $f$ is constant.
103125
Show that a non-constant holomorphic function has zeroes in a compact set
28120
Are intervally constant analytic functions always constant?
27772
Let $f$ be an analytic function such that $\operatorname{Im}(f(z))=\operatorname{Re}(f(z))^2$. Prove that $f$ is constant.
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 $D$
123249
An analytic function $f$ on $|z|<1$ with $\sup_{|z|<1} |f(z)|=1$ and $|f(0)|=\frac{1}{2}$ has no zeros in $|z|<\frac{1}{2}$