Show that if $f$ is holomorphic and $f'(a) \neq 0$ then $f$ is injective on a disk around $a$
62280
If $f:U\to V$ is holomorphic and $f'(z)\neq 0$ for all $z\in U$, then$f$ is locally bijective.
53848
$f$ holomorphic on $D\setminus \{0\}$ and takes no values in $(-\infty,0],$ then $0$ removable
147764
show that $f(z)+f(z^2)+\cdots + f(z^n)+\cdots$ converges locally uniformly to an analytic function in the unit disk.
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 $D$
23585
Show that a holomorphic function satisfying $|f(z)|\leq c|z|^n$ must be $f(z)=a\cdot z^n$ for some a.
39583
ii) Show that $u$ is not a real part of the function which analytic on $\mathbb{C} \backslash \lbrace 0 \rbrace$
44132
Let $f$ be an analytic isomorphism on the unit disc $D$, find the area of $f(D)$
35378
Prove that $f$ analytic, $f(x) \in \mathbb{R}$ for all $x \in \mathbb{R}$ implies $f(\overline{z})=\overline{f(z)}$
92056
An inequality on holomorphic functions
143354
Prove that there exists $c\in\mathbb{R}$ such that $f'(c)=f(c)$
56628
If $f: D(0,1)\to \mathbb C$ is holomorphic, then there is a sequence $\{z_n\}$ in $D(0,1)$ such that $|z_n| \to 1$ and $\{f(z_n)\}$ is bounded
150554
proof that $f'$ is continuous at $a$
88278
Show $S = f^{-1}(f(S))$ for all subsets $S$ iff $f$ is injective
31522
Proving $f$ has at least one zero inside unit disk
74527
Expressing the area of the image of a holomorphic function by the coefficients of its expansion
27251
Holomorphic function with $f\circ f=f$
102553
How to show that $f(z) = f(i)$