$f :\mathbb{D}\rightarrow \mathbb{D}$ is a holomorphic function and $z_0$ is a root of it, How can I prove $|z_0| \geq |f(0)|$
60520
Upper bound for $|f(0)|$ based on its behavior within the unit disk.
21921
Showing that $|f(z)| \leq \prod \limits_{k=1}^n \left|\frac{z-z_k}{1-\overline{z_k}z} \right|$
43177
How do I rigorously show $f(z)$ is analytic if and only if $\overline{f(\bar{z})}$ is?
73547
Prove that is $\bar{f}g$ is analytic then $f$ is constant or...?
75653
Prove that $|f ''|\ge 4$
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.
23585
Show that a holomorphic function satisfying $|f(z)|\leq c|z|^n$ must be $f(z)=a\cdot z^n$ for some a.
111635
Show that if $f(z)$ is a continuous function on a domain $D$ such that $f(z)^N$ is analytic for some integer $N$, then $f(z)$ is analytic on $D$.
82229
Show if complex function $f(z)$ contains all real coefficients then $f(\bar z)$ = $\bar {f(z)}$
125814
The function $f(z)=z+\frac{1}{z}$ maps $\Omega$ biholomorphically on the upper plane
147270
Entire function. Prove that $f(\bar{z})=\overline{f(z)}, \forall z\in C$
102086
Prove a property of divisor function
23663
Show that holomorphic $f_1, . . . , f_n $ are constant if $\sum_{k=1}^n \left| f_k(z) \right|$ is constant.
5365
Bound on $|f'(0)|$ of a holomorphic funtion in the unit disk
82367
Prove that a holomorphic function with postive real part is constant
14759
Prove that $ f(z) = 0 $ for all $ z \in \overline{D}(0,1) $.
84313
$|f(z)|\le 1-|z|\forall z\in D$, we need to show $f\equiv 0$