Complex monic polynomial $f(z)$ with $|f(z)|\leq 1$ for $|z|\leq 1.$
81063
If $p$ is a polynomial, then either $\,p(z)=z^n\,$ or $\,\max_{|z|=1}|p(z)|>1$.
158524
Complex numbers $z$ such that $|z|= 1$
89509
Suppose $z \neq -1$ is a complex number of norm 1. Prove that $(\frac{1+z}{|1+z|})^2 =z $
154837
mapping properties of $(1−z)^i$
33011
Prove that the polynomial $f_n(x)=nx^{n+1}-(n+1)x^n+1$ is divisible by $(x-1)^2$
69946
Do the polynomials $(1+z/n)^n$ converge compactly to $e^z$ on $\mathbb{C}$?
98399
Suppose $f$ and $g$ are entire functions, and $|f(z)| \leq |g(z)|$ for all $z \in \mathbb{C}$, Prove that $f(z)=cg(z)$.
139614
Polynomial bounded real part of an entire function
152347
Proof that a certain entire function is a polynomial
152982
Show that the range of $p(z)e^{q(z)}$ for nonconstant polynomials $p$ and $q$ is all of $\mathbb{C}$.
135577
Prove the existence of $c$ such that $f'(c) = 2c(f(c) - f(0))$
132431
Proving $|f'(z)|\leq\frac{\text{Re}(f(z))}{\text{Re}(z)}$
48619
Let $f$ be a non-constant analytic function on $\mathbb{D}$ satisfying $|f(z^2)| ≤ |f(z)|$ for $|z| < 1.$
80650
Complex differentiability of $f(z)=|z|$
73419
For $f$ entire and $\{f(kz) : k \in \mathbb{C}\}$ a normal family, prove that $f$ is a polynomial.
102553
How to show that $f(z) = f(i)$
58448
Complex:Schwarz lemma