For analytic function that is not univalent in a disk $|z|<R$, there exist $z_1$ and $z_2$ such that $|z_1| = |z_2| < R$, and $f(z_1) = f(z_2)$
65414
One-one analytic functions on unit disc
43177
How do I rigorously show $f(z)$ is analytic if and only if $\overline{f(\bar{z})}$ is?
15797
Holomorphic function having finitely many zeros in the open unit disc
101281
Does there exist a smooth function which is nowhere analytic?
28620
Function $f(z)$ such that $|f(z)| > |z|$ for every $z\in\mathbb{C}, 1/f(z)$
8039
an analytic function from unit disk to unit disk with two fixed point
69219
Prove there is a point $z\in[a,b]$ at which $f(z)=\frac{f(x_1)+f(x_2)+\cdots+f(x_k)}{k}$
127887
Existence of a meromorphic functions $f(z)$ such that $|f(z)|\geq |z|$.
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 $|z|<\frac{1}{2}$
18877
$f(z)$ and $\overline{f(\overline{z})}$ simultaneously holomorphic
62640
Show a given analytic function is constant
46514
Proving that if $f: \mathbb{C} \to \mathbb{C} $ is a continuous function with $f^2, f^3$ analytic, then $f$ is also analytic
27772
Let $f$ be an analytic function such that $\operatorname{Im}(f(z))=\operatorname{Re}(f(z))^2$. Prove that $f$ is constant.
10194
If $f,g$ are analytic in the unit disk, and $|f|^2+|g|^2=1$, then $f,g$ constant.
51625
Analytic functions close to $\bar{z}$
148247
Prove the existence of an analytic function
51577
Non-existence of a bijective analytic function between annulus and punctured disk