$f$ analytic on $\mathbb{D}$ with $|f'(z) - f'(0)| < |f'(0)|$, then $f$ is one-to-one
87723
Show a holomorphic function on unit disc with with $|f'(z) - f'(0)| < |f'(0)|$ is injective
32723
Open sets and holomorphic functions in Complex analysis
3349
For $f$ an analytic function, what is
85612
Let functions $ f: A \to B$ and $g: B \to A$, suppose that $ g \circ f = i_A$. Prove that if $f$ is onto, then $g$ is one-to-one
55138
An analytic function $f$ on $\{z \in\mathbb{C} : |z|>1 \}$
47748
Analytic function on an open disc.
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 $|z|<\frac{1}{2}$
147764
show that $f(z)+f(z^2)+\cdots + f(z^n)+\cdots$ converges locally uniformly to an analytic function in the unit disk.
62640
Show a given analytic function is constant
16555
If $\mathbb f$ is analytic and bounded on the unit disc with zeros $a_n$ then $\sum_{n=1}^\infty \left(1-\lvert a_n\rvert\right) \lt \infty$
53848
$f$ holomorphic on $D\setminus \{0\}$ and takes no values in $(-\infty,0],$ then $0$ removable
110219
Bound on the derivative of a holomorphic function at $0$
120118
The distinction between infinitely differentiable function and real analytic function
65414
One-one analytic functions on unit disc
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
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 $D$
55347
for two non zero complex polynomial $p(z),q(z)$ we have $p(z)\overline{q(z)}$ is analytic if and only if ?? CSIR - June $2013$