Theorems you wish you knew in complex analysis I recently stumbled upon a theorem of Landau, which states

Let $f(z)=\sum a_nz^n$ such that $f'(0)\neq 0$ and such that $f^{-1}(\{0,1\})=\emptyset$. The radius of
convergence of $f$ is bounded by a constant $C(f(0),f'(0))$

The theorem is not hard to prove once one knows the Ahlfors-Schwarz-Pick lemma, but it is an extremely surprising statement, and a quite elementary one. In particular, one can easily state and explain it to a student taking a first course in complex analysis. A similar situation occurs with Picard's theorems and Schottky's theorem.
My question is: What other "relatively unknown" theorems do you think fit this description?
Before this question gets closed as "opinion based", let me state some criteria for what I am looking for:

*

*The statement is elementary (as explained before)

*The theorem is not usually taught (or stated) in a first course in complex analysis

*The proof requires nontrivial machinery from more advanced math(e.g. A-S-P for Landau's theorem)

 A: A comment that got long:
In terms of usefulness I would name two such results:
1.(edited as per comments as is known also as Bloch's Heuristic principle) the Bloch/Robinson principle, namely that if a "property" of holomorphic or meromorphic functions would imply such functions to be constant when defined in the full plane, then a family of holomorphic/meromorphic functions satisfying that property on some domain is normal (eg bounded, missing two values for analytic, three values for meromorphic etc)
(can be made rigorous as a Theorem of Zalcman by defining the notion of property as usual on "elements" $(f,U)$ with some obvious properties of extension, affine invariance etc)
one freely available reference paper is on arxiv - W Bergweiler Bloch's principle, (pdf link) later published in Computational Methods and Function Theory,6 2006)-
2.An analytic function defined on the interior of a Jordan curve, continuous on the closed domain and injective on the curve is then injective on the full closed domain (note that the curve may be weird like an Osgood curve so $f$ can be far from differentiable on $J$, $f'$ can be quite unbounded inside $J$ etc; also another application is if $f$ is analytic beyond the boundary but has a critical point there; then it is of course not injective around the critical point, but if it is injective on the boundary, it is still injective inside which is a bit counterintuitive; $z^2-2z$ on the unit disc is a good example here)
(as noted in the comments this is known as Darboux-Picard Theorem)
In terms of "coolness" results like:
if two meromorphic functions in the plane share five values (or two entire functions share four finite values as they then share infinity as an empty set of course) they are equal - here share a value $a$ means that the sets $f(z)=a, g(z)=a$ are equal including multiplicity
($e^z, e^{-z}$ share $0,1,-1, \infty$)
an entire function can have at most two values for which $f(z)=a$ implies $f'(z)=0$ (in other words there are at most two values $a$ for which all the roots of $f(z)-a$ are multiple), while for plane meromorphic functions the number is $4$ where of course we include $\infty$ now so poles too)
(elliptic Weierstrass and $\sin, \cos$ show that the result is optimal)
A: One of mine that you might enjoy:
There is a rational function $f$ such that for every holomorphic function $g$ on the open unit disk $\mathbb D$, $g$ or $g−f$ has a zero in $\mathbb D$.
This is American Mathematical Monthly problem 6520, solution here
